Generalized Lyapunov exponents: monograph 9786010437890

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Generalized Lyapunov exponents: monograph
 9786010437890

Table of contents :
Recommended for publication by the Academic Council
(Protocol No.3 from 26.11.2018) and the Editorial Committee
of al-Farabi KazNU (Protocol No.2 from 20.12.2018)
Reviewers:
.
2) the blocks are exponentially separated relatively , i.e. there are the constants such, that

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AL-FARABI KAZAKH NATIONAL UNIVERSITY

T. M. Aldibekov

GENERALIZED LYAPUNOV EXPONENTS Monograph

Almaty «Kazak University» 2018 1

UDC 53 LBС 22.3 A 31 Recommended for publication by the Academic Council (Protocol No.3 from 26.11.2018) and the Editorial Committee of al-Farabi KazNU (Protocol No.2 from 20.12.2018) Reviewers: doctor of physical and mathematical sciences, professor D.S. Dzhumabaev doctor of physical and mathematical sciences, professor M.K. Dauylbaev doctor of physical and mathematical sciences, professor M.I. Tleubergenov

Aldibekov Т.М. A 31 Generalized Lyapunov exponents: monograph / Т.М. Aldibekov; trans. Zh. Zhunussova. – Almaty: Kazak University, 2018. – 232 p. ISBN 978-601-04-3789-0 The monograph is devoted to the study of some problems in the qualitative theory of differential equations and is related to the theory of Lyapunov exponents, the linear systems of differential equations under small perturbations and the stability of the asymptotic characteristics of the system are investigated. In the classical theory of Lyapunov exponents, the linear systems of differential equations with bounded coefficients are considered, in addition, linear systems of differential equations with zero exponents are not considered. In the monograph the generalized Lyapunov exponents, generalized central exponents and generalized singular exponents introduced in the work are effectively used for the study of a system of differential equations with unbounded coefficients, and also for the study of linear systems of differential equations with zero exponents. The results presented in the monograph have both theoretical and applied significance. It may be applied in the theory of stability, the theory of dynamical systems, control theory and in its applications.

UDC 53 LBС 22.3 ISBN 978-601-04-3789-0

© Aldibekov T.M., 2018 © Al-Farabi KazNU, 2018

2

INTRODUCTION The monograph is devoted to the study of some problems in the qualitative theory of differential equations and is related to the theory of Lyapunov exponents, the results of scientific research on linear systems of differential equations under small perturbations and the stability of the asymptotic characteristics of the system are presented. The foundations of the qualitative theory of differential equations were introduced in the late 19th century by A. Poincaré [1] and A.M. Lyapunov [2]. The main problem studied by Lyapunov is the stability of the solution of a differential equation (mainly a solution representing equilibrium or periodic motion) on an infinite interval of the time axis for small changes in the initial conditions that determine the solution [3], see also [4]. The creation of the theory of stability put forward him in the position of the most outstanding mathematicians of the 19th century. Lyapunov proposed two methods for solving this problem. Investigation of stability, which is based on the analytical construction of a family of integral curves, as noted in [5], Lyapunov called by the first method of investigation. The first method is based on the theory of characteristic exponents of linear systems, developed by Lyapunov. The general theory of Lyapunov exponents is described in detail in the monograph by B.F. Bylov, R.E. Vinograd, D.M. Grobman, V.V. Nemytsky [6]. Also there is information in the monographs of B.P. Demidovich [7], L.Ya. Adrianova [8], N.A. Izobov [9], I.V. Gayshun [10], in the reviews of V.M. Millionshchikov [11-13] and N.A. Izobov [14-18]. In the Lyapunov theory of characteristic exponents of linear differential systems (1) x = A(t )x the systems with bounded piecewise-continuous coefficients are considered. The Lyapunov characteristic exponent [2, p. 27] of the solution ( ) x t ≠ 0 of linear system (1) is a number λ ≡ lim

t →+∞

3

1 ln x(t ) t

.

Linear system can not have more n non-zero solutions with pairwise different exponents [2, p. 35]. The fundamental system of solutions (1) is called by normal [2, p. 35]), if the sum of the exponents of its solutions is minimal in the set of all fundamental systems. The exponents of a normal system are called by the characteristic exponents of the Lyapunov system (1). The Lyapunov exponent λk ( H , b) of homomorphism H ∈ Hom(G, Aut ( E , p, B)) is defined for any b ∈ B, k ∈ {1,..., n} by formula λ k (H , b ) =

____ R

n − k +1

min

(

∈Gn − k +1 p

−1

max

(b ))ξ ∈R∗n − k +1

lim

t → +∞

1 ln X t ξ t

.

Here Gi ( p −1 (b)) is the Grassmann manifold of i -dimensional vector subspaces of the layer

p −1 (b) ,

i ∈ {1,..., n} , R∗i is the set of all nonzero

vectors of the vector space R i , t ∈ G , where G is the group R or group Z , ⋅ is the norm induced by a fixed Riemannian metric vector bundle ( E , p, B ) . This definition is a modification of A.M. Lyapunov definition, proposed by V.M. Millionshchikov [19-20] for a family of morphisms of vector bundle that includes various constructions, including systems of linear differential equations, for which the exponents were originally introduced by A.M. Lyapunov that differ from the data sign. The first result on stability theory in the first approximation for a system with variable coefficients was established by A.M. Lyapunov [3, p. 52]. He highlighted a class of regular systems and showed that if the first approximation system is correct, then the negativity of all its characteristic exponents is sufficient to preserve the asymptotic stability. We note, that Lyapunov established this in the class of perturbations that are holomorphic functions whose expansions begin with terms not lower than the second order. Later, H.L. Massera [21] extended his result to a class of perturbations of a higher order of smallness. N.G. Chetaev [22] supplemented Lyapunov's theorem by establishing that the presence of at least one positive exponent in the regular linear system of the first approximation is sufficient to preserve the instability. The system (1) is called by regular [3, p. 38], if the sum of its Lyapunov exponents coincides with the lower limit of the mean value of the trace of the matrix of coefficient of the system, that is, equality 4

n

∑ λi i =1

t

1 SpA(u )du t ∫0

= lim t →+∞

is held and otherwise it is incorrect. There are criteria for the correctness by A.M. Lyapunov [3, p. 39], O.Perron [23], V.P. Basov [24], D.M. Grobman [25], Yu.S. Bogdanov [26], R.E. Vinograd [27], E.A. Barabanov [28]. There exist irregular systems with almost-periodic coefficients (V.M. Millionshchikov [29]). Generalization of regular systems. In the work of N.A. Izobov [30] the concept of a weakly irregular system is introduced. B.F. Bylov [31] highlighted the class of linear systems (1) whose asymptotics of solutions are determined with respect to functions of the form t

exp ∫ ρ (τ )dτ

, and the well-known Lyapunov, Perron, and R.E.

0

Vinograd criteria are obtained the generalizations by using instead of the exponents of system Λ1 < Λ 2 < ... < Λ q of a certain set of functions ρ1 , ρ 2 ,..., ρ q .

B.P. Demidovich [32] highlighted a class of regular systems which are completely regular systems. V.M. Millionshchikov introduced statistically regular systems [33]. Lyapunov [3, p. 52] and Massera [21, p. 81] theorem is fundamental in the theory of stability by the first approximation. In this direction, we note the works of Perron [34], K.P. Persidsky [35], I.G. Malkin [36], D.M. Grobman [25, p. 121], Yu.S. Bogdanov [37], V.M. Millionshchikov [38], Isobov N.A. [39], R.E. Vinograd, N.A. Izobov [40], N.E. Bolshakov [41]. The features of instability by the first approximation were constructed by N.G. Chetaev [22], N.P. Yerugin [42]. A general view of Lyapunov's exponents as discontinuous functions is reflected in the works of V.М. Millionshchikov [43-50]. In these papers, the typical properties in the sense of Baire are identified with respect to the theory of Lyapunov exponents. The concepts of a vector bundle, families of morphisms, indices of a family of morphisms and endomorphisms of a metrized vector bundle are the basis of the proposed by V.M. Millionshchikov constructions. A special case of a 5

family of morphisms of a vector bundle which is a linear extension of a dynamical system, was first considered by R. Sacker and J. Sell [51]. The similar objects were studied by I.U. Bronstein and V.F. Cherniy [52-53], as well as M.I. Rakhimberdiyev [54-55]. The study of the properties of vector subspaces [56-57] determined by Lyapunov exponents is of some interest from the point of view of the stability theory (especially from the point of view of the theory of conditional stability). The fact found in Millionshchikov's works is that the destruction with respect to certain properties of exponents in the sense of Baire categories is atypical. Stability of the exponents. The exponents of linear system (1) are called stable, if for any ε > 0 there exists such δ > 0 , that every exponent λ у of any perturbed system 𝑦𝑦 ′ = 𝐴𝐴(𝑡𝑡)𝑦𝑦 + 𝑄𝑄(𝑡𝑡)𝑦𝑦, ,

Q(t ) ≤ δ

, t≥0

satisfies to the inequality min λ y − λi ≤ ε . i

This concept arose from the work of Perron [34, p. 703], who first established that exponents can be unstable. K.P. Persidsky established the stability of the Lyapunov exponents under some condition (a). From this, in particular, the Poincaré-Perron theorem follows [58, p. 121], that the characteristic exponents of a system with constant coefficients are stable. The instability of the exponents of the regular systems as higher as small was discovered by R.E. Vinograd [59-61] (they are stable for reducible and almost reducible systems). After discovering that the Lyapunov exponents of a linear system of differential equations are generally unstable, the development of the theory of characteristic exponents began in the following directions: the first, to search for a class of systems for which the set of characteristic exponents is stable (continuously), and secondly, to search for more stable (continuous) characteristics of the system. The first paper, where the condition for the continuity of Lyapunov exponents appeared, was the work of Perron [62]. In this paper, the sufficient conditions for the stability of the exponents of the diagonal linear system were given as the diagonal separation condition. These 6

conditions were extended to the integral separation of the diagonal by B.F. Bylov [63] and R.E. Vinograd [64]. J. Lillo [65] considered arbitrary systems and formulated conditions for the separation of solutions. In the work [66], B.F. Bylov introduced a definition of the integral separation of linear system and proved the reducibility of such systems to Perron's divided-diagonal systems, thereby establishing the stability of the exponents of linear systems with integral separation. The definition of integral separation is closely related to the notion of an exponential dichotomy, the important use and history of which is in the book of D.V. Anosov [67] see also [68-70]. The integral separation of the solution subspaces, together with the integral closeness of the solutions in each of the subspaces are the necessary and sufficient condition for the stability of the entire set of exponents of a linear system. This result was obtained by V.M. Millionshchikov [71] and, independently, B.F. Bylov, N.A. Isobov [72-73]. In terms of sufficiency, the criterion for the stability of exponents is a corollary of the Perron-Bylov-Vinograd theorem (Theorem 15.2.1 of [6, p. 208]). The proof of the necessity in both papers is essentially based on a fundamentally new method in the theory of linear systems, which is called in the literature [9, p. 78; 14, p. 80] by the method of Millionschikov turns. From the stability of all Lyapunov exponents at a point in the space M n of linear systems with a uniform metric implies integral separation [14, p. 86] of subspaces of solutions of the system with different Lyapunov exponents. In the work of I.N. Sergeev [74-75] this result is clarified. Along with studies of the stability of exponents in general type of linear systems, there are also those that belong to a certain class of systems. These are almost periodic systems [76-78], systems with weak variation [79-80], almost-reducible systems [81-85], two-dimensional systems [86-89], Hamiltonian systems [90-91], regular systems [92 -95]. The investigation of the exponents’ stability with respect to other types of perturbations has independent meaning. These include perturbations tending to zero at infinity [96-97], exponentially decreasing [98-100], pulsed [101], random [102-103], close to the average constant [104], stabilizing [105]. One of the approaches to extend the method of characteristic Lyapunov numbers to systems with a leading nonlinear part is described by Yu.S. Bogdanov [106-110]. 7

One of the extensions of the theory of exponents for equations in Banach space is proved in the book by Yu.L. Daletskiy and M.G. Krein [111]. In the work of V.M. Millionschikov [112] the theory of linear systems of differential equations that is adequate to the metric theory of dynamical systems is studied. A similar approach is also available in the work of V.I. Osealedza [113]. K.P. Persidsky investigated a class of countable systems of differential equations, on which was extended to the notion of a characteristic number [114]. Finding of the characteristic Lyapunov exponents of linear systems of differential equations, in general, is a difficult problem. For equations with variable coefficients the computation of the characteristic exponents or their estimates involves great difficulties. For calculating of the exact lower bound of the mobility of the highest exponent for small perturbations of the coefficient matrix, see the works of I.N. Sergeev [115-120]. Special (general), central and exponential exponents of system (1) are stronger characteristics of the system in comparison with the Lyapunov exponents of a linear system of differential equations. Special exponents [6, p. 117] of linear systems (1) with bounded coefficients introduced by P. Bohl [121] as indices, remained undetected for a long time, and were rediscovered by K.P. Persidsky [122]. A detailed commentary on the history of this issue is contained in the book by Yu.L. Daletsky and M.G. Kerin [111, p. 211]. The instability to the downward of the upper special exponent and the instability to the upward of the lower exponent are found in the work [123]. R.E. Vinograd introduced the upper Ω and lower ω central exponents of linear systems (1) with bounded coefficients [124-126]. They set the estimations of the upper and lower Lyapunov exponents of the perturbed systems respectively [6, p. 104]. ϖ denotes the lower central exponent of a linear system with bounded coefficients, where, in the definition ω , the taking of the lower limit by V.M. Millionshchikov is replaced by taking the upper limit. From the example in [123, p. 750] follows the instability of the downward central exponent Ω and the upward instability of the lower central exponent ω introduced by R.E. Vinograd, as well as the upward instability of the lower central exponent ϖ . 8

The upper central exponent Ω and the lower central exponent ϖ give exact estimates of Lyapunov exponents of the perturbed systems. Their accuracy was proved in the general case by Millionshchikov [127]. The criteria for the stability of the central exponents of a linear system with almost periodic coefficients is contained in the work of V.L. Novikov [77, p. 151]. The properties of central exponents with respect to infinitesimal perturbations were studied by I.N. Sergeev [75, p. 438]. N.A. Izobov in the work [128] introduced the notion of exponential exponents of linear systems (1). Central, exponential and special (general) exponents are used in the study of the linear approximation of asymptotic stability and Lyapunov instability and uniform stability of the zero solution of differential systems with perturbations from various classes. The central exponent Ω k ( H , b) of homomorphism [129-134] H ∈ Hom(G, Aut ( E , p, B)) is defined for any b ∈ B, k ∈ {1,..., n} by formula Ω k (H , b ) =

m −1

___

1 R n − k +1∈Gn − k +1 ( p −1 (b ))τ ∈G∗ m→+∞ mτ inf

inf

∑ ln X τX τ R

lim

s =0

s

n − k +1

Here X Cτ denotes the constriction to the set C of the mapping

X τ . To study the conditional stability, the central exponents corresponding to vector subspaces of the solution space of equation 𝑥𝑥 ′ = 𝐴𝐴(𝑡𝑡)𝑥𝑥 are introduced, and their properties are studied. Let

{

}

En − k +1 (H , b ) = ξ ∈ p −1 (b) : λ ( H , ξ ) ≤ λk ( H , b))

be

a

vector

subspace

of

the

layer

p −1 (b) .

Ω ( k ) ( H , b) exponent of homomorphism H ∈ Hom(G, Aut ( E , p, B)) is defined for any b ∈ B, k ∈ {1,..., n} by formula

The

central

____

Ω (k ) (H , b ) = inf lim

1 τ ∈N m→+∞ mτ

9

m −1

∑ ln X τX τ E

s =0

s

n − k +1

( H ,b )

.

The central exponents of homomorphism H ∈ Hom(G, Aut ( E , p, B)) are modifications of definitions and results of R.E. Vinograd [6, p. 114]. If k > 1 , then the central exponents Ω k ( H , b) , Ω ( k ) ( H , b) of homomorphisms H ∈ Hom(G, Aut ( E , p, B)) , in general, are not equal [135-136]. If we consider central exponents on the space of Cauchy problems for nonlinear differential equations, the point of this space is a pair: a nonlinear system of differential equations (a set of such systems is endowed with topology) and a point of phase space. The nonlinear Cauchy problem is linearized, and the central exponent of the obtained system of equations in variations is considered as a function of the nonlinear Cauchy problem. In such spaces, not only the Lyapunov higher exponent, but also the upper central exponent is not everywhere upper semicontinuous. In the monograph the object of investigation are systems of differential equations defined on an infinite interval of the time axis. The subject of the study is the qualitative behavior of solutions of a system of differential equations by the first approximation method. In the first section of the monograph the Baire classes of central exponents of homomorphisms of a metrized vector bundle are considered. For convenience the detailed construction of homomorphisms in the concrete cases and some auxiliary materials are given. In the theory of Lyapunov exponents, the mainly, systems with bounded coefficients are considered. In the case of unboundedness, there are only a small number of special studies. Now the main sphere of application is the stability theory, the theory of exponents began to touch with other branches of mathematics, such as the theory of dynamical systems, the control theory, etc. In many theoretical problems and in a number of applications of the theory of differential equations, the systems of differential equations with unbounded coefficients, as well as linear systems with zero Lyapunov exponents arise. Therefore, the extension of the methods of the theory of Lyapunov exponents for the investigation of a system of differential equations with unbounded coefficients on an infinite interval, on the one hand, and studies of nonlinear systems of 10

the linear approximation of which have Lyapunov zero exponents on the other hand are topical problems. From the second to the fifth sections of the monograph, the theory of Lyapunov exponents are extended to systems of differential equations with unbounded coefficients. In the second section the generalized Lyapunov exponents of linear unbounded systems of differential equations with respect to nonlinear systems of differential equations are introduced. The class of a system of differential equations is indicated, when it is possible to calculate the generalized exponents by the coefficients of the system. In the third section, the concept of generalized-regular systems of differential equations is defined and the properties, the criterion of generalized-regular systems are given. An analog of Lyapunov's theorem on the stability of the trivial solution for perturbations over the first order for linear systems of differential equations with unbounded coefficients is established. A sufficient condition for the stability of the zero solution of an unbounded nonlinear system with respect to the coefficients of the system of the first approximation. In the fourth section, the generalized central and generalized singular exponents of linear systems of differential equations with unbounded coefficients are introduced. The uniform estimates for the solutions of nonlinear systems of differential equations by applying generalized central and generalized singular exponents are established. In the fifth section, the generalized exponents, generalized central and generalized singular exponents as functionals on the space of linear systems of differential equations with unbounded coefficients are studied. The exact bounds of mobility of the generalized Lyapunov exponents are established. A criterion for the continuity of generalized Lyapunov exponents is given. The results presented in the monograph have both theoretical and applied significance. It may be applied in the theory of stability, theory of dynamical systems, control theory and in their applications.

11

1. ON BAIRE CLASSES OF CENTRAL EXPONENTS OF HOMOMORPHISMS OF THE VECTOR BUNDLE

The properties of vector subspaces defined by Lyapunov exponents as functions based on a vector bundle are investigated. The study of the properties of such mappings is of some interest from the point of view of the stability theory (especially from the point of view of the theory of conditional stability). In order to be able to talk about the properties of this mapping, it is necessary to specify exactly the space of linear systems x = A(t )x on which this mapping is considered. This mapping, in general, is not continuous at every point. Further, in the first section, the central exponents of homomorphism of an abstract metrized vector bundle are investigated as Baire functions based on a vector bundle. We construct a homomorphism based on a vector bundle [170] with a compact-open topology. We consider a linear system of differential equations where

𝑥𝑥 ′ = 𝐴𝐴(𝑡𝑡)𝑥𝑥

(1.1)

(

x ∈ R n , A(⋅)R → Hom R n , R n

)

is a continuous operator-function defined on the entire infinite numerical axis. We transform the set of all systems (1.1) into a metric space CM n [4, p. 533] by introducing the distance 

1 t 

ρ ( A1, A2 ) = sup min  A1 (t ) − A2 (t ) , . t∈R



(1.2)

The space CM n defined by the distance is complete metric. We define a trivial vector bundle (E , p, B ) by the equalities B = CM n , E = B × R n , p = pr1,

12

(1.3)

where pr1 is a projection of the production B × R n on the first factor. In fact,

(E , p, B ) = (B × R n , pr1, B )

is n – dimensional vector bundle with the natural structure of the vector space on each layer b × R n = p −1 (b )

In detail: let be α , β ∈ R, b ∈ B for any ξ ,η ∈ p −1 (b ) we have: ξ = (b, x ),η = (b, x ′, )

where b ∈ B, x, x ′ ∈ R . Then we assume by definition n

αξ + βη = α (b, x ) + β (b, x ′) = (b, αx + βx ′)

thus for every b ∈ B on the layer p −1 (b ) the structure of a vector space over the field of the real numbers arises. Consequently, a definite structure of a vector space on every layer p −1 (b ) , and the second of formulas (1.3) transforms the bundle (E , p, B ) into a trivial vector bundle with layer R n . The Euclidean structure of the layer R n does not depend on b ∈ B and defines a Riemannian metric on the trivial vector bundle. Namely, for every b ∈ B and every ξ ,η ∈ R n by definition β ((b, ξ ), (b,η )) =< ξ ,η >

By

Χ A (θ ,τ )

we denote the Cauchy operator of system 𝜉𝜉 ′ = 𝐴𝐴(𝑡𝑡)𝜉𝜉 13

The Cauchy operator, by definition, the value of every solution ξ (t ) of the system in the point t = τ , puts in correspondence the value of the same solution in the point t = θ . For any A ∈ CM n , t ∈ R(B = CM n , b = A)

we prescribe a mapping χt : B → B

by formula

For

e∈E

χ t A(⋅) = A(t + (⋅)) .

(1.4)

we get: e = ( A, ξ ),

where A ∈ CM n and ξ ∈ R n . For any e ∈ E , t ∈ R we prescribe a mapping Xt :E → E

by the way

(

)

X t e = X t ( A, ξ ) = χ t A,Χ A (t ,0 )ξ .

(1.5)

Proposition 1. For any t ∈ R the pair of mappings

(X

t

,χt

)

is a morphism

(E , p , B ) → (E , p , B ) in the category of vector bundles. Proof. This means the following: for each t ∈ R the continuous mappings are given 14

X t : E → E, χ t : B → B

that pX t = χ t p

at every t ∈ R , moreover, at all

b ∈ B, t ∈ R

the mapping

( )

X t [b] : p −1 (b ) → p −1 χ t b ,

defined as a constriction (restriction) on layer X t is a linear mapping. In fact, let A ∈ CM n

p −1 (b )

of the mapping

{Am }+∞ m =1

and

be an arbitrary sequence, that ρ ( Am , A) → 0

at

m → +∞

,

then at every

(

)

t ∈ R, ρ χ t Am , χ t A = ρ ( Am (t + s ), A(t + s )) =  1 = sup min  Am (x ) − A(x ) ,  = ρ ( Am , A) x  x∈R 

(

)

i.e. ρ χ t Am , χ t A → 0 at m → +∞, consequently, χ t − is a continuous mapping. By the theorem of existence, the uniqueness and the continuous dependence of the solutions of the system on the mapping A , the Cauchy operator for every θ ,τ ∈ R is a non-degenerate linear mapping Χ A (θ ,τ )

: Rn → Rn

which is continuously depending on (θ ,τ , A) ∈ R × R × CM n . Hence, the mapping Xt :E → E

15

defined by (1.5) is a continuous mapping. For every e ∈ E we have: e = ( A, ξ ), where A ∈ CM n , ξ ∈ R n . For every t ∈ R and e ∈ E there is a chain of equalities

(

)

pX t e = pX t ( A, ξ ) = p χ t A, Χ(t ,0 )ξ = χ t A.

For any

t ∈ R, e ∈ E

(1.6)

the equalities are held χ t pe = χ t p ( A, ξ ) = χ t A

(1.7)

consequently, by (1.6) and (1.7), for any t ∈ R we have the equality pX t = χ t p.

For every ξ1 ∈ p −1 (b ), ξ 2 ∈ p −1 (b )

we get ξ1 = (b,η1 ), ξ 2 = (b,η 2 )

where b ∈ B,η1 ∈ R n ,η 2 ∈ R n .

Since for any α ∈ R, β ∈ R, αξ1 + βξ 2 = α (b,η1 ) + β (b,η 2 ) = (b, αη1 ) + (b, βη 2 ) = (b, αη1 + βη 2 ),

that there is a chain of equalities

(

)

X t [b](αξ1 + βη 2 ) = X t (b, αη1 + βη 2 ) = χ t b, Χ A (t ,0 )(αη1 + βη 2 ) =

(

) (

) (

)

= χ t b, αΧ A (t ,0 )η 2 = α χ t b, Χ A (t ,0)η1 + β χ t b, Χ A (t ,0)η 2 =

16

= αX t (b,η1 ) + βX t (b,η 2 ) = αX t [b]ξ1 + βX t [b]ξ 2 ,

consequently, for every

b ∈ B, t ∈ R

the mapping

X t [b] : p −1 (b) → −1 ( χ t b)

defined as a constriction (restriction) on the layer is a linear mapping. Proposition Xt For any

p −1 (b)

1

of mapping is proved.

t , s ∈ R, A ∈ CM n

the equalities are held χ t + s A(⋅) = A(t + s + (⋅)),

(1.8)

χ t χ s A(⋅) = χ t A( s + (⋅)) = A(t + s + (⋅));

(1.9)

therefore, by (1.8) and (1.9), for every

t ∈ R, s ∈ R

it is valid

χ t+s = χ t χ s .

(1.10)

For any t ∈ R, s ∈ R, ξ ∈ R n , A ∈ CM n

the equalities are held

(

)

X t X s ( A(⋅), ξ ) = X t χ s A(⋅), X A (s,0 )ξ =

(

)

= χ t χ s ( A(⋅)), X A(s +(⋅)) (t ,0)X A (s,0)ξ =

(

)

= χ t + s A(⋅), X A (t + s, s )X A (s,0 )ξ =

(

)

= χ t + s A(⋅), X A (t + s,0 )ξ = X t + s ( A(⋅), ξ ),

therefore, for every

t ∈ R, s ∈ R

it is valid

17

X t+s = X t X s .

(1.11)

In view of (1.10) and (1.11), for every t ∈ R the following formulas hold X t X −t = X 0 = 1E , X −t X t = X 0 = 1E

(1.12)

χ t χ − t = x 0 = 1B , χ − t χ t = x 0 = 1B ,

(1.13)

Proposition 2. For every t ∈ R, ( X t , χ t )

is an automorphism of the vector bundle ( E , p, B) . Proof. This means the following: X t is a homeomorphism E on E , χ t is a homeomorphism B → B , pX t = χ t p

and constriction

X t [b]

on the layer p −1 (b) of the mapping for every b ∈ B is an isomorphism (that is, a nondegenerate linear mapping) of the layer p −1 (b) on the layer p −1 ( χ t b) . Since the Cauchy operator Χ A (t , s ) : R n → R n

is a nondegenerate linear mapping that depends continuously on A ∈ CM n , t ∈ R, s ∈ R

that at every t ∈ R the mapping 18

X t [b] : p −1 (b) → p −1 ( χ t b)

defined as a constriction to the layer p −1 (b) of mapping X t is a nondegenerate linear mapping of the layer p −1 (b) on the layer p −1 ( χ t b) . By (1.12) and (1.13), we have the equalities ( X t ) −1 = X −t

( χ t ) −1 = χ − t .

and

It is proved in Proposition 1, that X t , χ t are continuous mappings for all t ∈ R , therefore, the mappings and X −t : E → E χ −t : B → B are continuous, i.e.

(X

−t

, χ −t

)

is morphism of the vector bundle (E , p, B ) which is inverse to the morphism ( X t , χ t ) . Therefore, ( X t , χ t ) is an isomorphism of the vector bundle. Proposition 2 is proved. We define the mapping H by the way: for any t ∈ R we suppose Ht = ( X t , χ t )

i.e. the value of the mapping H at the point t is a pair The multiplication of automorphisms is defined by

(X t , χ t ) .

( X t , χ t )( X s , χ s ) = ( X t X s , χ t χ s ),

where X t X s , χ t χ s are products (compositions) of mappings and, respectively, χ t , χ s . Proposition 3. For any t ∈ R, s ∈ R the equality holds H (t + s ) = HtHs

Proof. In fact, for any t ∈ R, s ∈ R we get H (t + s ) = ( X t + s , x t + s ) = ( X t X s , x t x s ) =

19

Xt,X s

= ( X t , x t )( X s , x s ) = HtHs

Proposition 3 is proved. Proposition 4. H ∈ Hom( R, Aut ( E , p, B ))

i.e. H is a homomorphism of the group R into the group of automorphisms of the vector bundle. Proof. For every t ∈ R of the image Ht of a point t ∈ R under a homomorphism H is a pair ( X t , χ t ) , where X t is a homeomorphism E on E , χ t is a homeomorphism B on B , moreover а)

pX t = χ t p

at every t ∈ R ;

б) at every b ∈ B, t ∈ R the constriction mapping X t is a linear mapping

X t [b]

on layer

p (b )

of

p −1 (b) → p −1 ( χ t b) .

By Proposition 2, t ∈ R, ( X t , χ t ) is an automorphism of the vector bundle ( E , p, B) . By Proposition 3 for any t ∈ R, s ∈ R , the equality holds H (t + s ) = HtHs

Hence, H is a homomorphism of the group R into the group of automorphisms of the vector bundle. Proposition 4 is proved. It is assumed, that  ___ 1  lim ln X t e , e ≠ θ , λ ( H , e) = t → +∞ t − ∞, e = θ 

where e ∈ E .

20

Lyapunov exponent λ n−k +1 ( H , b) [19-20] of the constructed is defined for any homomorphism H ∈ Hom( R, Aut ( E , p, B)) b ∈ B, k ∈ {1,..., n} by formula λ n − k +1 (H , b ) =

min

(

)

max λ ( H , e)

R k ∈Gk p −1 (b ) e∈R k ∗

Here Gi ( p −1 (b)) is the Grassmann manifold of i -dimensional vector subspaces of the layer p −1 (b) , R∗i is the set of all nonzero vectors of the vector space R i ; t ∈ R; ⋅ is the norm, induced by the Riemannian metric of the vector bundle fixed above ( E , p, B) . For any k ∈ {1,..., n}, b ∈ B

where

B = CM n

we define a set

{

}

E k ( H , b) = e ∈ p −1 (b) : λ ( H , e) ≤ λ n −k +1 ( H , b)

(1.14)

For every k ∈ {1,..., n}, b ∈ B the defined set is the vector space of the layer p −1 (b) . A set of all vector subspaces of the layer p −1 (b) of the vector bundle is denoted by E~b for every b ∈ B . We assume ~ E=

~

 Eb

b∈B

This set is naturally endowed with the structure of a topological __

space. For any λ ∈ R ( R is extended numerical line), b ∈ B we consider the set

{

}

E ( H , λ , b) = e ∈ p −1 (b) : λ ( H , e) ≤ λ .

It is known, that at every λ ∈ R, b ∈ B the defined set E ( H , λ , b) is the ~ vector subspace of the layer p −1 (b ) and, consequently E ( H , λ , b) ∈ E . Thus, at every λ ∈ R the mapping is determined 21

~ E ( H , λ ,⋅) : B → E .

(1.15)

i.e. H is the constructed where H ∈ Hom( R, Aut ( E , p, B )) homomorphism of the group R to the automorphism group of the vector bundle in Proposition 4. ~ As it is known, the mapping E ( H , λ ⋅) : B → E is semicontinuous ~ from below in the point b0 ∈ B , if for any neighborhood V ⊂ E of the point E (H , λ , b0 ) there exists a neighborhood W ⊂ B of the point b0 such, that for any point b ∈ W the vector space E (H , λ , b ) contains a subspace which is an element of the set V. The mapping (1.15) is not semicontinuous from below in the space B = CM n . We give an example, n = 1 . The equation dy =y at

has a solution y = ce t , and all solutions, in exception of the zero, have Lyapunov exponent which is equal to one λ ( A) = 1

Therefore, here 1 1  E ( A, ) = ζ ∈ R1 : λ (ζ ) ≤  = {0} 2 2 

is a zero subspace. We take the perturbed equation dz = (1 + θ (t − k )) z , ( k ∈ N ) dt

where 0, t ∈ [0, k ] θ (t − k ) =  − 1, t ≥ k

λ ( Ak ) = 0

22

Therefore, the entire solution has an exponent no greater than

1 2

,

therefore 1  1  E  Ak ,  = ξ = R ′ : λ (ξ ) ≤  = R 2  2 

Herewith 

1





ρ ( A, Ak ) = inf min  A(t ) − Ak (t ) ,  = t ≥0 t  1  1  = sup min  θ (t ) ,  = max  sup min θ , , t  t   t ≥0 0≤t ≤k 1  = sup min  θ (t ) ,  t  t ≥0

1  sup min  θ (t ) ,  t  t ≥0

}=

} = sup min θ (t ), 1  ≤ 1 t ≥k



t

k

Therefore ρ ( A , Ak ) → 0, k → ∞

here 1 1   dim E  A ,  = 0, dim E  Ak ,  = 1 2 2  

Hence it follows that the mapping (1.15) is not lower semicontinuous. Corollary 1.1. The mapping ~ E ( H , λ ,⋅) : B → E .

is not a Baire function of the zero class on the basis of the vector bundle with a compact-open topology. Theorem 1.1. For every λ ∈ R the mapping (1.15) everywhere is not lower semicontinuous in the space B = CM n , in which 0 < dim E (H , λ , b ) < n.

23

Proof. We assume, that it is not thrue, i.e. there exists a point and λ0 ∈ R

b0 ∈ B

0 < dim E (H , λ0 , b0 ) < n

and in the point the mapping (1.15) is semicuntinuous from below. By _

definition this means, that for any V ⊂ E neighborhood of the point E ( H , λ0 , b0 ) there exists a neighborhood W ⊂ B of the point b0 , that for any b ∈ W the vector space E ( H , λ0 , b) contains a subspace which is an element of the set V . Let b0 = A0 (t ) ∈ GM n and λ1 ( A0 ),..., λn ( A0 ) be Lyapunov’s exponents, ordered in the following way: λ1 ( A0 ) ≥ ... ≥ λn ( A0 ) It is known, that the vector subspace E ( H , λ0 , b0 ) coincides with one of the subspaces E k ( H , b0 ), k ∈ {1,..., n − 1}

Let

k = k0 , E ( H , λ0 , b0 ) = Ek 0 ( H , b0 ) where E k0 ( H , b0 ) = {η ∈ ρ −1 (b) : λ ( H ,η ) ≤ λ n−k +1 ( H , b0 )}.

From construction of the vector bundle ( E , p, B ) we get

Ek 0 ( H , b0 ) = {b0 } × E (λ0 , b0 ), where

E (λ0 , b0 ) = E k0 ( A0 )

24

is a set of the initial values of the solutions of the system A0 , which Lyapunov exponent not greater than λ n −k +1 ( A0 ) . E k0 ( A0 ) is a vector subspace of the space R n and has dimensionality

dim Ek 0 ( A0 ) = p0 and

k0 ≤ p0 ≤ n − 1 .

Let E ′ be an orthogonal addition of the subspace Ek 0 ( A0 ) dim E ′ = n − p0 , k 0 ≤ p0 ≤ n − 1 . Let

(

)

Vε = W0 × U ε E k0 (b0 )

be an arbitrary ε – neighborhood of the point E ( H , λ0 , b0 ) in the ~ space E . Here W0 is an arbitrary ε – neighborhood of the point b0 . U ε ( E k0 (b0 ))

is an arbitrary ε – neighborhood of the point E k0 (b0 ) , (at every

k ∈ {1,..., n} the Grassmann manifold Gk ( R n ) is compact and has a

countable basis) in the metric space G q (Rn ) . It is known, that in the space CM n Lyapunov's exponents are discontinuous in each points. Therefore, there exists A1 ∈ W0

such that

λ1 ( A1 ) > λ2 ( A1 ) = ... = λn ( A1 ), E1 ( A1 ) = E 2 ( A1 ) = ... = E n−1 ( A1 ) = E ′

E n ( A1 ) = R n

From the construction of the vector bundle it follows that 25

E (H , λ , b ) = Ei (H , b1 ) = {b1 }× E ′ __

at which i ∈ {1,..., n − 1}, λ ∈ R . The space E ( H , λ , b1 ) does not satisfy to the condition of the simicontiuity from below of the mapping (1.15) in the point b0 . Theorem 1.1 is proved. There is a stronger statement than Corollary 1.1. Corollary 1.2. The mapping ~ E ( H , λ ,⋅) : B → E .

is not a Baire function of the first class on the basis of a vector bundle with a compact-open topology. Let M n be a space of linear systems of the differential equations with continuous (piecewise continuous), bounded coefficients with uniform metrics. We consider a particular case of homomorphism H ∈ Hom( R, Aut ( E , p, B)) , where the base of the vector bundle

B = Mn. It is known from Perron's work, that the mapping (1.15) is not everywhere lower semicontinuous. We give an example. Let 𝑥𝑥 ′ = 𝐴𝐴(𝑡𝑡)𝑥𝑥

( )

(

x ∈ R n C n , A(⋅) : R + → Hom R n , R n

be a continuous bounded operator-function

{

( )

)

}

E ( A, λ ) = ξ ∈ R n C n : λ (ξ ) ≤ λ , (λ (0) = −∞ ) Ε( A, λ )

be a vector subspace. Therefore, a mapping arises A  E ( A, λ ).

We consider a system of equations 26

(1.16)

dy1  = − ay1  dt  dy  2 = (sin ln t + cos ln t − 2a )y 2  dt

(1.17)

( 0 < t 0 ≤ t < ∞ ). The general solution has the form y1 = c1е − at ,

y 2 = c 2 е t sin ln t − 2 at

If 1 1 1 < a < + e −π , 2 2 4

then Lyapunov exponents of the system (1.17) λ1 ( A) = 1 − 2a ,

since

λ 2 ( A) = − a

λ2 ( A) < λ1 ( A) < 0 ,

that

{

}

E ( A,0 ) = ξ ∈ R 2 : λ (ξ ) ≤ 0 = R 2 Now, if we choose a matrix as the perturbing matrix  0 Ak (t ) =  1 −at  e k

0  0 

(k ∈ N ) ,

then the perturbed system of equations has the form

𝑧𝑧 ′ = (𝐴𝐴 + 𝐴𝐴𝑘𝑘 )𝑧𝑧 dz1  = az1   dt  − at  dz 2 = (sin ln t + cos ln t − 2a )z + z e 2 1  dt k 

27

(1.18)

If 1 1 1 < a < + e −π , 2 2 4

then Luapunov exponents of the system (1.18) λ1 ( Ak ) > 0 ,

Solution of the system

λ2 ( Ak ) = −a .

( Ak ) is z1 = c1e − at

z 2 = e t sin ln t −2at (c 2 +

c1 t −τ sin lnτ dτ ) e k ∫0

λ2 ( Ak ) = χ [z1 ] = −a < 0

Thus, only such solutions, which have initial values from the axis z1 , have Lyapunov exponents not greater than zero. Consequently

{

}

E ( Ak ,0) = ξ ∈ R 2 : λ (ξ ) ≤ 0 = R

Here

and

(

Ak (⋅) : R + → Hom R 2 , R 2

ρ ( Ak , A) → 0 ,

)

(k ∈ N )

k →∞

In fact, since  0 Ak (t ) − A(t ) =  1 −at  e k

28

0 , 0 

that e − at 1 < → 0, k → +∞ k 0≤t < ∞ k

ρ ( Ak , A) = sup

although dim E ( A,0 ) = 2, dim E ( AK ,0 ) = 1(k ∈ N )

consequently,

A  E ( A, λ ).

i.e., the mapping (1.15) is not lower semicontinuous on the basis of the vector bundle. Now we consider the exponents of homomorphisms H ∈ Hom(G, Aut ( E , p, B))

of an abstract metrized vector bundle ( E , p, B) [129-134]. On vector bundle ( E , p, B ) a Riemannian metric is fixed and the +

existence of a function a (⋅) : B → R is required, at every b ∈ B, t ∈ G , G is a group R or Z , satisfying to the equality

a( χ t b) = a(b), and the inequality

)

(

X t [b] ≤ exp t a (b ) .

λk ( H , b) exponent of homomorphism H ∈ Hom(G, Aut ( E , p, B)) is defined at any b ∈ B, k ∈ {1,..., n} by formula Lyapunov's

λ k (H , b ) =

____ R

n − k +1

min

(

∈Gn − k +1 p

−1

max

(b ))ξ ∈R∗n − k +1

lim

t → +∞

1 ln X t ξ t

Here Gi ( p −1 (b)) is the Grassmann mapping of i -dimensional −1

i

vector subspaces of the layer p (b) , R∗ is denoted a set of all 29

nonzero vectors of the vector space R i ; t ∈ G; ⋅ is a norm induced by a fixed Riemann metrics of the vector bundle ( E , p, B ) .

Ω k ( H , b) of homomorphism The central exponent is defined [129-134] for any H ∈ Hom(G, Aut ( E , p, B )) b ∈ B, k ∈ {1,..., n} by formula Ω k (H , b ) =

___

1 R n − k +1∈Gn − k +1 ( p −1 (b ))τ ∈G∗ m→+∞ mτ inf

inf

lim

m −1

∑ ln X τX τ R s

s =0

n − k +1

Here by X Cτ is denoted the constriction on the set C of mapping

X τ . The norm of linear mapping is defined by the standard way; X τ i = sup  X τ ξ ξ R ξ ∈R i  ∗

−1 

 

It is assumed, that

where ξ ∈ E . Let

 ___ 1  lim ln X t ξ приξ ≠ 0, λ ( H , ξ ) = t → +∞ t − ∞приξ = 0 

{

}

E n −k +1 (H , b ) = ξ ∈ p −1 (b ) : λ (H , ξ ) ≤ λ k (H , b ) ,

En − k +1 ( H , b) be a vector subspace of the layer p −1 (b) . The central exponent [129-134] Ω ( H , b) of homomorphism H ∈ Hom(G, Aut ( E , p, B)) is defined for any b ∈ B, k ∈ {1,..., n} by formula (k )

____

Ω (k ) (H , b ) = inf lim

1 τ ∈N m→+∞ mτ

m −1

∑ ln X τX τ E

s =0

s

n − k +1

( H ,b )

.

The saturated homomorphisms are considered. We remind that the homomorphism H ∈ Hom(G, Aut ( E , p, B))

30

is called [129-134] saturated, if for every point b ∈ B such, that χ θ b ≠ b, at every θ ∈ G∗ (the star at the downward right everywhere is the throwing of the zero element) for every neighborhood W (b) of the point b (in the space B ) for every basis {ξ1 ,..., ξ n } of the vector space

p −1 (b) and every neighborhood U (ξi ) of the point E ) there exists δ ∈ R∗+ , such, that for

ξ i (i ∈ {1,..., n}) (in the space

every t ∈ N and every nondegenerated linear operators Ym : p −1 ( χ m−1b) → p −1 ( χ m b)(m ∈ {1,..., t }),

satisfying at every m ∈ {1,..., t } to the inequality

[

[

]

]

−1   Ym  X χ m −1b  − I + X χ m −1b Ym−1 − I < δ   point b′ ∈ W (b) and for every m ∈ {1,..., t }

there exists a isomorphism of the layers (as Euclidean spaces);

(

)

there exists

( )

ψ m : p −1 χ m b ′ → p −1 χ m b ,

moreover the requirements are satisfied 1) ψ 0−1ξ i ∈ U (ξ i ) at every

i ∈ {1,..., n}

2) at every m ∈ {1,..., t} the diagram

(

p −1 χ m −1b ′

↓ ψ m−1

(

p −1 χ m −1b

)

[



X χ

)

m −1

→ Ym

is commutative. 31

b′

]

(

p −1 χ m b ′



ψm

( )

p −1 χ m b

)

Definition 1.1. The subset X ⊂B

( B is a base of the abstract vector bundle ( E , p, B ) ) belongs to the class Fσ (Β) ,

if every Β is a measurable set (i.e. the Borel measurable set [171173]) with respect to B gives at the intersection with a set X of type Fσ relatively to B . Theorem 1.2. If for fixed k ∈ {1,..., n} the set

{

X k = b : Ω (k ) (H , b ) > λk (H , b ), b ∈ B

belongs to the class Fσ (Β) , then for this exponent

}

k ∈ {1,..., n}

the central

Ω ( k ) ( H , b) : B → R

is a Borel function of the second class. Proof. We fix k ∈ {1,..., n} and introduce the function ϕ k ( H , b) = Ω ( k ) ( H , b) − λ k ( H , b) ,

b∈B.

Then Ω (k ) (H , b ) − λ k (H , b ), b ∈ X k

ϕ k (H , b ) = 

0, b ∈ B \ X k

In the space B there is an everywhere dense set C of type Gδ such that for b ∈ C , the equality holds λ k ( H , b) = Ω ( k ) ( H , b)

32

We consider for any a ∈ R the sets E [ϕ k ( H , b) ≥ a ]

and

E [ϕ k ( H , b) ≤ a ]

We establish, that they are the sets of type Fσ ,δ . In fact, if a > 0 , then E [ϕ k ( H , b) ≥ a ] ⊂ X k ,

and by the condition of the theorem it is a set of type Fσ . If a ≤ 0 , then E [ϕ k ( H , b) ≥ a ] = B

The set B (as well as an empty set) is simultaneously opened and closed. As closed set it is a type of Gδ , consequently, the type Fσ ,δ , and as opened set it is a type of Fσ . If a > 0 , then from the equality E [ϕ k (H , b ) ≤ a ] = B \ E [ϕ k (H , b ) > a ] = B \ X k

follows, that E [ϕ k ( H , b) ≤ a ]

is a set of type Gδ , consequently, the type Fσ ,δ . The same, if

a = 0 , then E [ϕ k (H , b ) ≤ a ] = B \ E [ϕ k (H , b ) > a ] = B \ X k

Therefore, by condition of the theorem follows, that E [ϕ k ( H , b) ≤ a ]

is the set Gδ , consequently, the type Fσ ,δ . 33

If a < 0 , then E [ϕ k ( H , b) ≤ a ] =

Ø.

Thus, for any a ∈ R the sets E [ϕ k ( H , b) ≥ a ]

of the type Fσ ,δ . Consequently, at every

and

E [ϕ k ( H , b) ≤ a ]

ϕ ( H ,⋅) : B → R

is a function of the second Baire class, and therefore the class of functions Ω (k ) ( H ,⋅) : B → R

not higher than the second class. Theorem 1.2. is proved. Similarly it is proved the theorem. Theorem 1.3. If for fixed k ∈ {1,..., n} the set X k = {b : Ω k (H , b ) > λk (H , b ), b ∈ B}

belongs to the class Fσ (Β) , then in order to exponent

k ∈ {1,..., n}

the central

Ω k ( H ,⋅) : B → R

is Baire function of the second class. Let in the metric space B , ( B is the base of the vector bundle) Caratheodory measure be introduced [4, p. 456]. Definition 1.2. The subspace X ⊂ B of the base of vector bundle is called B scattered if, in the section with every set of measure zero, it gives the set not higher countable. Lemma 1.1. If the set X ⊂ B is B scattered, then it belongs to the class 34

Fσ (Β)

Proof. We remind, that the set X ⊂ B belongs to the class Fσ (Β) , if every Β is a measurable set respectively to B gives in the intersection with X the set of the type Fσ respectively B . We take any Β measurable set Q respectively to the base of the vector bundle, then: а) If Q is a set of the zero measure, then it gives in the intersection with X a countable set, i.e. set of the type Fσ . b) If Q is a set of the positive measure, then it can be represented as the unit Fσ set and the set of the zero measure. Consequently, we obtain the set of type Fσ in the intersection too. Lemma 1.1. is proved. Theorem 1.4. If for fixed k ∈ {1,..., n} the set

{

}

X k = b : Ω (k ) (H , b ) > λk (H , b ), b ∈ B ,

B – scattered, then for it k ∈ {1,..., n} the central exponent Ω ( k ) ( H ,⋅) : B → R

is Baire function of the second class. Proof. By lemma 1.1 the set X k belongs to the class Fσ (Β) . Consequently, from theorem 1.2 follows, that the central exponent Ω ( k ) ( H ,⋅) : B → R

is Baire function of the second class. Theorem 1.4 is proved. The statement is established by the same way. Theorem 1.5. If for any k ∈ {1,..., n} the set

{

}

Yk = b : Ω k (H , b ) > λk (H , b ), b ∈ B ,

B – scattered, then for this k ∈ {1,..., n} the central exponent Ω k ( H ,⋅) : B → R

is Baire function of the second order. 35

It is known, that for any k ∈ {1,..., n}, b ∈ B the inequalities hold λ k ( H , b) ≤ Ω k ( H , b) ≤ Ω ( k ) ( H , b).

As it is known, Lyapunov exponent of homomorphism H for any k ∈ {1,..., n} is Baire function of the second class. We fix k ∈ {1,..., n} and assume Z k = (b : Ω k (H , b ) = λk (H , b ), b ∈ R ).

Clearly, that the constriction of the central exponent Ω k ( H , b) of homomorphism H on Z k is the Baire function of the second class. The following statement holds. Theorem 1.6. The central exponent Ω ( k ) ( H , b) : B → R

of homomorphism H is the Baire function of the second class on the subset Z k ⊂ B of the base of vector bundle ( E , p, B) . Proof. In fact, if the equality holds Ω k ( H , b) = λk ( H , b)

for all fixed k ∈ {1,..., n} , then the following equalities are held: Ω (k ) ( H , b ) = Ω k ( H , b ) = λ k ( H , b )

If k = 1 , then for any b ∈ B the equality holds Ω (1) ( H , b) = Ω1 ( H , b) .

Therefore Ω (1) ( H , b) = λ1 ( H , b)

Consequently, Ω (1) ( H ,⋅) : Z1 → R

36

the central exponent of homomorphism H belongs to the second Baire class. We fix the index k ∈ {2,..., n} . If λk ( H , b) = λk −1 ( H , b) = ... = λ1 ( H , b)

then

Ω k ( H , b) = λk ( H , b) = λ1 ( H , b) = Ω1 ( H , b)

by the condition. By the expressions Ω (1) ( H , b) = Ω1 ( H , b) , Ω (k ) ( H , b) ≥ Ω k ( H , b) ≥ λ k ( H , b)

we get Ω (k ) ( H , b) ≥ Ω k ( H , b) = Ω1 ( H , b) = Ω (1) ( H , b) Ω1 ( H , b) = Ω (1) ( H , b) ≥ Ω (k ) ( H , b)

consequently

Ω k ( H , b) ≥ Ω (k ) ( H , b)

Therefore the equality holds Ω k ( H , b) = Ω (k ) ( H , b)

If λk ( H , b) = λk −1 ( H , b) = ... = λe ( H , b)

λe ( H , b) < λe −1 ( H , b) ≤ ... ≤ λ1 ( H , b) ,

then we get Ω k ( H , b) = λ k ( H , b) = λe ( H , b) = Ω e ( H , b) .

37

(k ≥ e > 1),

From definition Ω e ( H , b) follows, that for any ε > 0 , there exists such Rε n −e+1 − (n − e + 1)

– dimensional subspace, that 1 τ ∈G m→∞ mτ inf lim

m −1

∑ ln X τX τ Rε s

s =0

n − e +1

< Ω e ( H , b) + ε

.

If ε < λe−1 ( H , b) − λe ( H , b) ,

then it follows that for any ξ ∈ Rε n −e+1

the inequality is valid ____

lim

t →+∞

1 ln X еξ ≤ Ω e ( H , b) + ε ≤ λe ( H , b) + ε < λe−1 ( H , b) . t

Therefore in Rε n −e+1 , Lyapunov exponents do not exceed λe ( H , b) . This implies the inclusion Rε n −e+1 ∈ E n −e+1

(at

ε < λe −1 ( H , b) − λe ( H , b) ). From the assumption λe ( H , b) < λe−1 ( H , b)

and definition of the subspace E n −e +1 follows dim E n −e+1 = n − e + 1 = dim Rε n −e+1

38

.

Consequently Rε n −e +1 = En −e +1

i.e. the inequality holds Ω (e ) ( H , b) < Ω e ( H , b) + ε .

By virtue of arbitrariness ε ∈ (0, λe−1 − λe )

we get

Ω (e ) ( H , b) ≤ Ω e ( H , b)

Taking into account the inequality Ω (1) ( H , b) ≥ ... ≥ Ω (n ) ( H , b)

and

Ω (k ) ( H , b) ≥ Ω k ( H , b) ≥ λ k ( H , b)

we get Ω (k ) ( H , b) ≥ Ω k ( H , b) = λ k ( H , b) = λe ( H , b)

λe ( H , b) = Ω e ( H , b) ≥ Ω (e ) ( H , b) ≥ Ω (k ) ( H , b)

Consequently, for any k ∈ {2,..., n}, b ∈ Z k

the equality holds Ω ( k ) ( H , b) = λ k ( H , b)

Therefore, in this case the central index Ω ( k ) ( H , b) of homomorphism H is Baire function of the second class on the set Z k at any k ∈ {2,..., n} . Theorem 1.6 is proved. 39

It is known, that for every Baire function f there exists a such dense in the basis of the vector bundle set C of the type Gδ , that the function f :C → R

is continuous. We fix k ∈ {1,..., n} . Let C k be a set of the points of continuity of central exponent Ω k ( H , b) of homomorphism H ∈ Hom(G, Aut ( E , p, B)) Ω k ( H , b) : B → R .

We assume Dk = B \ C k

(1.22)

Definition 1.3. If every Β -measured set (i.e. Baire measured set [171-173]) respectively B ( B is the base of abstract vector bundle ( E , p, B) ) gives in the intersection with the set X a set of the type

Fi , i ∈ {1,2,3} (here F1 = Fσ , F2 = Fσδ , F3 = Fσδσ ) respectively B , then the set X is called the set of the class Fi (B ) . Theorem 1.7. If for fixed k ∈ {1,..., n} the set Dk is a set of the

class F1 ( B) , then for this k ∈ {1,..., n} the central exponent Ω k ( H , b), of homomorphism H ∈ Hom(G, Aut ( E , p, B)) is Baire function of the second class on the base B vector bundle ( E , p, B ) . Proof. We continue the function Ω k ( H ,⋅) : C k → R

prescribed on the set Ck to a defined on B function Ω k ( H ,⋅) : C k → R

of the first class. We define a function 40

ϕ k ( H , b) = Ω k ( H , b) − Ω k ( H , b) .

Then

Ω k (H , b ) − Ω k (H , b ), b ∈ Dk

ϕ k (H , b ) = 

0, b ∈ C k

We consider for any a ∈ R the sets E [ϕ k ( H , b) ≥ a ], E [ϕ k ( H , b) ≤ a ] .

(1.23)

We establish, that they are the sets of type Fσδ . In fact, if a > 0 , then E [ϕ k ( H , b) ≥ a ] ⊂ Dk

and by condition of the theorem it is a set of type Fσ . E [ϕ k ( H , b) ≤ a ] = B \ E [ϕ k ( H , b) > a ]

(1.24)

is a set of type Gδ , consequently, the type Fσδ . If a = 0 , then E [ϕ k ( H , b) ≤ 0] = B \ E [ϕ k ( H , b) > 0]

(1.25)

is a set of type Gδ , consequently, the type Fσδ . E [ϕ k ( H , b) ≥ 0] = B \ E [ϕ k ( H , b) < 0]

(1.26)

is a set of the type Gδ , consequently, the type Fσδ . If a < 0 , then E [ϕ k ( H , b) ≥ a ] = B \ E [ϕ k ( H , b) < a ]

is a set of type Gδ , consequently, the type Fσδ . The set, E [ϕ k ( H , b) ≤ a ] ⊂ Dk

41

(1.27)

i.e. by condition of the type Fδ . Consequently, ϕ k ( H , b) : B → R ,

is a Baire function of the second class, and therefore the class of the functions Ω k ( H ,⋅) : B → R

does not exceed of the second class. Theorem 1.7 is proved, Corollary 1.1. If the set D1 is a set of the class F1 ( B ) , then the central exponent Ω (1) ( H , ) : B → R

is a Baire function of the second class. We fix k ∈ {1,..., n} . Let C (k ) be a set of the points of contiuity of Ω ( k ) ( H , b) the central exponent of homomorphism H ∈ Hom(G, Aut ( E , p, B))

Ω ( k ) ( H , b) : B → R .

We assume D (k ) = B \ C (k )

.

Theorem 1.8. If for fixed k ∈ {2,..., n} and for fixed i ∈ {1,2,3} , the set D (k ) is a set of the class Fi (B) , then the central exponent Ω (k ) ( H ,⋅) : B → R

of homomorphism H ∈ Hom(G, Aut ( E , p, B)) is a Baire function correspondingly of the i + 1 -th class on the base B of vector bundle ( E , p, B ) . Proof. For fixed k ∈ {2,..., n} we introduce the function 42

Ω (k ) (H , b ) − Ω (k ) (H , b ), b ∈ D (k )

ϕ k (H , b ) = 

0, b ∈ C (k )

,

(k )

where Ω ( H , b) is a Baire function of the first class on the base B of vector bundle ( E , p, B ) , and for any b ∈ C (k ) the equality holds Ω

(k )

( H , b) = Ω ( k ) ( H , b)

If the set D (k ) is a set of the class Fi (B ) , then similarly, as in theorem 1.7, it is easy to establish, that ϕ ( k ) ( H ,⋅) : B → R

Baire function of the i + 1 -th class. Consequently, the central exponent Ω ( k ) ( H , b) = ϕ ( k ) ( H , b) + Ω

(k )

( H , b)

of homomorphism H ∈ Hom(G, Aut ( E , p, B)) is Baire function of the i + 1 -th class on the base of the vector bundle ( E , p, B) . Theorem 1.8 is proved. Definition 1.4. If the set X of class F1 ( B) is a set of type Gδ , then it is called by the set of the class F1′ ( B) . Theorem 1.9. If for fixed k ∈ {1,..., n} the sets Dk and D (k ) are the sets of the class F1′ ( B) , then the central exponents Ω k ( H , b) Ω

and

of homomorphism H ∈ Hom(G, Aut ( E , p, B)) are Baire functions of the first class on the base B of the vector bundle ( E , p, B ) . Proof. As in the theorems 1.2 and 1.3 at every we define ϕ k ( H , b) and ϕ ( k ) ( H , b) . For any a ∈ R we note, that the sets (k )

( H , b)

E [ϕ k ( H , b) ≥ a ], E [ϕ k ( H , b) ≤ a ] ,

[

] [

E ϕ ( k ) ( H , b) ≥ a , E ϕ ( k ) ( H , b) ≤ a

43

]

are the sets of type Fσ . In fact, if a > 0 in (1.24) the right hand side by condition of the theorem, as addition to the set of the type, is a set of the type Fσ . If a = 0 the right hand sides (1.25), (1.26), (1.27) are the sets of the types Fσ . Similarly, Lebesque sets of the functions ϕ k ( H , b) , ϕ ( k ) ( H , b) are the sets of type

Fσ . Consequently,

ϕ k ( H , b)

and ϕ ( H , b) are Baire functions of the first class. Therefore, the Baire classes of the functions (k )

Ω k ( H , b) = ϕ k ( H , b) + Ω k ( H , b) Ω ( k ) ( H , b) = ϕ ( k ) ( H , b) + Ω

(k )

and

( H , b)

are not higher than the first class. Theorem 1.9 is proved. Corollary 1.2. If for fixed k ∈ {1,..., n} the sets Dk and D (k ) are rarefied, then the central exponents Ω k ( H , b) and Ω ( k ) ( H , b) of homomorphism H ∈ Hom(G, Aut ( E , p, B)) are Baire functions in the exactness of the first class on the basis B of the vector bundle ( E , p, B ) . Proof. In fact, if the sets Dk and D (k ) are rarefied, then they are not empty by definition, consequently, the central exponents Ω k ( H , b) and Ω ( k ) ( H , b) are not Baire functions of the zero class. On the other hand, any rarefied set [172] is decomposed. Therefore, by completeness of the base of vector bundle the decomposable sets Dk and D (k ) coincide with the sets which are simultaneously the type Fσ and Gδ , i.e. Dk and D (k ) are σ –spaces. Therefore, at every ϕ k ( H , b) and ϕ ( k ) ( H , b) are Baire functions of the first class. It follows that the central exponents Ω k ( H , b) and Ω ( k ) ( H , b)

are Baire functions in the exactness of the first class on the base B of the vector bundle ( E , p, B ) . Corollary 1.2 is proved.

44

2. GENERALIZED LYAPUNOV EXPONENTS OF THE SYSTEM OF DIFFERENTIAL EQUATIONS

The central concept of the first Lyapunov's method is the Lyapunov exponent. Usually the exponents are determined for linear systems of differential equations that satisfy a condition of boundedness of the coefficients (uniformly or on the average) that ensure the finality of the Lyapunov exponents of any nonzero solution and imply the finality of the Lyapunov exponents of the system under consideration. For linear systems with continuous and unbounded matrices, the exponents may not have a finite value, and therefore do not describe the properties of the solutions of the system. In this section we consider generalized Lyapunov exponents of linear systems of differential equations with unbounded coefficients. We note, that the requirement for boundness of the coefficients of the system for Lyapunov exponents is very important, since if they are not bounded, that the conclusions may be incorrect. We consider, for example, the system dy = −t 2 y dt

dx = tx , dt

a fundamental system of solutions for it be x1

t2 =e 2

x 2 = 0, y 2 = e

, y1 = 0 ;



t3 3

The characteristic exponent of the first solution is + ∞ , and the second is − ∞ . The characteristic exponent of the fundamental system is + ∞ . We can not determine the Lyapunov characteristic exponents for the system

45

 et x2  x1 = − ln(t + 1) x1 +  (t + 1) t + 2  1   x 2 = t + 1 x 2 

Now we consider the following linear homogeneous system of differential equations with unbounded coefficients  t  t  t ln (1+t )−t   x1 + 2 ln (1 + t ) + e x2  x1 = − ln (1 + t ) + 1 + t  1 + t      x = x 2  2

It is easy to verify, that the linear system has the following linearly independent solutions x

(1)

[

= colon e −t ln (1+t ) ,0

]

and

x

(2 )

[

= colon e t ln (1+t ) , e t

]

It is directly verified, that the lowest upper Lyapunov exponent is equal to − ∞ , the highest upper Lyapunov exponent is equal to + ∞ and the upper central exponent of the system is equal to one. As it is known, in the classical theory of Lyapunov exponents, the upper central exponent is the exact upper bound for the Lyapunov exponents of linear system. Thus, we note, that for linear systems with unbounded coefficients the basic theory of Lyapunov exponents can not be applied. Consequently, it would be natural to compare the growth of λt solutions not with an exponential scale e , but with some other, relatively to which to each solution of the system a numerical characteristic would be assigned. For this purpose, generalized exponents are considered. Let f (t ) be a complex function in the general case, determined in I = [t 0 ,+∞ ) . Q be a class of positive monotonically increasing to + ∞ at t ↑ +∞ piecewise continuous functions determined in I . We present the definition of the generalized exponent f (t ) relatively to q (t ) ∈ Q . 46

Definition 2.1. [4, p. 186-187]. The number (or symbol − ∞ or

+ ∞ ), defined by formula

χ [ f , q ] = lim

t → +∞

1 ln f (t ) q (t )

(2.1)

where q(t ) ∈ Q , f (t ) > 0 , t ∈ I is called by generalized upper characteristic Lyapunov exponent, otherwise, the generalized exponent of the function f (t ) with respect to q(t ) . If f (t ) = 0 , t ∈ I then it is considered, that χ [0, q ] = −∞ . If q(t ) = t , then we obtain the ordinary definition of the Lyapunov exponent in the form of Perron. We note, that for a non-zero function there is always a function q(t ) ∈ Q , such that the generalized exponent χ [ f , q ] takes a finite value. The following properties of generalized exponents are easily verified: 10 .χ [cf , q ] = χ [ f , q ] at c ≠ 0.

2 0. If

f (t ) ≤ g (t )

at t > T , T ∈ I , then

χ [ f , q ] ≤ χ [g , q ].

3 0 .χ [ f + g , q ] ≤ max{χ [ f , q ], χ [g , q ]}. 4 0 .χ [ f ⋅ g , q ] ≤ χ [ f , q ] + χ [g , q ].

5 0. For any

t k , t k → +∞ lim

k →∞

at k → ∞ 1 ln f (t k ) ≤ χ [ f , q ]. q (t k )

Let B n be a set of all continuous vector-functions defined on I with values in linear space R n . A generalized exponent of a continuous vector- function x(t ) relatively q(t ) ∈ Q is called the general exponent of its norm. 47

We note, that the solutions x above (p. 52)

(1)

and

x

(2 )

of the linear system

 t  t  t ln (1+t )−t   x1 + 2 ln (1 + t ) + e x2  x1 = − ln (1 + t ) + 1 + t  1 + t      x = x 2  2

have generalized exponents relatively q (t ) = t ln (1 + t ) which equal to − 1 and + 1 . Let F (t ) = ( f jk (t ))

be continuous finite dimensional matrix defined on [t 0 ,+∞ ), q(t ) ∈ Q . Then we suppose by definition χ [F , q ] = χ [ F , q ]

where the norm ⋅ of the matrix is understood to be one of three norms. 6 0. Generalized characteristic exponent of the finite dimensional matrix

[

]

χ [F , q ] = max χ f j ,k (t ), q , where j ,k

F (t ) = ( f jk (t ))

Proof: Since f j ,k (t ) ≤ F (t ) for any j, k ∈ {1,..., n} , that

χ [ f j , k (t ), q ] ≤ χ [F , q ] , for any j , k ∈ {1,.., n}. Consequently,

[

]

max χ f j ,k (t ), q ≤ χ [ F , q ] j ,k

On the other hand, we get 48

F (t ) ≤ ∑ f j ,k (t ) j ,k

Therefore, by the property of the generalized exponents we have

[

χ [F , q ] ≤ max χ f j ,k , q j ,k

]

Consequently, the equality holds

7 0. If χ [x, q ] = α ≠ ±∞, then for any ε > 0 1) x(t ) = 0(exp(α + ε )q(t )), т.е. _____

2) lim

t → +∞

x(t )

exp((α − ε )q (t ))

lim

t →∞

x(t )

exp((α + ε )q (t ))

=0

= +∞

and inversely, if 1) and 2) are satisfied, then χ [x, q ] = α .

8 0. Vectors x ∈ B n with different finite generalized exponents are linearly independent. In fact, let χ [x, q ] = α , χ [ y, q ] = β and α ≠ β . We assume, that for definiteness, x = cy, c ≠ 0 , i.e. vectors x and y are linearly dependent. Then, by the property of generalized exponents, we get α = χ [x, q ] = χ [cy, q ] = χ [ y, q ] = β

i.e. have obtained a contradiction. Theorem 2.1. If a vector function F (t , x ) is continuous in the n domain G = I × R and satisfies to the condition

F (t , x ) ≤ Kψ (t ) x

(2.2) t

where ψ (t ) ≥ 0 is continuous function t ∈ I , K > 0 , q(t ) = ∫ψ (τ )dτ , t0

49

q(t ) ↑ +∞ at t ↑ +∞ . Then any nonzero solution of the vector equation dx = F (t , x ) dt

has finite generalized exponents relatively q (t ) . Proof. By calculating the derivative x

2

= ( x, x )

and applying the Bunyakovsky inequality, we obtain d (x, x ) = 2 Re(F , x ) ≤ 2 Kψ (t )(x, x ). dt

(2.3)

Since condition (2.2) ensures the uniqueness of the zero solution, that every other solution does not vanish for any t ∈ I , and for such solution, dividing inequality (2.3) by 2

x , we find − Kψ (t ) ≤

therefore

d ln x ≤ Kψ (t ), dt

− K ≤ χ [x, q ] ≤ K ,

where t

q (t ) = ∫ ψ (τ )dτ . t0

Theorem 2.1 is proved. In particular, this is always true [4, p. 187, Theorem 6] for solutions of the linear system 50

x = A(t )x

(2.4)

with continuous (piecewise continuous) matrix A(t ) ≤ Kψ (t )

(2.5)

Example 1. We consider the system t

x =

− 1 (1 + ln t )x + t 2 y , y = (ln t ) y . 2

(t ≥ 1)

It is directly verified, that t

x1 (t ) = t 2

, y1 (t ) = 0

(

)

t

x 2 (t ) = 1 − e1−t t 2 ,

and

y 2 (t ) = e1−t t t

they are solutions of the system. A(t ) ≤ K (1+ ln t ) , ψ (t ) = 1+ ln t

, q(t ) = t ln t .

Consequently, the generalized exponents of the system λ1 (q ) = 1 , λ 2 (q ) =

1 2

.

Characteristic exponents are equal to + ∞ . 2 2 2 Example 2. x = e t x Here A(t ) = e t A(t ) = e t , K =1 t

t

0

0

q (t ) = ∫ ψ (τ )dτ = ∫ eτ dτ 2

Solution of the equation is t

x(t ) = x(0 )e

Lyapunov’s exponent 51

∫e 0

τ2



.

ψ (t ) = e t , 2

t

1 χ [x ] = im n x(t ) = im t → +∞ t t → +∞

2 ln x(0 ) + ∫ eτ dτ

°

t

1 t2 2 e et 2 = im t = im 2 = +∞ . t t → +∞ t → +∞ 2t

Generalized exponent relatively q (t ) is t

χ [x, q ] = im

t → +∞

1 ln x(t ) = im q (t ) t → +∞

ln x(0 ) + ∫ eτ dτ °

t



ι

=1

τ2

e dτ

°

q (t ) i.e. the solution increases with a characteristics relatively e .

Example 3.

(t ≥ 1) .

x = 1 + t x

3

A(t ) = 1 + t

χ [x, q ] =

q (t ) = t 2

,

ψ (t ) = K t ,

2 . 3

The solution x increases as a function 3

2 2 t e3 ,x

Example 4. а)

(t ) =

3 2 (1+ t ) 2 3 ce

(t ≥ 1) .

x = 1+ t 2 x t

q (t ) = ∫ Kτdτ = 0

A(t ) = 1 + t 2 ≤ Kt ,

Kt 2 , 2

K =2.

χ [x, q ] =

1 . 2

Then q (t ) = t 2 ,

Solution 52

x(t ) =

increases as a function e б)

1 2 t 2

t 1+ t 2 ce 2 4

(1 < m < 2)

A(t ) = 1 + t m ≤ Kt

q (t ) = K ∫ τ

m 2 dτ

1+ t2 − t

.

x = 1 + t m x

t

1+ t2 + t

=

°

m 2

(t ≥ 1) .

ψ (t ) = Kt

,

m +1 2

m 2

m+2 2 Kt , q (t ) = t , K= 2 m+2

m+ 2 2

t



x(t ) = ce 1

(If

m≠

k = ±1,±2,...;

2 k

x(t )

, then

1+ t1m dt1

3 2



χ [x, q ]∈  ,2 . 

is not elementary function, where

).

Solution x increases as a function

e

λt

m+2 2

,

where λ = χ [x, q ] .

Example 5. а) x = (ln t ) x (t ≥ 1) . The coefficient A(t ) = ln t is a unbounded continuous function. Solution has the form

x(t ) = ce ∫

ln tdt

= cet ln t −t

The ordinary Lyapunov exponent (q (t ) = t ) is equal to χ [x, t ] = im

t → +∞

1 ln x = +∞ . t

53

If we substitute τ = q(t ) , where t

t

1

1

q (t ) = ∫ ψ (τ )dτ = ∫ ln τdτ = t ln t − t + 1

then the equation is obtained

(

) (

dx q −1 (τ ) = x q −1 (τ ) dτ

) , (t = q

−1

(τ ))

where Lyapunov exponent = 1 . But we can not say how the solution is growing. Now we consider the generalized exponent t

A(t ) = ln t , ψ (t ) = ln t

q (t ) = ∫ ψ (τ )dτ = t ln t − t + 1

i.e.

1

consequently χ [x, q ] = im

t → +∞

ln x t ln t − t + 1

____

ln c + t ln t − t

t → +∞

t ln t − t + 1

= lim

= 1.

Solution is growing as a function et ln t −t +1

б) х = (ln n t ) x

at

t → +∞

t

χ [x, q ] = 1 ,

(t ≥ 1) . q(t ) = ∫ ln n t1dt1 , 1

Solution is growing as a function e t ln

Example 6. х = t 3 ln 3 tx

n

t

at t → +∞

(t ≥ 1) . χ [x, q] = 1 , where 4

q (t ) = t ln t 4

3

Solution is growing as a function 54

∀n ∈ N

1 4 3 t ln t 4 e at x =

Example 7.

e 2t 1+ et

t → +∞

x . χ [x, q ] = 1 ,

where

q (t ) = e e

t

( )

− ln 1+ et

x = (t − sin t )3 x . χ [x, q ] =

Example 8 . where q(t ) = t 4 Example 9.

x =

t 4 arctgt 1+ t

2

x . q (t ) = t 3 , χ [x, q ] = 6 1

Example 10.

x =

1 , 4

8 (1 + t 5 ) 2 x 5

8 5

π . 6

6 1

, y = − (1 + t 5 ) 2 y . 8

χ [ x, q] ≥ 1 , χ [ y, q] ≤ −2 , q(t ) = t 5 .

Example 11. y 2 (x ) = xe − x

2

(

)

y ′′ + 4 xy ′ + 4 x 2 + 2 y = 0

−x , y1 ( x ) = e , 2

.

q( x ) = x 2 , λ1 (q ) = −1 , λ 2 (q ) = −1 . Example 12.

x = 2tx

, y = y , z = −2tz .

χ [ x, q] = 1 , χ [ y, q] = 0 , χ [ x, q] = 1 , q(t ) = t 2 .

As it is known, a linear homogeneous system [4, p. 187, Theorem 6] with a continuous matrix has no more than n generalized exponents with respect to q(t ) ∈ Q . Let 55

n

[

σ ( X , q ) ≡ ∑ χ x (i ) , q i =1

]

be sum of generalized exponents relatively to q(t ) ∈ Q the fundamental system of solutions x (1) ,..., x (n ) , of system (2.4). A fundamental system of solutions and the corresponding fundamental matrix are called normal relatively q (t ) ∈ Q , if the sum of their generalized exponents is the smallest in comparison with other fundamental systems of solutions. For a normal fundamental system of solutions, i.e. for a normal basis of the solution space of system (2.4) the sum σ ( X , q ) is denoted σ (q ) .Thus, σ (q ) = min σ ( X , q ).

Here the minimum always exists, since the set of generalized exponents relatively to q (t ) of system (2.4) forms a finite set. We note, that the generalized exponents of a normal fundamental system of solutions do not depend on the choice of a normal fundamental system of solutions. The generalized exponents of a normal fundamental system of solutions are called generalized Lyapunov exponents of the system (2.4) with respect to q (t ) and are arranged in the following order −∞ < λ n (q ) ≤ ... ≤ λ1 (q ) < +∞,

λ1 (q ) is called the highest generalized exponent of system (2.4), λ n (q ) is called the lowest upper generalized exponent. To distinguish the generalized Lyapunov exponents of system (2.4) from other systems, we use the notations λ k ( A) . If the matrix A(t ) is real, then the normal fundamental system can always be assumed to be real.

56

For every k ∈ {1,..., n} we consider the set En − k +1(q ) of all

solutions, x(t ), including trivial of system (2.4), which generalized exponents do not exceed the number λ k (q ) i.e. E n − k +1 (q ) = {x(t ); χ [x, q ] ≤ λ k (q )}.

Proposition 2.1. For every k ∈ {1,..., n} the set E n − k +1 (q ) is a linear subspace of the space of solutions of system (2.4), in particular, when k = 1 coincide with it. Moreover, the chain of inclusions hold E1 (q ) ⊆ E 2 (q ) ⊆ ... ⊆ E n (q ).

Proof. In fact, if x1 ∈ En − k +1(q), x2 ∈ En − k +1(q) , then χ [x1 , q ] ≤ λ k (q ) , χ [x 2 , q ] ≤ λ k (q ) .

By the property of generalized exponents the inequalities hold χ [c1 x1 + c 2 x 2 , q ] ≤ max{χ [x1 , q ], χ [x 2 , q ]} ≤ λ k (q) .

Therefore E n − k +1 (q ) is a linear subspace of the space solutions of system (2.4). Further, if x ∈ E n − k +1 (q ) ,

k ∈ {1,..., n}

then χ [x, q ] ≤ λ k (q ) . By the inequalities λ k (q) ≤ λ k −1 (q) , k ∈ {2,..., n}

we get

χ [x, q ] ≤ λ k −1 (q ) .

Consequently, x ∈ E n − k + 2 (q ) , i.e. the inclusion holds 57

k ∈ {2,..., n} .

En − k +1 (q ) ⊆ En − k + 2 (q ) ,

Therefore, the chain of inclusions holds E1 (q ) ⊆ E 2 (q ) ⊆ ... ⊆ E n (q ).

Proposition 2.1 is proved. Lemma 2.1. Let a system (2.4) have finite generalized exponents relatively q(t ) ∈ Q . Them the generalized Lyapunov inequality respectively q(t ) holds, i.e. for any fundamental system of solutions x (1) ,..., x (n ) of system (2.4) the inequality is satisfied

[

t

n 1 SpA(τ )dτ ≤ ∑ χ x (i ) , q ∫ t → ∞ q (t ) i =1 t ___

lim

]

(2.6)

0

Proof. In fact, from the condition follows, that the Wronskian

[

W (t ) = det x (1) ,..., x (n )

]

has a finite generalized exponent χ [W , q ] . By the property of the generalized exponents for sum and production we get n

[

]

χ [W , q ] ≤ ∑ χ x (i ) , q . i =1

(2.7)

On the other hand, from the Ostrogradsky-Liouville formula it follows that ___

χ [W , q ] = lim

t →∞

t

1 SpA(τ )dτ q (t ) t∫

.

(2.8)

0

Consequently, from (2.7) and (2.8) we get (2.6). Lemma 2.1 is proved. We establish a class of a system of differential equations, when it is possible to calculate the generalized exponents by the coefficients of the system. 58

We consider a linear system of differential equations n dy i = p i (t ) y i + ∑ p ik (t ) y k dt k =1

(2.9)

in general, with complex continuous coefficients p i (t ), p ik (t ), i ∈ {1,..., n}, k ∈ {1,..., n} , t ∈ J ≡ [0,+∞ ) .

Lemma 2.2. If in the system (2.9) for any i ∈ {1,..., n}, k ∈ {1,..., n} the condition is satisfied lim

t → +∞

pik (t )

ψ (t )

=0

(2.10)

where ψ (t ) is a continuous positive function on J , then for any α > 0 there exists such T1 ∈ J , that for any t ≥ T1 the inequalities hold Re p i (t ) 2 α n 1 d yi − ∑ y i y k ≤ 2ψ (t ) dt y i2 2n k =1 ψ (t )

where i ∈ {1,..., n} .

1 d 2 Re p i (t ) 2 α n yi + yi ≤ ∑ yi y k 2n k =1 2ψ (t ) dt ψ (t )

(2.11)

(2.12)

Proof. We multiply each of equations of system (2.9) by y i , where y i is a complex value, adjoined with y i . We obtain n

y i y i′ = y i p i (t ) y i + y i ∑ p ik (t ) y k k =1

hence, by condition yi yi = y i

59

2

we get y i y i′ = p i (t ) y i

2

n

+ ∑ p ik (t ) y i y k k =1

whence we obtain n

Re( y i y i′ ) − Re( p i (t ) y i ) ≤ ∑ p ik (t ) y i y k 2

k =1

consequently, by using the equality 1 d 2 y i = Re( y i y i′ ) 2 dt

we get 1 d 2 y i − Re p i (t ) y i  2 dt

2

n

 ≤ p ik (t ) y i y k  k∑ =1

(2.13)

Whence by dividing on ψ (t ) , we obtain 1 d 2 Re( p i (t )) yi − yi ψ (t ) 2ψ (t ) dt

2

n

≤∑

k =1

p ik (t )

ψ (t )

yi y k

(2.14)

From the condition (2.10) follows, that for any α > 0 there exists such T1 ∈ J , that for any t ≥ T1 and for any i ∈ {1,..., n}, k ∈ {1,..., n} p ik (t )

ψ (t )



α 2n

(2.15)

holds. Consequently, for any t ≥ T1 from (2.14) and (2.15) follows, that at any i ∈ {1,..., n}, k ∈ {1,..., n} the inequality holds 1 d 2 Re( p i (t )) 2 α n yi − yi ≤ ∑ yi y k 2ψ (t ) dt ψ (t ) 2n k =1

60

Then we obtain the inequalities (2.11) and (2.12). Lemma 2.2 is proved. For α > 0 , the number T1 ∈ J is fixed which is found in Lemma 2.2 Lemma 2.3. Let for system (2.9) the conditions are satisfied 1) Re p1 (t ) > Re p i (t ) + αψ (t ), t ∈ J , i ∈ {2,..., n},

2) lim

t → +∞

p ik (t )

ψ (t )

= 0, i ∈ {1,..., n}, k ∈ {1,..., n} ,

where α > 0 , ψ (t ) is a continuous positive function on J . Then there exists a solution y1 , y 2 ,..., y n of system (2.9) such, that for any t ≥ t 0 the inequality holds y1 (t )

2

> y i (t ) , i ∈ {2,..., n}, t 0 > T1 2

Proof. Let y1 , y 2 ,..., y n be a solution of system (2.9) satisfying to the initial conditions y i (t 0 ) = η i , i ∈ {1,..., n} ,

where

η1 > η i , i ∈ {2,..., n} , t 0 > T1 .

We prove, that the solution is original. We assume the contrary. Let the inequality y1 (t )

2

> y i (t ) , i ∈ {2,..., n}, t 0 > T1 2

be satisfied in a semi-interval t 0 ≤ t < t 2 and in the moment t = t 2 is not satisfied, i.e. for an index k ∈ {2,..., n} we get y1 (t 2 )

2

= y k (t 2 )

2

moreover 61

≥ y i (t 2 )

2

, i ≠ 1, i ≠ k

(2.16)

d d 2 2 y1  yk  ≤  dt dt t = t 2  t = t 2 

Since t 2 > T1 , that from inequality follows, that

(2.17)

(2.11) of Lemma 2.2 at

i = 1, t = t 2

Re p1 (t 2 ) 2 α n 1 d  y1 (t 2 ) − ∑ y1 (t 2 ) y k (t 2 ) ≤ 2ψ (t )  dt y12  ψ (t 2 ) 2n k =1  t =t 2 2 

Hence by (2.16) we obtain the inequality Re p1 (t 2 ) 2 1 d α n y1 (t 2 ) − ∑ y (t 2 ) y k (t 2 ) ≤ 2ψ (t )  dt y k 2n k =1 1 ψ (t 2 ) 2 

2

   t =t 2

(2.18)

As well as from inequality (2.12) of Lemma 2.2 at i = k , t = t 2 follows, that 1 d yk  2ψ (t 2 )  dt

2

Re p k (t 2 ) 2 α  y k (t 2 ) + ≤  ψ (t 2 ) 2n  t =t 2

n

∑ y k (t 2 ) y s (t 2 )

s =1

By (2.16) for any s ∈ {1,..., n} we get y k (t 2 ) y s (t 2 ) = y k (t 2 ) y s (t 2 ) = y1 (t 2 ) y s (t 2 ) ≤ y1 (t 2 )

2

Therefore, the inequality holds 1 d yk  2ψ (t 2 )  dt

2

Re p k (t 2 )  2 2 α ≤ y 1 (t 2 ) + y1 (t 2 )  ψ (t 2 ) 2  t =t 2

(2.19)

Consequently, taking into account (2.16) from inequalities (2.18) and (2.19) we obtain, that Re p k (t 2 ) Re p1 (t 2 ) 2 α 2 2 α 2 y1 (t 2 ) − y1 (t 2 ) ≤ y1 (t 2 ) + y1 (t 2 ) ψ (t 2 ) ψ (t 2 ) 2 2

62

Since y1 (t 2 ) ≥ y i (t 2 ) , i ≠ 1;

that y1 (t 2 ) ≠ 0 ,

therefore the inequality can be divided into y1 (t 2 ) 2 . Then we get Re p1 (t 2 ) α Re p k (t 2 ) α − ≤ + ψ (t 2 ) 2 ψ (t 2 ) 2

or Re p1 (t 2 ) ≤ Re p k (t 2 ) + αψ (t 2 ), k ∈ {2,..., n}, t 2 ∈ J

the inequality directly contradicts to the condition 1). Lemma 2.3 is proved. Lemma 2.4. Let for system (2.9) the conditions be satisfied 1) Re p1 (t ) > Re p i (t ) + αψ (t ), t ∈ J , i ∈ {2,..., n};

2) lim

t → +∞

p ik (t )

ψ (t )

= 0, i ∈ {1,..., n}, k ∈ {1,..., n} ,

where α > 0 , ψ (t ) is a continuous positive function on J . Then there exists a solution y1 , y 2 ,..., y n of system (2.9) such, that lim

t → +∞

yi y1

= 0, i ∈ {2,..., n}

Proof. In Lemmas 2.2, 2.3 the fixed number T1 are taken by using the condition 2) , that for any t > T1 the inequality holds p ik (t )

ψ (t )


0

(2.21)

This implies, that there exists an arbitrary large t = τ > T1 , for which the inequalities simultaneously hold y k (τ ) y1 (τ )

2

>

 1  d yk  2 ψ (τ )  dt y1 

γ

;

2

 αγ  >−  2  t =τ

(2.22)

In fact, at first, obviously, that beginning with a certain t ∈ J , the inequality is not held 1 d yk ψ (t ) dt y1

2

−  2  t =τ

.

If for these 2

y t = τ > T1 , k y1

>

γ 2

then the statement is proved. We assume, that for inequality is satisfied yk y1

2


T1 the

.

Since lim

t → +∞

yk y1

2



,

that among the numbers t = τ > T1 , there exists at least one τ > T1 for which y k (τ ) 3 ≥ γ y1 (τ ) 4

and for this τ > T1 we get   d yk  dt y 1 

2

 αγ  ψ (τ ) . >−  2  t =τ

Consequently, for this τ > T1 the inequalities are satisfied

65

y k (τ ) γ > 2 y1 (τ )

  d yk  dt y 1 

,

2

 αγ  >− ψ (τ ) 2   t =τ

Thus, for a fixed, but arbitrary large t = τ > T1 (2.29) are simultaneously held. We note, that d yk dt y1

2

=

1 y1

2

d yk dt

2



yk y1

2 4

d y1 dt

inequalities

2

(2.23)

In the proof of Lemma 2.2, inequality (2.13) is obtained 1 d 2 y i − Re( p i (t ) y i 2 dt

2

n

≤ ∑ p ik (t ) y i y k k =1

This implies, that n 1 d 2 y i ≤ Re( p i (t ) y i2 + ∑ p ik (t ) y i y k 2 dt k =1 n 1 d 2 y i ≥ Re( p i (t ) y i2 − ∑ p ik (t ) y i y k 2 dt k =1

(2.24)

(2.25)

From inequality (2.24) at i = k we get n 1 d 2 y k ≤ Re( p k (t ) y k2 + ∑ p ks (t ) y k y s 2 dt s =1

(2.26)

From inequality (2.25) at i = 1 after multiplying by − 1 we obtain −

n 1 d 2 y1 ≤ ∑ p1s (t ) y 1 y s − Re( p1 (t ) y12 2 dt s =1

(2.27)

Consequently, from (2.23) by (2.26) and (2.27) follows, that

66

1 d yk 2 dt y1

2

=

1 y1

2

1 d yk 2 dt

n 1  Re p k (t ) y k2 + ∑ p ks (t ) y k y s 2  y1  s =1



n

=∑

s =1

p ks (t ) y k y s y1

2

2

+

yk y1

 y + k   y1

2 4

2 4

 1 d 2 y1  ≤ −  2 dt 

 n  ∑ p1s (t ) y y s − Re p1 (t ) y12 1   s =1

2   y   n p1s (t ) y 1 y s  + Re p k (t ) k   + ∑ yk 4 y1   s =1   y 1  

2

2   y    − Re p1 (t ) k     y1   

or 2 2 n pks (t ) y k y s n p1s (t ) y1 y s y  1 d  yk    + Re( p1 (t ) − pk (t )) k  ≤ ∑ +∑ yk     2 4 2 dt  y1  y1 y1  y1  s =1 s =1

2

(2.28) y i < y1 , i ∈ {2,..., n} , then from (2.28) follows, that

By Lemma 2.3, 2

2

n n y  1 d  yk    + Re( p1 (t ) − p k (t )) k  ≤ ∑ p ks (t ) + ∑ p1s (t )     2 dt  y1   y1  s =1 s =1

Hence, by condition 1) of Lemma леммы the inequality holds 2

2

n n y  1 d  yk    + αψ (t ) k  ≤ ∑ p ks (t ) + ∑ p1s (t )     2 dt  y1   y1  s =1 s =1

Dividing by ψ (t ) we get 2

2

n p (t ) n p (t ) y  1 d  yk  1s   + α  k  ≤ ∑ ks + ∑  y  2ψ (t ) dt  y1  ψ ψ ( t ) (t )  1 s =1 s =1

67

 =  

(2.29)

Then from (2.29) at t = τ > T1 by (2.22) follows, that −

αγ 4

+

αγ 2

n

≤∑

s =1

p ks (τ )

ψ (τ )

n

+∑

s =1

p1s (τ )

ψ (τ )

Therefore, taking into account (2.20) we get αγ 4

i.e.,

n

≤∑

αγ

s =1 16n

n

+∑

αγ

s =1 16n

1 1 ≤ . The obtained contradictions proves the lemma. Lemma 2.4 4 8

is proved. Lemma 2.5. Let for system (2.9) for a continuous positive function ψ (t ) on J , the conditions be held 1) Re p1 (t ) > Re p i (t ) + αψ (t ), t ∈ J , i ∈ {2,..., n}, α > 0,ψ (t ) ≥ β > 0,

2) lim

t → +∞

p ik (t )

ψ (t )

= 0, i ∈ {1,..., n}, k ∈ {1,..., n} ,

then there exists a solution y1 , y 2 ,..., y n of system (2.9) such, that lim

1

t → +∞ ψ (t )

y1′ p1 (t ) − =0 y1 ψ (t )

Proof. The first equation of system (2.9) has the form y1′ = p1 (t ) y1 + p11 (t ) y1 + p12 (t ) y 2 + ... + p1n (t ) y n

We consider a solution y1 , y 2 ,..., y n of Lemma 2.4. By substituting the solution into the equation, and taking into account, that y1 (t ) ≠ 0,ψ (t ) ≠ 0, t ≥ t 0 ∈ J

the inequality is obtained 68

n p (t ) y p11 (t ) 1 y1′ p1 (t ) 1k k − ≤ +∑ ψ (t ) y1 ψ (t ) ψ (t ) ψ ( ) t y 1 k =2

Consequently, by Lemma 2.4 and from the condition 2) follows the statement of the Lemma. Lemma 2.5 is proved. Further, we consider the system (2.9) with the real coefficients. Lemma 2.6. Let for system (2.9) for a continuous positive function ψ (t ) on J , the conditions be satisfied 1) p1 (t ) > p i (t ) + αψ (t ), t ∈ J , i ∈ {2,..., n}, α > 0,ψ (t ) ≥ β > 0,

2) lim

t → +∞

3) lim

t → +∞

1 q (t )

t



p ik (t )

ψ (t )

= 0, i ∈ {1,..., n}, k ∈ {1,..., n}

t

λ1 (q) ∈ R; where q(t ) = ∫ ψ (τ )dτ ,

p1 (τ )dτ = λ1 (q ),

0

0

y = {y1 , y 2 ,..., y n } such, that then the system (2.9) has a solution χ [ y1 , q] = λ1 (q) , where χ [ y1 , q] is a upper characteristic Lyapunov exponent of the first coordinate function y1 (t ) with respect to q(t ) .

Proof. We consider a solution y = {y1 , y 2 ,..., y n } of Lemma 2.5. Then by Lemma 2.5 we get for any ε > 0 , there exists such T ∈ J , that for any t > T the inequality holds 1

ψ (t )

(ln y1 )′ − ψp1((tt)) < ε

or for any t > T the inequalities are satisfied

(

p1 (t ) − εψ (t ) < ln y1

)′ < p1 (t ) + εψ (t )

Consequently, by integrating we get 69

t

t

0

0

∫ p1 (τ )dτ − ε ∫ψ (τ )dτ ≤ ln

y1 (t ) y1 (0)

t

t

0

0

≤ ∫ p1 (τ )dτ + ε ∫ ψ (τ )dτ

or t t y1 (t ) 1 1 1 τ τ ε ( ) ln p d p1 (τ )dτ + ε − ≤ ≤ 1 q (t ) ∫0 q (t ) y1 (0) q (t ) ∫0

Now by transferring to the upper limits by using the conditions

3) , we obtain λ1 (q) − ε ≤ lim

t → +∞

1 ln y1 (t ) ≤ λ1 (q ) + ε q (t )

.

By definition χ [ y1 , q] = lim

t → +∞

1 ln y1 (t ) q (t )

Hence, by arbitrariness ε > 0 we get λ1 (q) = χ [ y1 , q] . Lemma 2.6 is proved. Theorem 2.5. Let for system (2.9) with continuous real coefficients n dy i = p i (t ) y i + ∑ p ik (t ) y k dt k =1

t ∈ J = [0,+∞) , i ∈ {1,..., n} .

For a continuous positive function ψ (t ) , t ∈ J the conditions are satisfied 1) p1 (t ) > p i (t ) + αψ (t ), t ∈ J , i ∈ {2,..., n}, α > 0,ψ (t ) ≥ β > 0,

2) lim

t → +∞

p ik (t )

ψ (t )

= 0, i ∈ {1,..., n}, k ∈ {1,..., n}

70

3) lim

t → +∞

1 q (t )

t

∫ p1 (τ )dτ = λ1 (q), где

t

q (t ) = ∫ ψ (τ )dτ

0

,

0

then the system (2.9) has a solution y = {y1 , y 2 ,..., y n } such, that χ [ y, q] = λ1 (q) .

Proof. We consider a solution y = {y1 , y 2 ,..., y n } of Lemma 2.6. By Lemma 2.3, for any t > t 0 , i ∈ {2,..., n} the inequality holds y1 (t ) > y i (t )

Therefore, from the property of exponents with respect to q(t ) the equality holds χ [ y1 , q] = χ [ y, q] .

Hence, from Lemma 2.6 the required equality is followed. Theorem 2.5 is proved. Corollary 2.1. In the conditions of Theorem 2.5 the system (2.16) has λ1 (q) - upper exponent with respect to q(t ) , which is calculated by formula λ1 (q) = lim

t → +∞

1 q (t )

t

∫ p1 (τ )dτ 0

Theorem 2.6. Let for a system n dy i = p i (t ) y i + ∑ p ik (t ) y k dt k =1

(2.30)

where the coefficients are continuous real functions, defined on the semi-axis J = [0,+∞) , i ∈ {1,..., n} , for a continuous positive function ψ (t ) the conditions be satisfied

71

1) p i (t ) > p i +1 (t ) + αψ (t ), t ∈ J , i ∈ {1,2,..., n − 1}, α > 0,ψ (t ) ≥ β > 0,

p ik (t )

2) lim

t → +∞

ψ (t )

= 0, i ∈ {1,..., n}, k ∈ {1,..., n}

then the system (2.30) has n linearly independent solutions y k = {y1k , y 2k ,..., y nk }, k = 1,2,..., n;

satisfying to the equalities a ) lim

yµk

t → +∞

ykk

= 0, µ ≠ k

′  1 y kk p (t )  b) lim  − k  = 0 t → +∞ ψ (t ) y kk ψ (t ) 

Proof. We use a induction k = 1,..., n. Let be k = 1. . Then by Lemma 2.4 and Lemma 2.5 the system (2.30) has a solution y1 = y11 ; y 2 = y 21 ;...; y n = y n1 ;

(2.31)

and the statements a ) and b) of Theorem are held. Let the statements a ) and b) of Theorem be satisfied for any k < n. We assume y1 = y11 ∫ udt , y 2 = y 21 ∫ udt + z1 ,..., y n = y n1 ∫ udt + z n −1

(2.32)

where y 1 = {y11 , y 21 ,..., y n1 } is a solution (2.31). Then the system (2.30) transfers to the system n  dzλ −1 y  = pλ zλ −1 + ∑  pλµ (t ) − p1µ (t ) λ1  zµ −1 , dt y11  µ =2 

72

(λ = 2,3,..., n)

( 2.33)

We note, that (λ = 2,3,..., n − 1) .

p λ (t ) > p λ +1 (t ) + αψ (t ),

Further,

lim

y pλµ (t ) − p1µ (t ) λ1 y11

ψ (t )

t → +∞

pλµ (t )

= lim

ψ (t )

t → +∞



p1µ (t ) yλ1 =0 ψ (t ) y11

Thus, for system (2.33), the conditions of Theorem 2.6 are satisfied. System (2.33) consists of k = n − 1 equations. Therefore, by proposition the system (2.33) has a fundamental system of solutions

{

}

z s = z1s , z 2 s ,..., z n −1, s ; s = 1,..., n − 1;

satisfying to the statement of Theorem, i.e. lim

zµs

t → +∞ z ss

= 0; µ ≠ s.

 1 lim 

t → +∞ ψ (t )

z′ss ps +1 (t )  =0. − zss ψ (t ) 

We assume t

ψ s = ∫ u s (τ )dτ ; s = 1,..., n − 1, t0

where u s (τ ) =

p1k (τ ) z k −1, s (τ ); s = 1,..., n − 1. k = 2 y11 (τ ) n



Then the functions y1k = y11ψ k −1, y2 k = y21ψ k −1 + z1, k −1,..., ynk = yn1ψ k −1 + zn −1, k −1; k = 2,..., n.

73

together with the original solution y11 , y 21 ,..., y n1 form a fundamental system of solutions of the initial system (2.30). Theorem 2.6 is proved. Theorem 2.7. Let for a system n dy i = ∑ p ik (t ) y k , k = 1,..., n; dt k =1

(2.34)

where the coefficients are continuous real functions, defined on the semi-axis J = [0,+∞) . ψ (t ) is a continuous positive function on J , the conditions are satisfied 1) p k −1,k −1 (t ) − p kk (t ) ≥ αψ (t ), t ∈ J , k ∈ {2,..., n}, α > 0,ψ (t ) ≥ β > 0,

2) lim

p ik (t ) = 0, i ∈ {1,..., n}, k ∈ {1,..., n}, i ≠ k , ψ (t )

3) lim

1 q (t )

t → +∞

t → +∞

t

∫ p kk (τ )dτ = λ k (q); k = 1, n. , где

t

q (t ) = ∫ ψ (τ )dτ

0

,

0

then the system (2.34) has a fundamental system of solutions y 1 , y 2 ,..., y n such, that χ [ y k , q] = λ k (q), k = 1, n.

Proof. By condition of Theorem and from Theorem 2.6 follows, that there exists a fundamental system of solution of system (2.34), y k = {y1k , y 2k ,..., y nk }, k = 1, n.

such, that

74

yik t → +∞ ykk

a ) lim

= 0, i ≠ k

(2.35)

 1 ykk ′ p (t )  b) lim  − kk  = 0 t → +∞ ψ (t ) ykk ψ (t ) 

(2.36)

This implies, that for any ε > 0 there exists T ∈ J , that for any t > T , k = 1,..., n; the inequalities hold p kk (t ) − εψ (t )
0 we obtain the required equality. Theorem 2.7 is proved. Remark. Theorem 2.7 is a generalization of the Perron theorem for system with unbounded coefficients. 75

Corollary 2.2. In the conditions of Theorem 2.7 the fundamental system of solutions y 1 , y 2 ,..., y n forms a normal basis of system (2.34) i.e. λ k (q), k = 1, n. ,

are generalized exponents of system (2.34). Proof. In fact, from the condition 1) of Theorem follows, that t



t

t

0

0

p k −1,k −1 (τ )dτ ≥ ∫ p kk (τ )dτ + α ∫ ψ (τ )dτ , k = 2, n.

0

1 q (t )

t

∫ 0

p k −1,k −1 (τ )dτ ≥

1 q (t )

t

∫ p kk (τ )dτ + α , k = 2, n. 0

λ k −1 (q) ≥ λ k (q) + α , k = 2, n.

By positiveness α follows, that λ1 (q),..., λ n (q) are different, that is why the fundamental system of solutions y 1 , y 2 ,..., y n forms a normal basis, and λ n (q) < ... < λ1 (q) are generalized exponents of system (2.34). Corollary 2.3. In the conditions of Theorem 2.7 the generalized exponents of system (2.34) are calculated by formula λ k (q) = lim

t → +∞

1 q (t )

t

∫ p kk (τ )dτ , k = 1, n. 0

Corollary 2.4. Let for a system n dy i = ∑ p ik (t ) y k , k = 1,..., n; dt k =1

where coefficients are continuous, real functions which are defined on the semi-axis J = [0,+∞) . ψ (t ) is a continuous positive function on J , the conditions are held 76

1) p k −1,k −1 (t ) − p kk (t ) ≥ αψ (t ), t ∈ J , k ∈ {2,..., n}, α > 0,ψ (t ) ≥ β > 0,

2)

p ik (t ) , i ∈ {1,..., n}, k ∈ {1,..., n}, i ≠ k , ψ (t )

it is sufficient the small bounded functions 3) lim

t → +∞

1 q (t )

t

∫ p kk (τ )dτ = λ k (q); k = 1, n. , где

t

q (t ) = ∫ ψ (τ )dτ

0

,

0

then the system (2.34) has a fundamental system of solutions y 1 , y 2 ,..., y n such, that χ [ y k , q] = λ k (q), k = 1, n.

Example.

  4t 3  t t4 t2 2007t 2007 πt 3  x =  − + + + 1 x + y arctg 2  3  t +1 3 6t 2 + 2t + 1 4  et    2  y = −2007t ln tx − 2007t y

Here we get p1 (t ) =

t3 3

b(t ) =

 t  4arctg t − π +  1 + t (t + 1)2 + t 2 

2007t 2007 e

t

t2

q (t ) = ∫ ψ (τ )dτ = 0

c(t ) = −2007t ln t ,

, t3 3

 t2 + +1 ,  4 

t

,

∫ p1 (τ )dτ 0

77

=

p 2 (t ) = −2007t 2 .

α = 2008 , ψ (t ) = t 2 ,

t4  t π  t3 − + +t  arctg t + 1 4  12 3 

 3 t4  t π  3 t3 3  −  + ⋅ + ⋅t = lim  ⋅  arctg t → +∞  t 3 3  t + 1 4  t 3 12 t 3  

 t π 1 3 1 1 1 = lim t  arctg −  + +  = − + = − = λ1 3 t → +∞   t +1 4  4 t  2 4 4

∫ (− 2007t t →∞ t 3 lim

3

t

2

)dt = lim − 3 ⋅t2007 ⋅ t3 = −2007 = λ 3

3

0

2

It is easy to verify, that for system  x = p1 (t )x + b(t )y   y = c(t )x + p 2 (t )y

the conditions are satisfied p1 (t ) ≥ αψ (t ) + p 2 (t ) ,

b(t )

ψ (t )

→ 0,

c(t )

ψ (t )

→0

moreover, ____

lim

t → +∞

____

t

1 1 p1 (τ )dτ = − q (t ) ∫0 4 t

1 p 2 (τ )dτ = −2007 t → +∞ q (t ) ∫ 0 lim

Consequently, the system have the generalized exponents λ1 (q ) = −

1 4

and λ 2 (q ) = −2007 .

According to the work [19, p. 1408] we consider the generalized exponents, in general, the case for a family of endomorphisms of a metric vector bundle. 78

Let (E , p, B ) be a metric abstract vector bundle with standard layer R and a Riemann metrics < ⋅,⋅ > . Let M ⊂ R be a set which have +∞ as its limiting point. The main particular cases: + M = R, M = R , M = Z , M = N . Let Q be a set of positive monotonically increasing functions, defined in M . Let H be a mapping of the set M to the set of endomorphism of an abstract vector bundle (E , p, B ) . We remind, that by the imagine Ht of the point t ∈ M at the mapping H is a pair ( X t , χ t ) , where X t is a mapping E → E , and χt is a mapping B → B , moreover pX t = χ t p and for every b ∈ B the constriction n

Χ t [b]

of the mapping Χ t on the layer p −1 (b ) is a linear mapping of

the layer p −1 (b ) to the layer p −1 (χ t b ) . We require, that there exists a function a(.) : b → R + , such, that for every t ∈ M = N and the given q (t ) ∈ Q the inequality is held max X t [b] , 

[X t [b]]−1

 ≤ exp(q(t )a (b )). 

(2.38)

This implies, that for every b ∈ B and every k ∈ {1,..., n} λk [b, q ] =

min

(

)

max lim

R n − k +1∈Gn − k +1 p −1 (b ) ξ ∈R n − k +1 t →∞

1 ln X t [b]ξ q (t )

is a number belonging to [−a(b ), a(b )] . Here X t [b]ξ

is the norm of the vector X t [b]ξ ∈ p −1 (χ t , b ),

(

)

Gn − k +1 p −1(b )

of the Grassmann manifold of n − k + 1 – dimensional vector subspaces of the layer p −1 (b ) . The number is called by k -th generalized 79

exponent of the family of endomorphisms of the metric vector bundle with respect to q(t ) . Obviously, that for every b ∈ B the chain of the inequalities holds λ1 [b, q ] ≥  ≥ λ n [b, q ]

By the equality ____

lim at = lim lim

max

t → ∞ s → ∞ m∈{0,1,..., s}

t →∞

at + m

we get λk [b, q ] =

min

(

max

)

R n − k +1∈Gn − k +1 p −1 (b ) ξ ∈ R

ξ =1

lim lim

1

max

{0,1,..., s} q(t + m )

n − k +1 t → ∞ s → ∞ m∈

ln X t + m [b]ξ

(2.39) We suppose µk [b, q ] = lim lim

min

(

)

t →∞ s →∞ R n − k +1∈Gn − k +1 p −1 (b )

max{0,1,..., s}

max

ξ∈ R

n − k +1

ξ =1

1 ln X t + m [b]ξ . q (t + m )

(2.40) We note, that at every b ∈ B, k ∈ {1,..., n} and q(t ) ∈ Q the equality holds λ k [b, q ] = µ k [b, q ]

(2.41)

In fact, we fix b ∈ B, k ∈ {1,..., n} and q(t ) ∈ Q . Let min

(

)

R n − k +1∈Gn − k +1 p −1 (b )

in formula (2.39) a min be reached in the point

(

)

R0n − k +1 ∈ Gn − k +1 p −1(b ) .

80

Then from (2.39) follows, that for every ξ ∈ R0n − k +1 and for every ε > 0 there exists Dε (ξ ) > 0 such, that for every t ∈ N the inequality is satisfied X t [b ]ξ ≤ Dε (ξ )exp[(λk [b, q ] + ε )q (t )]

Since X t [b] : p −1 (b ) → p −1 (χ t b )

is a linear mapping, that X t [b ]R0 n −k +1

is a vector subspace of the layer p −1 (χ t b )

Considering the isomorphism of layers as Euclidean spaces, using the Banach-Steinhaus theorem for a family of linear operators

{X t [b]}, t ∈ N we obtain, that for every ε > 0 there exists Dε > 0 such, that for every t ∈ N is held X t [b ]R0n − k +1 ≤ Dε exp[(λ k [b, q ] + ε )q (t )]

where

X t [b]R0n −k +1

is a constriction of the linear operator X t [b] to the linear subspace R0 n −k +1 ∈ p −1 (b ) .

81

Consequently, for every ε > 0 at every t ≥ ε −1 Dε and every s ∈ N holds 1

max

m∈{0,1,..., s} q

(t + m )

ln X t [b]R0n − k +1 ≤ λ k [b, q ] + 2ε .

This implies, that for every ε > 0 at t ≥ ε −1 ln Dε

and s ∈ N the inequality is satisfied

R

n − k +1

min

(

∈Gn − k +1 p

−1

max

max

(b ))m∈{0,1,..., s}ξ ∈R

n − k +1

ξ =1

1 ln X t + m [b]ξ ≤ λk [b, q ] + 2ε . q (t + m )

Transferring to the limits at first on s , then on t we get µ k [b, q ] ≤ λ k [b, q ]

(2.42)

On the other hand, for every ε > 0 , from (2.40) follows, that there exists t ε ∈ N and for every s ∈ N there exists

(

)

Rtnε−, sk +1 ∈ Gn − k +1 p −1 (b )

at every m ∈ {0,1,..., s} the inequality is satisfied 1 ln X tε + m [b]Rtnε−, sk +1 < µ k [b, q ] + ε . q (t ε + m )

Using the compactness [167] of the Grassmann maniforld

(

)

G n − k +1 p −1 (b )

we choose a subsequence 82

{R

}

n − k +1 ∞ tε , si i =1.

Let Rεn −k +1

be a limit of the subsequence. Consequently, at every m ∈ {0,1,...} is held 1 ln X t ε q (tε + m )

+m

[b]Rε

n − k +1

≤ µ k [b, q ] + ε .

Thus, for every ε > 0, ξ ∈ Rεn − k +1 , ξ = 1

the inequality holds ____

lim

t →∞

1 ln X tε + m [b]ξ ≤ µ k [b, q ] + ε . q (t ε + m )

Hence, by definition of generalized exponent follows, that for every b∈B

, k ∈ {1,..., n} и

q (t ) ∈ Q

for every ε > 0 the inequality is satisfied λk [b, q ] ≤ µ k [b, q ] + ε

Consequently, λ k [b, q ] ≤ µ k [b, q ]

83

(2.43)

for every b ∈ B , k ∈ {1,..., n} and q(t ) ∈ Q . From (2.42) and (2.43) follows the equality (2.41). Theorem 2.8. In the space В there exists an everywhere dense set C of type Gδ , such for each k ∈ {1,..., n} the constriction of the function λ k [b, q ] : B → R

on the set C ⊂ B is continuous. Proof. From equality (2.41) follows, that the generalized Lyapunov exponents of a family of endomorphisms of a metrized vector bundle with respect to q (t ) are Baire functions of the second class. Therefore, the proof follows directly from Hausdorff's theorem [171]. Theorem 2.8 is proved. Theorem 2.9. In the space B there exists an everywhere dense set C of type Gδ , such, that at each k ∈ {1,..., n} the function λk [b, q ] : B → R

is upper semicontinuous at each point b ∈ C. Proof. At every k ∈ {1,  , n}, t ∈ N the function µ k(t )[b, q ] = lim

s →∞ R

n − k +1

min

(

∈Gn − k +1 p

−1

max

max

(b ))m∈{0,1,..., s}ξ ∈R

n − k +1

ξ =1

1 ln X t + m [b]ξ q (t + m )

is a Baire function of the first class on the base of a vector bundle E , p, B . Let C tk ⊂ B be a set of the continuity points of the function µ k(t ) [b, q ] : B → R.

At each k ∈ {1,, n} and each t ∈ N the set C tk is everywhere dense in B B of type Gδ . Then 84

n

C =   Ctk t∈N k =1

is a set of type Gδ , everywhere dense in B at every

k ∈ {1,, n}, t ∈ N , the function is continuous in C . Since the function µ k [b, q ] : B → R

at every k ∈ {1,, n} is the limit of a monotonically nonincreasing sequence µ k(t ) [b, q ] : B → R

,

that in each point b ∈ B , in which an every function µ

(t ) k

[b, q ] : B → R, k ∈ {1,  , n},

t∈N

is continuous, an every function µ k [b, q ] : B → R, k ∈ {1,  , n}

is upper semicontinuous. Theorem 2.9 is proved.

85

,

3. GENERALIZED REGULAR LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS WITH UNBOUNDED COEFFICIENTS AND STABILITY OF SOLUTIONS OF THE SYSTEM OF DIFFERENTIAL EQUATIONS

Let x = A(t )x

(3.1)

be a system with continuous real coefficients in I. Definition 3.1. System (3.1) is called generalized regular by Lyapunov with respect to q(t ) ∈ Q , if there exists its fundamental system of solutions x (1) ,..., x (n ) , for which the numeric equality holds

∑ χ [x (i ) , q ] = lim q(t ) ∫ SpA(τ )dτ . n

i =1

t

1

t →∞

(3.2)

t0

Theorem 3.1. If system (3.1) has the finite generalized exponents with respect to q(t ) ∈ Q , then for general regularity of system (3.1) with respect to q(t ) necessary the existence of the exact limit S (q ) ≡ lim

t →∞

t

1 SpA(τ )dτ q (t ) t∫

.

(3.3)

0

Proof. If system (3.1) is generalized regular with respect to q(t ) ,

then there exists its fundamental system of solutions, x (1) ,..., x (n ) , for which the equality (3.2) holds. Therefore from (3.2) and from generalized inequality we get

86

lim t →∞

t

t

0

0

___ 1 1 SpA(τ )dτ ≥ lim ∫ ∫ SpA(τ )dτ . t →∞ q (t ) q (t ) t t

Consequently, there exists the exact limit (3.3). Theorem 3.1 is proved. Let y = − AT (t )y. (3.4) be adjoint system for system (3.1). Lemma 3.1. If system (3.1) is generalized regular with respect to q (t ) ∈ Q , then the adjoint system (3.4) has a fundamental system of solutions y (1) ,..., y (n ) , for which the equality holds

[

] [

]

χ y (k ) , q + χ x (k ) , q = 0, k = 1, n,

(3.5)

where x (1) ,..., x (n ) , is a fundamental system of solution of system (3.1). Proof. If system (3.1) is generalized regular with respect to q(t ) , then there exists a fundamental system of solutions x (1) ,..., x (n ) of system (3.1), for which by theorem 3.1 the equality holds

∑ χ [x (i ) , q ] = tlim ∫ SpA(τ )dτ . →∞ q (t ) n

t

1

i =1

t0

Let

[

]

____

X (t ) = x jk (t ) , j , k = 1, n,

be a fundamental matrix consisting of solutions ___

x (k ) = colon[x1k ,..., x nk ], k = 1, n,

of system (3.1). We assume

[

]

Y (t ) = X −1 (t ) .

Then 87

T

,

(3.6)

]

[

___

Y (t ) = y jk (t ) , j , k = 1, n,

is a fundamental matrix of the adjoint system (3.4), where y jk (t ) =

(t ) , X jk (t ) det X (t ) X

jk

is an algebraic addition of the element x jk (t ) of Wronskian W (T ) = det Χ(t ) . Further,       X  X jk  jk , q = χ y jk , q = χ  , q = χ     t   W      det X (t 0 ) exp ∫ SpA(τ )dτ      t0 

[

]

      X jk , q ≤ χ X = χ    t     ( ) SpA d exp τ τ     ∫    t0 

[



jk , q





t

 





t0

 

]+ χ [X jk , q]+ χ exp − ∫ SpA(τ )dτ , q ≤



t



n





t0



j =1 j≠k



t



t0



    − ∫ SpA(τ )dτ  = ∑ χ [x ( j ) , q ]+ lim  − ∑ χ [x ( j ) , q ]+ tlim ∫ SpA(τ )dτ  = →∞ q (t )  t →∞ q (t )  n

___

1

j =1 j≠k

=

∑ χ [x ( j ) , q ]− ∑ χ [x (i ) , q ] = − χ [x (k ) , q ] n

n

j =1 j ≠k

i −1

88

1

Finally, for any

[

]

[ ]

[

[

]

j ∈ 1,..., n, χ y jk , q ≤ − χ x (k ) , q .

Consequently,

]

χ y k , q ≤ − χ x (k ) , q . .

Thus,

[

] [

]

χ y (k ) , q + χ x (k ) , q ≤ 0

(3.6)

On the other hand, as it is known, a scalar product of any two ___

solutions x (i ) , y ( j ) , i, j = 1, n is a number different from zero, in particular,

(y ( ) , x ( ) ) = c, c ≠ 0. k

k

Therefore, from inequality y (k ) x (k ) ≥ c ,

by transferring to the generalized exponents we get

[

] [

]

χ y (k ) , q + χ x (k ) , q ≥ 0.

(3.7)

From (3.6) and (3.7) we obtain (3.5). Lemma 3.1 is proved. Theorem 3.2. The adjoint system for generalized regular with respect to q(t ) ∈ Q linear system as well as is generalized regular linear system with respect to q (t ) . Proof. If system (3.1) is generalized regular with respect to q(t ) , then there exists a fundamental system of solutions, x (1) ,..., x (n ) , for which by Lemma 3.1 the equality (3.5) holds. Consequently, for the 89

fundamental system of solutions y (1) ,..., y (n ) , the adjoint system (3.4) by Lemma 3.1 we get

∑ χ [y (i ) , q ] = −∑ χ [x (i ) , q ] = − tlim ∫ SpA(τ )dτ = →∞ q (t ) n

n

i =1

i =1

= lim

t →∞

t

1

t0

t

t

0

0

[

]

1 [− SpA(τ )]dτ = lim 1 ∫ Sp − AT (τ ) dτ . ∫ t →∞ q (t ) q (t ) t t

Thus, the adjoint system is generalized regular with respect to q(t ) . Theorem 3.2 is proved. Lemma 3.2. If the mutually conjugate systems (3.1) and (3.4) have fundamental systems x (1) ,..., x (n ) , and y (1) ,..., y (n ) , for which at any k ∈ {1,..., n} the equality (3.5) holds, then there exists the exact limit (3.3). Proof. By generalized regular inequality of Lyapunov the inequalities are satisfied ___

lim

t →∞

[

t

]

n 1 SpA(τ )dτ ≤ ∑ χ x (i ) , q , ∫ q (t ) t i =1 0

t

___

lim

t →∞

[

[

]

]

n 1 Sp − AT (τ ) dτ ≤ ∑ χ y (i ) , q . ∫ q (t ) t i =1 0

Adding these inequalities term by term, by (3.5) we have ___

lim

t →∞

t

t

0

0

[

]

___ 1 1 T SpA(τ )dτ + lim ∫ ∫ Sp − A (τ ) dτ ≤ 0. t →∞ q (t ) q (t ) t t

This implies, that ___

t

t

1 1 lim SpA(τ )dτ ∫ SpA(τ )dτ ≤ ___ t →∞ q (t ) (t ) t∫ q t lim

t →∞

0

90

0

.

Therefore, there exists the exact limit (3.3). Lemma 3.2 is proved. Theorem 3.4. If the mutually conjugate systems (3.1) and (3.4) have fundamental systems of solutions x (1) ,..., x (n ) , and y (1) ,..., y (n ) , for which the equality (3.5) holds, then the systems (3.1) and (3.4) are generalized regular with respect to q(t ) ∈ Q . Proof. By Lemma 3.2 there exists the exact limit (3.3). If it is assumed, that the systems (3.1) and (3.4) are not generalized regular with respect to q(t ) ∈ Q , then there are strict inequalities

∑ χ [x (i ) , q ] > tlim ∫ SpA(τ )dτ , →∞ q (t ) n

t

1

i =1

t0

T ∑ χ [y (i ) , q ] > tlim ∫ Sp[− A (τ )]dτ . →∞ q (t ) n

1

i =1

t

t0

Adding these inequalities term by term and taking onto account (3.5), we get the contradiction 0 < 0. Theorem 3.4 is proved. The generalized exponents of the adjoint system (3.4) are denoted ___

µ i (q ), i = 1, n,

but we arrange them in the reverse order, by virtue of their convenience, i.e. −∞ < µ1 (q ) ≤ ... ≤ µ n (q ) < +∞.

Definition 3.2. By generalized Lyapunov coefficient with respect to q (t ) is called the number n

Λ (q ) = ∑ λi (q ) − lim i =1

___ t →∞

From theorem 3.1 follows, that always Λ (q ) ≥ 0.

91

t

1 SpA(τ )dτ . q (t ) t∫ 0

If Λ (q ) = 0, then the system (3.1) is a generalized regular with

respect to q(t ) . Definition 3.3. By generalized Perron coefficient is called П (q ) = max{λi (q ) + µ i (q )} i

where

µ i (q ), i ∈ {1,..., n} ,

are generalized exponents of the adjoint system Numbering of exponents µ i (q ), i ∈ {1,..., n} is the opposite to the numbering of the exponents λi (q ), i ∈ {1,..., n} . Theorem 3.5. In order to the system (3.1) be generalized regular with respect to q(t ) necessary and sufficiently, that П (q ) = 0.

Proof. In fact, from Lemma 3.1 and Theorem 3.4 follows, that is proved. Notes 1 The name "generalized regular systems" is used in the work [31, p.575]. In this paper the systems with piecewise-continuous and bounded coefficients on the semiaxis t ≥ 0 are considered [31, p. 576]. П (q ) = 0. Theorem 3.5

2. If q (t ) = t , then the generalized Lyapunov and Perron coefficients turn into ordinary Lyapunov and Perron coefficients. Theorem 3.5 in Perron's theorem. Theorem 3.6. For general regularity of the system of differential equations (3.1) with respect to q(t ) ∈ Q necessary and sufficiently, that the Gramians composed of any its solutions, have exact generalized exponents with respect to q(t ) ∈ Q . In particular, each solution of the generalized regular system with respect to q(t ) ∈ Q has the exact generalized exponent q(t ) ∈ Q . 92

Proof. Theorem is proved enough for a triangular system of differential equations. In fact, we use Perron's theorem, there the transformation is unitary, and therefore does not change either the Gramians or the lengths of the vectors, i.e. saves all the studied exponents. The basis of the system (3.1) becomes a triangular basis. Necessity. The Gramian of linear dependent solutions are equal to zero. Therefore, we consider only linearly independent solutions. Let x11 ,..., x1m be some arbitrary linear independent solutions m ≤ n . We assume, that the chosen solutions coincide with first m columns of the triangular basis. Then they form the basis of the truncated system. For the truncated system the generalized regularity with respect to q(t ) ∈ Q is saved. Therefore, there exists the exact limit S m (q ) ≡ lim

i →∞

t

1 Sp Am (r )dr q (t ) t∫ 0

where Am (r ) is a coefficient of the truncated system, m = n coincides to the coefficient of the original triangular system. Then from the Liouville formula

it

1

∫ SpAm (r )dr

det X 1m (t ) = det X 1m (0)e 0

[

]

where Χ1m = x11 ,..., x1m is a fundamental matrix of the truncated system, it follows, that there exists the exact limit lim

t →∞

1 ln det X 1m (t ) q (t )

This implies, that Gramian of solutions x11 ,..., x1m . G1...m = det X 1m

2

has the exact generalized exponent with respect to q(t ) ∈ Q . A norm of solution as a particular case of the Gramian volume 93

x = G ( x) = Г ( x)

has also the exact generalized exponent with respect to q(t ) ∈ Q . Sufficiency. Let the Gramians composed from any solutions have the exact generalized exponents with respect to q(t ) ∈ Q . By Perron’s theorem the diagonal coefficients p k (t ) of the triangular system are expressed by formulas p k (t ) =

G 1 d ln k (k = 1,..., n), 2 dt G k −1

where G0 = 1, and Gk is a Gram determinant composed from the first k solutions. Therefore, the diagonal coefficients have the exact generalized exponents with respect to q(t ) ∈ Q . Consequently, the system is generalized-regular with respect to q(t ) ∈ Q by generalized Lyapunov theorem. Theorem 3.6 is proved. Note. If q(t ) = t and the linear system with bounded coefficients are considered, then from system 3.6 we obtain a particular case, i.e. theorem of Vinograd [6, p.283]. Theorem 3.7. (coefficient feature). Let for a system n dy i = ∑ p ik (t ) y k , k = 1,..., n; dt k =1

(3.8)

where coefficients are continuous real functions, defined on the semiaxis J = [0,+∞) . ψ (t ) is a continuous positive function on J , the conditions are held 1) p k −1,k −1 (t ) − p kk (t ) ≥ αψ (t ), t ∈ J , k ∈ {2,..., n}, α > 0,ψ (t ) ≥ β > 0,

2) lim

t → +∞

p ik (t )

ψ (t )

= 0, i ∈ {1,..., n}, k ∈ {1,..., n}, i ≠ k ,

94

3) lim

t → +∞

1 q (t )

t

∫ p kk (τ )dτ = λ k (q); k = 1, n. где

t

q (t ) = ∫ ψ (τ )dτ

0

,

0

then the system (3.8) is generalized regular by Lyapunov with respect to q (t ) . Proof. We consider a fundamental system of solutions

y 1 , y 2 ,..., y n in Theorem 2.7. Then n

n

k =1

k =1

∑ χ [ y k , q] = ∑ λ k (q) =

=

n

1

k =1

t

t

∑ t →lim+∞ q(t ) ∫ p kk (τ )dτ = 0

lim t → +∞

t

n 1 1 p kk (τ )dτ = lim ∑ ∫ ∫ SpP(τ )dτ q (t ) 0 k =1 q t → +∞ (t ) 0

i.e. the system (3.8) is generalized regular by definition. Theorem 3.7 is proved. Example 1.   4t 3  2007t 2007 πt 3 t t4 t2  x =  arctg − + + + 1 x + y 2  3  3 t +1 6t 2 + 2t + 1 4  et    2  y = −2007t ln tx − 2007t y

Here p1 (t ) =

t3 3

 t  4arctg t − π +  t +1 (t + 1)2 + t 2 

b(t ) =

2007t 2007 et

2

 t2 + +1 ,  4 

p 2 (t ) = −2007t 2 ,

c(t ) = −2007t ln t

,

By example from the section 2 the system has the exact generalized exponents 95

 3 t4  t π  3 t3 3  ⋅  arctg −  + ⋅ + ⋅t = 3 t → +∞  t t + 1 4  t 3 12 t 3  3  

λ1 = lim 

 1 1 1 t π 1 3 − + +  = − + = − = lim t  arctg t → +∞   4 2 4 t +1 4  4 t 3 

λ 2 = lim

3

t →∞ t 3

− 3 ⋅ 2007 t ⋅ = −2007 ∫ (− 2007t )dt = lim t 3 t

3

2

3

0

The conditions of Theorem 3.7 are satisfied, consequently, the system is generalized regular. Example 2. We consider a system  et x2  x1 = − ln(t + 1) x1 +  (t + 1) t + 2  1   x 2 = t + 1 x 2 

Let  − ln(1 + t )  A(t ) =   0

et (t + 1) 1 t +1

t +2

      

 a (t )   0 

=

b(t )    g (t )

where a (t ) = − ln(1 + t )

, b(t ) =

et

(t + 1)

t +2

,

g (t ) =

The equalities are held a (t ) = − ln (1 + t ) = ln (1 + t ) ,

96

t≥0.

1 t +1

lim

t → +∞

et

(t + 1)t + 2

=0

,

lim

t → +∞ t

1 =0 +1

.

There exists K 1 > 0 , b(t ) ≤ K 1 a (t )

, g (t ) ≤ K1 a(t ) .

A(t ) = a 2 (t ) + b 2 (t ) + g 2 (t ) ≤ a 2 (t ) + K 1 2 a 2 (t ) + K 12 a 2 (t )

≤ a(t )

≤ 1 + 2 K 12

We assume K = 1 + 2 K 12 , ψ (t ) = ln (1 + t ) ,

then A(t ) ≤ Kψ (t ) t

t

0

0

q (t ) = ∫ ψ (τ )dτ = ∫ ln (1 + τ )dτ

q (t ) = (t + 1) ln (1 + t ) − t

It can be directly verified, that x

(1)

= colon[

et

(1 + t )t +1

,0]

is a solution of the system. By the similar way, it may be verified, that x

(2 )

= colon[

is a solution of the system. 97

te t

(t + 1)t +1

, t + 1]

 et   (1 + t )t +1  X (t ) =  0   

tet

       

(t + 1)t +1 t +1

is a fundamental matrix of the system, moreover it is a normal basis in the space of solutions of the system. (1) χ [ x , q ] = lim

t → +∞

(1) t − (t + 1) ln (1 + t ) 1 = −1 ln x (t ) = lim t → +∞ (t + 1) ln (1 + t ) − t q (t )

 t 2 e 2t 2 ln (1 + t ) + ln1 +  (t + 1)2(t +1)+ 2 (2 ) (2 ) 1  χ [ x , q ] = lim ln x (t ) = lim t → +∞ q (t ) t → +∞ 2[(t + 1) ln (1 + t ) − t ]

   

=0

Consequently, the generalized exponents of the system are λ1 (q ) = 0

and

λ 2 (q ) = −1

In the system SpA(t ) = − ln (1 + t ) +

1 t +1

consequently t

lim t → +∞

t

1 1 1   SpA(τ )dτ = lim  − ln (1 + t ) + dτ ∫ ( ) ( ) 1 ln 1 + + − + q (t ) ∫0 t t t t 1  t → +∞ 0

t − t ln (1 + t ) = −1 ( ln 1 + t ) + ln (1 + t ) − t t t → +∞

= lim

98

=

Thus, the equality holds λ1 (q ) + λ 2 (q ) = lim

t → +∞

t

1 SpA(τ )dτ q (t ) ∫0

i.e. the system is generalized regular. We consider a linear system (3.1) with continuous (piecewise continuous) matrix A(t ) ≤ Kψ (t )

Theorem 3.8. If the highest generalized exponent of linear homogeneous system (3.1) is negative with respect to t

q (t ) = ∫ ψ (τ )dτ

, q(t ) ↑ +∞ при t ↑ +∞ ,

t0

then the linear homogeneous system (3.1) is asymptotically stable. Proof. Let x1 (t ), x 2 (t ),..., x n (t ) be a fundamental system of solutions of linear homogeneous system (3.1). We take ε > 0 such, that λ1 (q) + ε < 0

where λ1 (q) is the highest generalized exponent of system (3.1). There exists a constant Dε > 0 , such, that the inequality holds xi (t ) ≤ Dε e(λ1 ( q ) +ε ) q (t ) , i ∈ {1,..., n} .

Let x(t ) be an arbitrary nonzero solution of the system (3.1). This solution admits a decomposition n

x(t ) = ∑ ci xi (t ) i =1

where

c1 ,..., c n

are some constants. 99

Consequently, x(t ) =

n

n

i =1

i =1

∑ ci xi (t ) ≤ ∑ ci Dε e (λ (q)+ε )q(t ) 1

An arbitrary nonzero solution x(t ) of system (3.1) tends to zero, at t → +∞ . This implies, that linear homogeneous system (3.1) is asymptotically stable. Theorem 3.8 is proved. We consider a system of differential equations x = F (t , x ),

(3.9)

where a vector function F (t , x ) is continuous in the domain G = I × R n and satisfies to the condition F (t , x ) ≤ Kψ (t ) x

where ψ (t ) is a continuous positive function in I, K is a positive constant. t

q (t ) = ∫ ψ (τ )dτ

, q(t ) ↑ +∞

at t ↑ +∞ .

t0

We denote by

Λ (q ) = sup λ [x, q ]

{x}

a largest generalized exponent of system (3.9). It assesses the estimation x(t ) ≤ Dε (x0 ) x(t0 ) e( Λ[q ]+ε )( q (t ) − q (t0 ))

(3.10)

for all solutions of system (3.9). Here the constant Dε (x 0 ) is depended

on the data ε > 0 and the initial point x(t 0 ) = x 0 of the solution x(t ) . If the constant Dε (x 0 ) depends only on ε > 0 , then the estimation (3.10) is a uniform with respect to the solution of system (3.9). 100

Theorem 3.9. If the estimation (3.10) is uniform in a neighborhood of zero and the largest generalized exponent Λ (q ) < 0 , then the zero solution of system (3.9) is asymptotically stable. Proof. Any nonzero solution of the system (3.9) admits an estimation x(t ) ≤ Dε x(t0 ) e( Λ[q ]+ε )( q (t ) − q (t0 ))

where Dε depends on ε > 0 . We take ε > 0 such, that Λ (q ) + ε < 0

where Λ(q ) is the highest generalized exponent. Let the inequality be held x(t ) ≤ Dε x(t0 ) e( Λ[q ]+ε )( q (t ) − q (t0 ))

for all solutions with the initial data x(t 0 ) < R

Then all these solutions are satisfied to the estimation x(t ) ≤ Dε Re( Λ[q ]+ε )( q (t ) − q (t0 ))

and uniformly tends to zero. In addition, with a moment tε >t 0 will be x(t ) < ε

at

t ≥ tε

.

In view of the continuous dependence of the solutions on the initial data, a neighborhood of zero can be indicated such that the solutions beginning in it will be also in ε - neighborhood t ∈ [t 0 , t ε ] . This means that the zero solution of system (3.9) is asymptotically stable. Theorem 3.9 is proved. 101

We consider a quasilinear system .

x = A(t ) x + f (t , x),

(3.11)

where a vector function f (t , x ) is continuous in the domain G = I × R n , f (t ,0 ) = 0 ;

A(t ) is continuous matrix on I. In addition, f (t , x ) satisfies to the smallness condition

f (t , x ) ≤ δ (t ) x ,

(3.12)

where δ (t ) is defined on I and integrated on any finite interval in I function. Let λ [q ] is a generalized exponent of the linear part of system (3.11). The estimation x(t ) ≤ x(t0 ) Dε e(λ [q ]+ε )( q (t ) − q (t0 ))

(3.13)

is called uniform with respect to the initial moment t 0 , if for a given set of solutions the constant Dε (x 0 ) can be chosen, independently on t0 (for all t ≥ t 0 and the entire set of solutions considered, so that the definition includes uniformity with respect to the initial point). Theorem 3.10. If for all solutions of the linear part of system (3.11) the uniform estimation (3.13) is held, then all solutions of quasilinear system (3.11) with condition (3.12) admit the uniform estimation t

x(t ) ≤ Dε x(t0 ) e

(λ [q ]+ε )(q (t )− q (t0 ))+ Dε ∫ δ (τ )dτ t0

(3.14)

Proof. For Cauchy matrix X (t , s ) of the linear system x = A(t )x at any fixed t , s ∈ I the expression is valid

102

X (t , s ) = max

x(t )

x(s )

,

where the maximum is taken by all nonzero solutions x(t ) of the system. Hence from condition of the theorem follows, that for Cauchy matrix X (t , s ) of the linear system x = A(t )x the uniform estimation is held X (t , s ) ≤ Dε e(λ [q ]+ε )( q (t ) − q ( s )) ,

t≥s

(3.15)

All solutions of quasilinear system (3.11) satisfy to the equation t

x(t ) = X (t , t 0 ) x(t 0 ) + ∫ X (t , s ) f ( s, x( s ))ds, t0

therefore by (3.12) and (3.15) it follows t

x(t ) ≤ D x(t0 ) e(λ [q ]+ε )( q (t ) − q (t0 )) + ∫ Dε e(λ [q ]+ε )( q (t ) − q ( s ))δ ( s ) x(s ) ds. t0

Consequently, we get x(t ) e−(λ [q ]+ε )(q (t )− q (t0 )) ≤ Dε x(t0 ) + t

+ ∫ Dε δ (s ) x(s ) e − (λ [q ]+ε )(q (t )− q (t0 ))+ (λ [q ]+ε )(q (t )− q (s ))ds = t0 t

= Dε x(t0 ) + ∫ Dε δ (s ) x(s ) e−(λ [q ]+ε )(q (s )− q (t0 ))ds. t0

We denote

y (t ) = x(t ) e −(λ [q ]+ε )( q (t ) − q (t0 )) .

Then the estimation is held

103

t

y (t ) ≤ Dε x(t 0 ) + ∫ Dε δ ( s ) y ( s )ds. t0

Hence, by the Gronwall-Bellman lemma, we obtain t

y (t ) ≤ Dε x(t0 ) e



Dε δ ( s ) ds t0

or t

x(t ) e − (λ [q ]+ε )( q (t ) − q (t0 )) ≤ Dε x(t0 ) e



Dε δ ( s ) ds t0

that proves inequality (3.14). Theorem 3.10 is proved. Theorem 3.11. Let for a system n dy i = ∑ p ik (t ) y k , k = 1,..., n; dt k =1

(3.16)

where coefficients are continuous real functions, defined on the semiaxis J = [0,+∞) . ψ (t ) is a continuous positive function on J , the conditions are satisfied 1)

pk −1, k −1 (t ) − pkk (t ) ≥ αψ (t ), t ∈ J , k ∈ {2,..., n}, α > 0,ψ (t ) ≥ β > 0,

lim

pik (t )

ψ (t )

= 0, i ∈ {1,..., n}, k ∈ {1,..., n}, i ≠ k ,

2)

t → +∞

3)

1 ∫ pkk (τ )dτ = λk (q); k = 1, n. , t → +∞ q (t ) 0

t

lim t



where q (t ) = ψ (τ )dτ , 0

104

4)

lim

t → +∞

1 q (t )

t

∫ p1 (τ )dτ < 0 0

then the system (3.16) is asymptotically stable. Proof. In fact, by theorem 2.7 the system (3.16) has a fundamental system of solutions y 1 , y 2 ,..., y n such, that χ [ y k , q] = λ k (q), k = 1, n.

In addition, by the condition 4) the highest generalized exponent of system (3.16) is negative. Then the system (3.16) is asymptotically stable by theorem 3.8. Theorem 3.11 is proved. Corollary 3.1. Let for a system (3.16) the conditions are satisfied 1)

pk −1, k −1 (t ) − pkk (t ) ≥ αψ (t ), t ∈ J , k ∈ {2,..., n}, α > 0,ψ (t ) ≥ β > 0,

lim

pik (t ) = 0, i ∈ {1,..., n}, k ∈ {1,..., n}, i ≠ k , ψ (t )

2)

t → +∞

3)

1 ∫ pkk (τ )dτ = λk (q); k = 1, n. , t → +∞ q (t ) 0

t

lim

4) λ k (q) = lim

t → +∞

1 q (t )

t

where q(t ) = ∫ψ (τ )dτ , 0

t

∫ p kk (τ )dτ < 0 , for a fixed

k ∈ {2,..., n}

0

then the system (3.16) has k − dimensional conditional stable subspace of solutions. Example 3.

1 1 x = − x , y = y t t

.

The system with respect to q(t ) = ln t have generalized exponents λ1 (q ) = 1 , λ 2 (q ) = −1 . E1 (q ) = {x(t ); χ [x, q ] ≤ λ 2 (q )}.

of onedimensional conditional stable subspace to the space of solutions 105

E 2 (q ) = {x(t ); χ [x, q ] ≤ λ1 (q )}.

Example 4.   4t 3  2007t 2007 πt 3 t t4 t2  x =  arctg − + + + 1 x + y 2 2   3 t +1 6t + 2t + 1 4  et  3    y = −2007t ln tx − 2007t 2 y

Here we get p1 (t ) =

t3 3

 t  4arctg t − π +  1 + t (t + 1)2 + t 2 

2007t 2007

b(t ) =

e

t2

t

q (t ) = ∫ ψ (τ )dτ = 0

c(t ) = −2007t ln t ,

, t3 3

 t2 + +1 ,  4 

t

∫ p1 (τ )dτ

,

=

0

p 2 (t ) = −2007t 2 .

α = 2008 , ψ (t ) = t 2 ,

t t4  π  t3 − + +t  arctg t + 1 4  12 3 

 3 t4  t π  3 t3 3  lim  ⋅  arctg −  + ⋅ + ⋅t = 3 t → +∞  t 3  t + 1 4  t 3 12 t 3  

 t π 1 3 1 1 1 = lim t  arctg −  + +  = − + = − = λ1 t → +∞   t +1 4  4 t 3  2 4 4

lim

3

t →∞ t 3

∫ (− 2007t t

0

2

)dt = lim − 3 ⋅t2007 ⋅ t3 = −2007 = λ 3

3

It is easy to verify, that for system  x = p1 (t )x + b(t )y   y = c(t )x + p 2 (t )y

106

2

the conditions are satisfied p1 (t ) ≥ αψ (t ) + p 2 (t ) ,

b(t )

ψ (t )

→ 0,

c(t )

ψ (t )

→0

moreover, t

____

lim

t → +∞

1 1 p1 (τ )dτ = − q (t ) ∫0 4 t

____

1 ∫ p2 (τ )dτ = λ2 = −2007 t → +∞ q (t ) 0 lim

the system has generalized exponents λ1 (q ) = −

1 4

and

λ 2 (q ) = −2007 .

Consequently, the system is asymptotically stable. Example 5. dy1 dy = −2aty1 , 2 = (2t sin ln t + t cos ln t − 4αt )y 2 dt dt

(3.17)

General solution of system (3.17) has the form y1 = e −αt

2

+ c1

, y2 = et

sin ln t − 2αt 2 + c 2

(3.18)

If as the perturbing matrix we choose the matrix  0.....0  , B (t ) =   e −αt 2 0   

then the perturbed system of equations has the form: 2 dz1 dz = −2αtz1 2 = (2t sin ln t + cos ln t − 4αt )z 2 + z1e −αt dt dt

General solution of system (3.19) has the form 107

(3.19)

z1 = c1e −αt , z2 =  c2 + c1 ∫ e −t  2

2

sin ln t

dt et 

2

sin ln t − 2αt 2

(3.20)

Now we find generalized characteristic exponents of solutions (3.18) and (3.20) with respect to q(t ) = t 2 ____  −αt 2 + c   −αt 2 ec1  = χ [c f ] = χ [ f ] = χ e−αt 2  = lim ln e−αt 2 / t 2 = −α . χ [ y1 ] = χ e 1  = χ e       t →+∞  

χ [ y2 ] = χ et

2



____

2 2 = χ et (sin ln t − 2α )  = lim ln et (sin ln t − 2α ) / t 2 = 1 − 2α   t → +∞

sin ln t − 2αt 2 + 2α 



Finally, χ [ y1 ] = −α , χ [ y 2 ] = 1 − 2α . 1 2

If α > , then the generalized exponents χ [ y1 ] < 0, χ [ y 2 ] < 0,

and the general solution of the system (3.17) is stable. ____

2 2 2 χ [z1 ] = χ c1e −αt  = χ e −αt  = lim ln e −αt / t 2 = −α .









t → +∞

____

2 2 2 2 2 2 χ [z2 ] = χ  c2 + ∫ e −t sin ln t dt et sin ln t − 2αt  = lim ln et sin ln t − 2αt  c2 + c1 ∫ e −t sin ln t dt  / t 2 ;      t → +∞

Now we estimate the integrand. We set

tn = e

2 nπ +

π 2

Since −

t

−t sin ln t ∫ e 1 1 dt1 > 0

2

tne



2π 3

t n e −π



e−t1 sin ln t1 dt1 ≥ 2

tne



2π 3

t n e −π

1 2 t1 e 2 dt1

(

)

1 −π 2 t n e tne > e2

108





2π 3

t n e −π

1

dt1 = e 2

(t e )

−π 2

n

 − 2π    tn  e 3 − e−π ;    

1

t

−t sin ln t1 dt1 > e 2 ∫e 1 2

(t e π ) −

2

n

0

 − 2π    tn  e 3 − e −π     

We estimate the expression  t 2 sin ln t − 2αt 2  − t 2 sin ln t    c2 + c1 ∫ e  e    1

et n sin ln t n − 2αt n e 2 2

2

(t e π ) −

n

2

 − 2π    tn  e 3 − e −π  =    

(

1  = c 3 t n exp t n2 sin ln t n − 2αt n2 + t n e −π 2 

(

 t e −π  = c 3 t n exp  sin ln t n − 2α + n  2t n2 

)

2

)  = 2

   e − 2π  2  t n  = c 3 t n exp 1 − 2α + 2    

Finally, 1  t 2 sin ln t − 2αt 2  −t 2 sin ln t   ln e dt   >  c2 + c1 ∫ e  2   t → +∞ t  ____

χ [z2 ] = lim

 e − 2π 1  ln c 3 t n exp1 − 2α +  t → +∞ t 2  2  n  ____

> lim

 2 t n  

 =  

= 1 − 2α + e −2π / 2;

χ [z1 ] = −α ,

In the system (3.19)

χ [z 2 ] = 1 − 2α + e −2π / 2. B (t ) → 0

if a parameter α satisfies to the inequality 109

 2 t n     

1 1 1