The description for this book, Lectures on Differential Equations. (AM14), Volume 14, will be forthcoming.
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Table of contents :
PREFACE
TABLE OF CONTENTS
Chapter I. SOME PRELIMINARY QUESTIONS
§1. Matrices
§2. Vector spaces
§3. Analytic functions of several variables
Chapter II. DIFFERENTIAL EQUATIONS
§1. Generalities
§2. The fundamental existence theorem
§3. Continuity properties
§4. Analyticity properties
§5. Equations of higher order
§6. Systems in which the time does not figure explicitly
Chapter III. LINEAR SYSTEMS
§1. Various types of linear systems
§2. Homogeneous systems
§3. Nonhomogeneous systems
§4. Linear systems with constant coefficients
Chapter IV. CRITICAL POINTS AND PERIODIC MOTIONS QUESTIONS OF STABILITY
§1. Stability
§2. A preliminary lemma
§3. Solutions in the neighborhood of a critical point (finite time)
§4. Solutions in the neighborhood of a critical point (infinite time) for systems in which the time does not figure explicitly
§5. Critical points when the coefficients are periodic
Chapter V. TWO DIMENSIONAL SYSTEMS
§1. Generalities
§2. Linear homogeneous systems
§3. Critical points in the general case
§4. The index in the plane
§5. Differential systems on a sphere
§6. The limiting sets and limiting behavior of characteristics
Chapter VI. APPLICATION TO CERTAIN EQUATIONS OP THE SECOND ORDER
§1. Equations of the electric circuit
§2. Lienard’s equation
§3. Application of Poincare's method of small parameters
§4. Existence of periodic solutions for certain differential equations
Index
LECTURES ON DIFFERENTIAL EQUATIONS By SOLOMON LEFSCHETZ
P r in c e t o n U n iv e r s it y P r e s s
Princeton, New Jersey
Published by Princeton University Press 41 William Street Princeton, New Jersey 08540
Copyright 1946 by Princeton University Press; copyright © renewed 1976 by Princeton University Press All Rights Reserved Library of Congress Card No. 1578 ISBN 0691083959
Princeton University Press books are printed on acidfree paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources
Printed in the United States of America
PREFACE The subject of differential equations in the large would seem to offer a most attractive field for further study and research. Many hold the opinion that the class ical contributions of Poincare, Liapounoff and Birkhoff have exhausted the possibilities. This is certainly not the opinion of a large school of Soviet physico mathe maticians as the reader will find by consulting N. Minorsky's recent Report on NonLinear Mechanics issued by the David Taylor Model Basin. In recent lectures at Princeton and Mexico, the author endeavored to provide the necessary background and preparation. The material of these lectures is now offered in the present monograph. The first three chapters are selfexplanatory and deal with more familiar questions. In the presentation vectors and matrices are used to the fullest extent. The fourth chapter contains a rather full treatment of the asymptotic' behavior and stability of the solutions near critical points. The method here is entirely inspired by Liapounoff, whose work is less well known that it should be. In Chapter V there will be found the PoincareBendixson theory of planar characteristics in the large. The very short last chapter contains an analytical treat ment of certain nonlinear differential equations of the second order, dealt with notably by Lienard and van der Pol, and of great importance in certain applications. The ^uthor wishes to express his indebtedness to Messrs. Richard Bellman and Jaime Lifshitz for many valu able suggestions and corrections to this monograph. The responsibility, however, for whatever is still required along that line is wholly the authors.
TAHIJi OF CONTENTS Page Chapter I. SOME PRELIMINARY QUESTIONS........ §1 . Matrices......................... §2 . Vector sp aces....... §3 . Analytic functions of several variables.
1
1 10
14 21 21 23 30 34 37
Chapter II. DIFFERENTIAL EQUATIONS.......... $1 • Generalities...................... §2 . The fundamental existence theorem . . . §3* Continuity properties......... §4. Analyticity properties............. §5« Equations of higher o r d e r .......... §6. Systems in which the time does not figure explicitly.................
38
Chapter III. LINEAR SYSTEMS................. 5 1 . Various types of linear systems . . . . §2. Homogeneous systems......... 53* Nonhomogeneous systems........... §4. Linear systems with constant coefficients
47 47 49 62 65
Chapter IV. CRITICAL POINTS AND PERIODIC MOTIONS QUESTIONS OF STABILITY............. §1 . Stability....................... 52. A preliminary lemma........ §3 . Solutions in the neighborhood of a critical point (finite time) . . . . . . . 54.
72 72 82
84
Solutions in the neighborhood* of a* critical point (infinite time) for systems in which the time does not figure explicitly............. . . 91 §5 . Critical points when the coefficients are periodic........................ 109
vl_________________ TABLE OP CONTENTS____________ Chapter V. TWO DIMENSIONAL SYSTEMS . . . . . . . §1. Generalities........................ §2. Linear homogeneous systems........... §3 . Critical points in the general case . . §4. The index in the p l a n e ............... §5 . Differential systems on a sphere . . . . §6. The limiting sets and limiting behavior of characteristics...................
117 117 119 125 133 142 162
Chapter VI. APPLICATION TO CERTAIN EQUATIONS OP THE SECOND ORDER . ................. §1. Equations of the electric circuit . . . 52. Lienard’s equation................... §3• Application of Poincare^ method of small parameters.......................... §4. Existence of periodic solutions for cer tain differential equat i o n s .........
204
I n d e x ....... ...............................
210
185 185 188 194
CHAPTER I SOME PRELIMINARY QUESTIONS §1. MATRICES 1. The reader Is assumed familiar with the elements 0 of matrix theory. The matrices I I, I I, are written A, X, ... . The transpose of A is written A 1. The matrix diag (A^ ..., Ap ) is
where the A^ are square matrices and the zeros stand for zero matrices. Noteworthy special case: diag(a1,...,0^) •denotes a square matrix of order n with the scalars down the main diagonal and the other terms zero. In par ticular if a1 ■ ... «  1, the matrix is written or E and called a unitmatrix. The terms of En are written 6 ^. and called Kronecker deltas. 2.Suppose now A square and of order n. The deter minant of A is denoted by A. When AI « 0, A is said to be Singular. A nonsingular matrix A possesses an inverse A*1 which satisfies AA~1 « A 1A « E. The trace of a square matrix A written tr A, is the expression 2 If An  0, A is called nlipotent. We recall the relations (AB)'1  B"1A1
(A1)'  (A*)1,
where A, B are nonsingular. Iff(A) ■ aQ + a^A + ... + flpA then &0E + a,A + ... + a^Ar has a unique meaning and is written f (A). The poly nomial 6(A)  AAE is known as the characteristic poly
2
DIFFERENTIAL equations
nomial of A, and its roots as the characteristic roots of A. (See Theorem (35) below.) 3. (3*1) Two real [complex] square matrices A, B of same order n are called similar in the real [complex ] domain if there can be found a nonsingular square real [complex] matrix P of order n such that B = PAP"1. This relation Is clearly an equivalence. For if we denote it by ** then the relation is symmetric; A ~ B — *B ~ A, since B  PAP’ 1 =* A  P‘ 1 BP reflexive: A
A, sinceA * EAE’1,
transitive: A ~
B, B ^ C — 4 A ~ C.
For if A * PBP’1,
B * QCQ" 1 then A « PQQQ'V 1 = (PQ)C(PQ) ’ 1 . (3 .2 ) If A ^ B and f(A) is any polynomial then f (A) ~ f (B). Hence f(A) ■» 0—* f(B) = 0 . For if B  PAP"1 then Br «.PArP~1, kB = P(kA)p’\ and P(At + A^P"1  PA^’1 + PAgP’1. (33) Similar matrices have the same characteristic. polynomial. For B « PAP"1 — » B  A E “ P(AAE)P 1, and therefore also IBAEI  IAAEI. Since the characteristic polynomials are the same, their coefficients are also the same. Only two are of in terest: the determinants, manifestly equal, and the traces. If A^, ..., An are the characteristic roots then a ready calculation yields ** 2aii * tr A Therefore (3.4) Simn«p matrices have equal traces. For the proof of the following two classical theorems the reader Is referred to the standard treatises on the subject: (3.5) Theorem. If d>(A) is the characteristic poly nomial of A, then • An ). 2 and A we By way of illustration when n have the two distinct types ", V/ 1, A (3 .7 ) Real Matrices. When A is real the A^ occur in conjugate pairs Aj, Aj and hence the matrices occur likewise^ in conjugate pairs Aj, Aj where Aj is like Aj with Aj instead, of Aj. Thus they may be disposed into a sequence A^, •••, 9 *** 9 ^2k+l9 *** 9 ^s ^*©re the A2jc+i correspond to the real A .. We will then say that the canonical form is real. k. Limits. Series. C^.1) Let JApI, Ap  K ajj I be a sequence of matrices of order n such that aij « lim Ja^ji exists for every pair i, j. We then apply the customary "limit" tenninology to the sequence Apl and call A «  a^j II its limit. As a consequence we will naturally say that the infinite series ^ Ap is convergent if the n2 series (*.2)
aij
2 aP lj
k
DIFFERENTIAL EQUATIONS
are convergent and the sum of the series is by definition the matrix A  I . If the a^j are functions of a parameter t and the n series ]> ©jj ar© uniformly convergent as to t over a certain range then 2 Ap is said to be uniformly convergent as to t over the same range. (^•3) Let us apply to the Ap's the simultaneous operation Bp  PApQ where P, Q are fixed. If we set «mr _
§
p
then clearly the corresponding to the by
• for the B's la related
*= 1 pih^^cjNow a n.a.a.c. of convergency of (4.2) may be phrased thus: for every t > o there la an N such that m > N =♦ ISI < t whatever r. If a *» sup j I then 2 IT"jl < na«2 2 ls” : .
i,J
3
i.4
J
Hence the convergence of (4.2) Implies the convergence of 2 Bp  I p/yi whose limit Is clearly B = PAQ. In particular (4.4) If ApI converges to A and If Bp = PApP” 1 then BpI converges to B = PAP 1. 5 . Consider a power series with complex coefficients (5 .1 )
f(z) “ a0 + u^z + a2 z2 + ...
whose radius of convergence f > 0 . If (5.2)
X = I xti 1
is a square matrix of order n we may form the series
I. SOME PRELIMINARY QUESTIONS (53)
aQE +
1
+ agX2 + ...
and If It converges its limit will be written f(X). . (5.*0 Suppose that X  diag (X,,^). If g(z) is a  scalar polynomial then g(X)  diag (g(X1), g(X2)). Hence in this case (5*3) converges when and .only when the same series in the X^^ converge and its limit is then f (X) « diag (f(X1), f and setting for conven ience ■» D, we obtain for any matrix X the analogue of the well known elementary relation: (77)
(D  A)X « eAt . D . e"At X.
As an application consider the matrix differential equa tion (78)
jy « AX, A constant.
Owing to (7*7) it reduces to eAt . D . e‘AtX  0 . Multiplying both sides by e~At (see 6.2) we have DeAt . X  o At
and hence e . X = C, an arbitrary constant matrix. At Hence the complete solution of (7*6) is X «* e . C. We will return to this later. §2. VECTOR SPACES 8. We will assume familiarity with the first con cepts: dimension, base, coordinates relative to a base. When the scalars are all real [complex] the space is said to be a real [complex] vector space. The only vector spaces which we shall encounter are finite dimensional. Let V be such a space. Its vectors will be denoted by an arrow over small latin characters with possible super
I. SOME PRELIMINARY QUESTIONS
11
scripts as: &, x, a1, ... . Let in particular fe1, ..., e11! be a base forV. If we denote by x any element of V then we will have (8.1)
x  5E^e1 + . . . + x^e11.
The x^ are the coordinates or components of x. If we adopt x, x^ for the arbitrary vectors and their coordin ates we will often denote Vby V . Similarly If say vectors and coordinates were u, ul, we would write for lb the space. In the coordinates of x will be usually written x1h (exceptionally and then explicitly stated xhi ^" The metrization of V will be done in the customary manner by means of a norm I x II. We choose here for con venience (8.2)
I ? II— 2 I I
and accordingly define the distance in V (8.3)
as
d(x,xf)  I (xx» )  2 x±  x . r.
As is well known this distance has the usual properties: d(x,x *)  0 — 4x « x 1; &(x,x')  d(x*,x) ^ d(x,x") from the real space Vx to the real space Vy represented by
(101 )
.7±  4 ^ ,
***, XQ).
We will say that i> is regular at x° whenever the analytic at the point and the Jacobian
are
J
does not vanish at the point. The implicit function the orem for the system (1 0 .1 ) yields immediately the follow ing property: (1 0 .2 ) Let be regular at x° and let y° « is a topological mapping of into a pcell Ep through x° in V x . The cell Ep is known as an elementary analytical pcell through x°. When p = 1 we also speak of an elementary analytical arc through the point. An analytical curve A in Vx is a con nected set such that every point has a neighborhood which is an elementary arc. That is to say every point x° of A has a neighborhood E 1 which admits a parametric repre
18
DIFFERENTIAL EQUATIONS
sentation (1 0 .3 *2 )
x±  x° * ^(v), ^ ( 0 ) = 0 ,
Iv)< a,
where the are analytical on their range and the deriv atives tij(v) never vanish simultaneously on the range. When the analytical curve A is compact it is a Jordan curve, otherwise it is an arc. The following intersection property will be found useful later. (10.4) Let and the analytical curve A have an infinite number of intersections with a limitpoint P contained in both. Then P has a neighborhood v in the intersection h which is a subarc of A . We may suppose the representations as above with P as x°. Under the assumption we may solve (1 0 .3 . 1 ) for u.j, ..., Up in terms of p of the differences x^  x£, say in terms of the first p differences and thus replace the system in the vicinity of P by another of the form * V i
'
" *i(*i*?»•••» V * p }
where the fi(w1,... ,wp ) are analytic at the origin w * 0 . As a consequence the analytic functions of v pi(v) “
•••» *p(v)) ~ *p+i*v)
have an infinity of zeros in the vicinity of v * 0 . It follows that F^ vanishes on some interval v < cl^. If a Is the least then P has for neighborhood in the subarc of A corresponding to v < a. (1 0 .5 ) Application. Two analytical curves A, m which have an infinite number of intersections with a limitpoint P in both A and ji, necessarily coincide. Let £ be the intersection. The point P is in an elementary subcell of A, and /a intersects in an in finite number of points. Hence P Is contained in a subarc 1/ of which Is in the intersection § and is a neighbor
I. SOME PRELIMINARY QUESTIONS
19
hood of P In both A and >*. Let ^ he the largest subarc of both A and fi which contains v and suppose that v has an end point Q. Since 5 Is closed In both A and p, Q Is In 5. By the above argument Q Is contained In some sub arc v' of both A and /i. Since this contradicts the maxi mal property of ^ , Q cannot exist. Hence  A » j*. This proves (1 0 .5 ). (10.6) Bv an.analytical (ni 1manifnlri m11" 1 in V ^ we shall understand a locus represented by an equation (1 0 .6 .1 )
f(x1 ,...,xn)  0
where f Is real analytical In a certain region £L of V_ and the partlals g—~ do not vanish simultaneously In II. Under this assunqptlon If x° is a point of Up 1 and say ^ 0 , one may solve (1 0 .6 .1 ) In the neighborhood of x°
d*n
« for xn ~ *n “ “ ^n■“ *^X1r
xni ^
where Is analytical In the vicinity of (x°,.. ). Thus the x.  x? may be expressed in the form (1 0 .3 .1 ) 0 with p =* n1 and * x^  xh, h « 1,2,... ,ni. In other words x°, i.e. any point of M111 has a neighborhood in M11"1 which is an elementary analytical (nl.)cell. When n * 2 , M 1 is merely an analytical curve in the plane Vx . 11. On a question of reality. Largely as a conse quence of the possible complex characteristic roots of real matrices we shall find it convenient to describe certain systems as real in a broader sense than commonly understood • Consider an ndimensional vector space wnose elements are sets x^, x^, •. •, x^, x^,, ^gr+i'’ *** 9 ^2r+j ^ real. The resulting vector space V x is still referred to as a real n dimensional vector space.
so
DIFFERENTIAL EQUATIONS .p. A function f(x^, ..., * f(x) ia said to be real whenever f(x,, x 1, ..., xn ) « f ^ , x1, x2, x2, ...,.3^). *
It la said to be analytic at a point x° whenever f ia analytic in
x 1, ..., xn> (all considered as independ
ent VuTxables) about (x°, xlj*, ..., x£). A vector function T(x) =* (f^(x), ..., fn(x)) is said to be real whenever the fp+i are real and fjL(x1, x,, ..., Xn) «
x,, ..., Xn),
i £ r.
It is said to be analytic whenever the f^ are analytic. To illustrate our general meaning, take r = 1 , n = 3 . Thus the coordinates may be named x, x, y. The function f(x, x, y)  f(x, x, y). It is analytic at (x°, x°, y°) whenever f is analytic in x, x, y in the ordinary sense at the point. A real vector function ? is a triple U* T, g) of functions of x, x, y where g is real and Tlx, x, y)  f(x, x, y). 12. Topological and related considerations. Free use will be made in the sequel of certain symbols and con cepts of set theory and topology. A few words concerning these may help the reader unfamiliar with them. (12.1) First as to set theoretic symbols. The fol lowing should be kept in mind. If Aj, ..., Ap or lA®! are sets or a collection of sets then A1 u ... uAj or uAa denotes their union (Jll the elements in one of the sets); A1 n ••• n ^ or OA II is bounded under certain conditions," "II if has an upper bound M". §2* THE FUNDAMENTAL EXISTENCE THEOREM. k. In order to facilitate the statement it will be convenient to introduce a certain preliminary concept. We shall be considering a region II of the space V of (x, t) and a point (f*, tj ) of that space • We shall understand by a box B(a,T) of center (f, ) an open set of hf which is a product Ua * 1^. where Ua is the set H"x"f  < a and Ij. is the time interval lt^1 < t . We may now state the (^.1) Existence Theorem of CauchyLinschitz for real functions. Let SI be a region of the space \jJ of (x,t) such that: (Jf.1.1) 3(x.t) Is continuous in St. (i*.i .2) for every pair of points (x,t ),(xf,t) in XL there is fulfilled a Llpschltz condition (^.1.3)
II P(x,t)  p(x',t)  < k  xxf ,
k > 0.
Correspond 1np; tn any in SL and the various boxes being of center there exists a B(2cu2r) C fl, where r depends upon at., and with the following properties: For every (x°.tQ) € B(ol.t ) there exists a unique solution x(t) of (3.2) defined over IT such that (x(t),t) remains in B(2a,r) when t Is in 1^ , and that x(t°) » x°. The proof will rest upon Picard1s classical process of successive approximations. We begin with a preliminary remark. We know that has a neighborhood in St of the form B(2gl,2t). Now let M be an upper bound for pt^t) in SL. It is clear that
2k
DIFFERENTIAL EQUATIONS o.
Since we may replace r by a smaller number, we may take it 1 n]f .such that 4>(2t ) < a, that Is to say take r < log (1+~jj)* We sow pass to the proof proper. We shall take (x°,t°) as an arbitrary point .in B(a,T). Thus I< lt°nl < r. (a) Existence of 5c(t). Consider the sequence of vectors x ^ t ) defined in succession by (it.2^
3m+1  x° + S p(S°»t)dtt to
t € IT.
We will show that for t € Ir, the sequence has a limit which satisfies (3 .2 ). For t € Ir, we have first of all tt°l £ 11—  + T“t° < r +r  2T*
II. DIFFERENTIAL EQUATIONS
25
Then A x 0?1 II £ M tt° < m . 2r < 4>(2t ) I, t € ir, is a Cauchy sequence. Since"9 is complete, the sequence con verges to a limit x(t) such that IIx(t)x°  ^4>(2t ) x°. We may suppose r £ r1, and so the second solution is de fined likewise over I*.. Since ^(t), ^(t) are differenti able they are continuous. Hence given any P > 0 there will exist a positive cr < r such that tt°l < cr—  xx° , I II < p and therefore IIxy II < 2 p. Now by (4.1.2):
(4.5)
II xy  £ k
S II xy  dt < 2Pk tt° . *0
By repeated substitution in (4.5) we obtain
II xy II < 2 p C. ¥.t^t. °.l.ff
0 wlth i
#
Hence {xy II =» o’and therefore x « y for;tt° < cr. Thus if x(t), y(t) assume the same value tor any t° € IT, they coincide on an interval containing t° and contained in I T•
Now consider the maximal interval I1 : t1 < t < t" containing t°, contained in IT and such that ?(t) » x(t)  y(t)  0 on I*. If I*  IT, then x(t) * y(t) on lr, and the proof is complete. In the contrary case, one of t 1 or t", say t1 is in IT. Since ?(t) is continuous in and is zero in I1, it is also zero at t», since t1 € I1. It follows from the earlier argument that *z(t) vanishes in an interval I" containing t" and contained in IT. Thus we can augment I1 by I", and so we get a contradiction. This completes the proof of the existence theorem.. 5 . Complementary Remarks. (5.1) Of the two basic conditions (4.1.1), (4.1.2) in the statement of the exist ence theorem the second is less natural than the first, and would seem more difficult to verJLfy. The following
28
differential equations
property will therefore be useful in this direction, and also will suffice for later applications. (5.8) If the partial derivatives dp^/dxj exist and are continuous and bounded in H then (4.1.3 ) holds♦ For we may clearly replace H by a box B(p,r). Ap plying now the mean value theorem to p^, with x, x' € Up we find p . ^ t )  PjtjcSt)
(x*,t)(x,xi), 9 x
j
J
J
^ 3Pi I where x* € Up. Hence if A is an upper bound of ths for all p^, we find ^ II p(x,t)  p(x',t) I < nA . Ilxx'  . Thus (4.1 .2 ) holds with k — nA. (53) It is worth while to exhibit functions which . do not satisfy a Lipschltz condition, e.g. (a) p(£,t) with
^ for 0 < Xjl < 1 } (b) iftx,t)with p^  V"x^
for 0 < x^. (5 .4 ) Domain of continuity. The union of all the sets such as 11 is an open set of W . We choose one of its Components •&' ^nd call it the domain of continuity of the differential equation £3 .2 ). Henceforth the equation is supposed to be taken together with a definite domain jy and all our operations will be generally restricted to that domain. (5.5) oCf is an open set. For every point of has a connected neighborhood N which is in H and hence in . Hence e & is open. 6 . Extension of the solution. Trajectories. (6 .1 ) The existence theorem yields a solution x(t) = x°, and valid over a certain interval IT containing t°, and such that ^(t°). Let t*° be any point of IT with x(t*°) » x'°. There is a similar solution it'(t) valid over an interval IT, » t»° and such that ^(t*0)  x'°. ‘ Moreover by the
II.
DIFFERENTIAL EQUATIONS
29
uniqueness property x(t) » x'(t) for t € I_ n I_t. We call x'(t) the continuation of x(t) to I,., and we now de fine 5?(t) on IT. = Xp u Iji by assigning to It the values jT*(t ) for t In If, * Ij. We thus extend the definition of x{t) to If... By repetition of this process we thus arrive at a maximal interval t 1 < t < t" over which x(t) is de fined. It may be of course that t' «* co or t" « + cd or both. The analogy with the classical process of analytical continuation is obvious. (6 .2 ) If we start from (x°,t°) €«Cf, the set of points (x,t) reached is such that from (x°,t°) to any one of them there may be constructed a chain of boxes B1, ..., Br each in a domain such as J7 and withB^, Bi + 1 overlap ping. Since the boxes are connected so is their union andsince JO is a component of all the sets JCf,_ every (t)  (x(t),t), t € IQ. It Is clear that Is continuous, univalent, and i) I0 « T, Hence 4> Is oneone and onto. It follows already that If I is any closed subinterval of I0,(hence T is com pact ), the values of i on T give rise to a topological mapping + of T onto a subset J of r. There remains to prove open. Consider (x°,t°) € r with 4>_1 (3°>t0) « t° € IQ. Let 2r be a lower bound for the distance from t° to the end points of I0. Then if U => r n ^ (5^0 ,t°,r), the set 4>1U Is contained in an interval I such that I C I0. Since 4> is topological, 4>_1D is an open set of T and since It Is in I, it is an open set of I and hence of I0 Therefore 4> is open, hence It is topolog ical "Tiri r is an arc.
30
DIFFERENTIAL EQUATIONS §3
CONTINUITY PROPERTIES
7* If (x°,t°) € JO then we may enclose it in a box. B(a,T). It will be recalled that for any t,t° € ^and € Ua there was obtained a solution (7.1)
x(t) = x° + (x1x°) + ...
where under the conditions just stated the series is uni formly convergent. Let us consider 5?(t) as a function of t, x°, t°, and write it accordingly x(t,x°,t°). Then (72) If t, t° € Ij. and x° € Ua, then x(t,x°,t°) is a continuous function of (t.x°.t°). ' Owing to the uniform convergence it is only necessary to prove that xm+1  x®1 is continuous over the range under consideration, or in the last analysis that has this property. This is trivial for x° so we assume it for 3P and prove it for 5m+1. In the proof of the existence theorem it has been shown that under the conditions under consideration all the x®1are confined in a certain box B* C G in whose points p(x,t) is continuous and bounded. It follows that in (^.2 )m the integrand is continuous and bounded under our conditions. Since the integral is thus a continuous function of (t,30,t0), the same holds for 5P+1 and (7*2) follows. We now remove the restrictions on t, x°, t°, save of course that only points (x,t) € Juf are to be con sidered and we will prove continuity under the same condi tions. More precisely: (7*3) If (^°,t°) € JO and x(t,x°,t°) is the solution such that 2U°,20,t°) * *° then x(t,1x°,t°) is. continuous in (t,x°,t°). Consider the trajectory rQ through (x°,t°). Combining successive approximations with the process of continuation we find that the arc of rQ from (5f°,t°) to (5c,t), being a compact set, may be covered by a finite number of boxes such as B, say B1, ..., Bp, where the first contains
II.
DIFFERENTIAL EQUATIONS
31
(x°,t°), the last (3c,t), and consecutive boxes overlap. More precisely If B.  I_ * then I_ , I overlap 1 i 1 T1 Ti+1 and so we may choose a value t^ in their common part. We *0 0 will now keep (x ,t ) fixed and consider a new variable initial point (xf,tf) in B1. If r is the trajectory through this point then it meets the set t =» t1 at a cer tain point (x1,t1) whose coordinate 3c1 is a continuous function of (xf,tf) by (7.2). It follows that we can choose a sphere ]> (x°,t°,p) C B1, with p so small that when (xSf) € 2 then (x1,t1) € Bg. Now for. (x1,t*) € Bg and t € the solution x(t)such that xtt1)  x1, is a continuous function of x f,tV and hence of (t^xSt*) for t. € IT and (x1,t1) € ]>. Confining now (x* ,tf) to r will cSntain the point (x1,t1) € Bg, and so it will con tain a point (£2,t2), where x2 = "x(t2). The same contin uity argument shows that when p is sufficientlysmall (x2,t2) € B^ and we maynow proceed as before. After a finite number of steps we shall find a p so small that for (5T*,t1) € 2 > x(t,5^ ,tf) will remain in Bp and be continu ous in (tyxSt1) for t € IT and (x%,t) • This im plies (7*5)• r (7M Several noteworthy consequences may be drawn from the preceding results. The notations being the same corresponding to M(x°,t°) we may choose t, 6 such that the sets t° *>/(x°,€.), 2 (*M ) C B1. Consider now 5c(t,x*,*t1)> (S^jt1) € 2 . By the existence theorem the function is defined for every t € IT and hence for t 1
t°. Let x(t,x',tf) « x"(t), so x(t°,xf,tM = x"(t°). By (7*3) x"(t) is a continuous function of (xf,tf) Therefore 6 may be chosen such that (x"(t°),t°) € t° * In other words any trajectory passing near enough to (5f°,t°) inc# will cross t° * */(x°.,£): and of course the converse will hold if 6 ^ We may thus say that the trajectories passing close to a given point M(x°,t°) € JO at time t° include all those passing quite close to M in Xf. Or ex plicitly:
32
differential equations
(753 Corrftanrmdlng to any M € 0 there exists a 6 > o such that a trajectory crossing 2(M, 6 ) Intersects the hroerolane t  t° within >/(x >£) (i.e.,? it meets t° » J (x°.fc.)). (7.6) An extension of the continuity theorem in a new direction is the following. Suppose p is in fact a ' function ]£(x,t,y) which is continuous and bounded in a regional of W « V^. We suppose moreover that with re spect to fl the Lipschitz condition (4.1.3) is still ful filled in the same form as before. The domain of contin uity A is defined as before (5.M as a component of the union of all the sets SL. We have now the following stronger result which extends (7 .3 ) and may understand ably be stated in the brief form: (77) The solution is continuous in (t.x0^ ^ ) when (X°,t°y*) ranges over A. The proof may be related to (7 .3 ) by a well known de vice. Enlarge (3 .2 ) by adding the differential equation
so that (3 .2 ) and (7 .8 ) form a system such as (3.2): for the vector (x,y). Owing to the special form of (7 .8 ) the Lipschitz conditional .3 ) still suffices to prove the existence theorem and hence all its corollaries, including among them (7>3)> which in the present instance becomes (7.7) (7*9) Consider again the trajectory H. and an arc MN of r , where (x ,t ) are the coordinates of M and (x ,t ) are those of N. Introduce the two closed sets Sp * t°« */(x2^>0), S1 « t1 « where p 0, f 1 are so chosen that the two sets are In il. Let M v € SQ ‘and let H : 3?(t,Mf) be the trajectory through M f. It is clear from the argument proving (7*3) that for p Q small enough 5?(t,NP ) may be extended throughout the whole closed In terval I : t° £ t £ t1 • Consider a mapping 4 of the com pact product T * SQ Into JO sending I * M into lit and
II. DIFFERENTIAL EQUATIONS
33
I « M 1 Into M'N'. This napping 4> Is univAlart. since (x',t’) and (x",tn) certainly have different Images if t 1 t" or x 1 f x", (the latter since otherwise distinct trajectories would meet). Since T « SQ is compact is topological. Take now a small ncell E? containing M and contained In SQ. Applying 4> merely to E” or to I « E^ and denoting the hyperplanes t  t1 by we find: (7 .1 0 ) Given a sufficiently small ncell E^ In ttq ftnnt.ftining m, a tra jectory passing through a point M* of E^ Intersects ^ In a single point. jp such that M' — *■JP defines a topological mapping ® of E^ onto a simlTen Elj1 C ir^, enntAining N and of course N »* OM. (7.11) The r.lTvnimgt.anftftg remaining the same and ^ being small enough, let A be the arc M'N* of the traject ory through M', and let A(t) be the point of A corres ponding to any t £ T, T : t° £ t £ t1. Then (M' ,t)— *A(t) defines a topological mapping 4> of the cylinder I « E^ such that i ( 1 « M) « HI, 6 (I * M') « A . .This last result embodies essentially the socalled "field” theorem for minimizing arcs In the Calculus of Variations. (7 .1 2 ) General solution. This concept may now be Introduced with reasonable clarity. Let V Q be ndimensional and ?(t,ct) a function such that: (a) for t l n a certain region A of Vc, T is a solution of (3 .2 ) In the domain J&; (b) If Mq = 7(t®,c®), M “ ?(t,"3), where are In A, thero are ncells E11 In ?/ * t° and £ n In A, such that for c € A the correspondence c — ► M is a topolog ical mapping ♦ of f n onto E11, such that 4>c°  MQ. In other words may be chosen in £n so as to yield any soluwith its initial value at t° in eP. The function ?(t/c) is known as a general solution. By (7 «1 0 ) the concept of general solution is manifestly independent of the parti cular point M0 chosen on the trajectory T(t,t°), i.e. it does not depend upon t° but solely upon the trajectory itself.
DIFFERENTIAL EQUATIONS
3b
(713) To avoid undue repetitions we may state here and now that in dealing with complex functions the defini tion of general solution will be the same save that If c will be complex ndimensional, the cells 2ndimensional ‘and the mapping r\ analytical. An explicit reformulation will not be necessary. §4.
m i a a i G m erqeerties
8. (8.1) The argument in deriving the continuity properties of x(t) rests essentially upon the following three propositions: (a) A continuous function of a continuous function is continuous • (b) If ?(x,t) is continuous in (?,t) over a suitable range so is ?(x*,t)dt. o (c) A uniformly convergent series of continuous functions is continuous. These three propositions made it possible to "trans fer" continuity from the approximations ’im(t) to the solu tion x(t) itself. Since these three properties hold also with "continuous" replaced by "analytic" or "holomorphic" the same transfer will operate for analyticity or holomorphism. It will be necessary however to distinguish care fully between analyticity as to x°, as to t, t°‘ , or as to all three. The difference arises from the fact that we may wish to consider not only p(x,t) analytic in (x,t) but also merely in x alone, or in certain additional par ameters that may be present in p. (8.2) Suppose p(x,t) in (32) analytic in x and con tinuous in (x,t) in a certain region Jl. If (?,'*?) € SL we may choose a box B(2cl,2t) of center (?>>?) contained in JC1. As a consequence the p* and their partial derivatives g— are continuous on the compact set B, and so bounded in B.^ Hence (4.1.1), (4.1.2) hold in B (5*2) and the exist ence theorem is applicable in B.
II.
DIFFERENTIAL EQUATIONS
35
(8 .3 ) We now proceed as before replacing continuity by analytlcity wherever need be and obtain first an ana logue A of & which we call in any case domain of analy tlcity, then we have the following properties, which we merely state since the proofs are unmodified. The space TjT and the values of the variables and functions may be real or complex unless otherwise restricted. (8.4) If ]?(x,t) is analytic in both variables and A is. the domain of analytlcity then the solution x(t,x°,t°) such that (x(t),t) € A, x(t°,x°,t°) « x° is analytic in all three arguments, (see (73). (85) If p(x,t) is merely continuous in t (t real) then x(t,x°,t°) is merely analytic in x°. (see T•3)• (8.6) If p(x,t,y) is analytic In y also and A is defined accordingly as in (7 .6 ) then x(t,x°,t°,y) is ana lytic in all arguments in the case (8.4) and in (x,y) alone in the case (8.5)* (see 7*7) 9 . (9 .1 ) Poincare^ expansion theorem. Let the dif ferential equation with t real aaad the domain of analyticit (9.1.1)
= p(x,t,y)
possess for y  0 a solution 5f(t) on the closed interval T : t° £ t £ t1 such that the p^(x,t,y) may be expanded in power series of the (x^^tt)) and the j convergent in some range
(9.1.2)
< a, i “ 1 > 2 , ••• 9 n;
ly^l < a ..., r,
(r « dim V ) and this for every t € T. Then setting "f(t°) the equation (9 .1 .1 ) has a solution ^(t,*0,y) such that f(t,t°,o)  f(t) and that the $±(t,x°,Y) may be erjvmHad in power aerlea of the (x°°) and the y^ conver gent for t € I and (x°,y) in a certain toroid
36
(9.1.3)
DIFFERENTIAL EQUATIONS < a>
lyj* < a *
In particular If ^(t,?°,y) — ^(t,y) then the latter la a solution such that T(t,o) »= lf(t) and that ?*(t,y) may be expanded in power series of the y^ convergent In a toroid
(9.1 A)
yjI < a.
This theorem has been utilized by Poincare, Picard and others in many fruitful ways In questions of approxi mations, likewise in the search for solutions with special properties (periodicity among others) neighboring speci fied solutions. If we make the change of variable x  ^(t) ~ ?(t), then ?(t) satisfies the differential equation (9.2)
f'q(?»t,y)
which behaves like (9 .1 .1 ) save that now ^(t) Is replaced by 0 and that q^ may be expanded In a power series in the z^, y^ In the toroid 3 (a) :  I < a, y^ < a. (1 «■ 1 , 2 , ..., n; j = 1 , 2 , ..., r). We merely need toprovenow for (9 .2 ) the analogue of (9*1) with ^T(t) =?° « 0 , and the existence of a solution ?*(t,z0,y) possessing a series expansion in powers of the z°, y^ valid for t € 7 and (z°,y) In a certain toroid J(P). Let (9 .2 ) be amplified by
Thus (9.2), (9*3) form a system in the unknown (y,t) with the right sides holomorphlc In 7(a) for every t 6 7. The associated domain of analyticlty A 1 in * 2^ * L contains the point (0 ,0 ,t1 ) and the product 3( the same as (1.3)'. All the systems considered are then said to be selfadiolnt. Prom (4.2) and (5*2) there follows: (5«3)
Y ff + H t X ~ °
and therefore (54)
YXC,
where C is a matrix of scalars. Let Y be a solution of (5*2) such that Yl^o and set X  Y1C. Then (5.4) holds, and hence also (5*3). Consequently (55)
y
+YAX.
Since lYljtoy Y1 exists and so from (5*5) follows (4.;). Consequently every solution of (4.2) is of the form Y1C. This proves: (5.6) If Y Is a nonsingular solution of the joint equation to (4.3) then every solution X of (4.3) Is. rep resented by Y_1C where C Is. an arbitrary scalar matrix. The nonsingular solutions correspond to IC ^ 0. Taking C  E' we have YX  E; and so X*1 is a special solution of (52). Since Y1 « X, we recognize in (5.6) proposition (3.3); in another formulation. (5*7) Fbr obvious reasons of synmetry the same holds with (4.3) replaced by (52), Y by X and Y_1C by CX1. (58) In the applications the Important associated types are really (1.3) and (4.2). For this reason It will be worth while to be a little more explicit regard ing the relation of a system (1 .3 ) to its adjoint. If we write (1 .3 ) as d*i (5.8.1)
' 2
56
DIFFERENTIAL EQUATIONS
then the adjoint (5 .1 ) assumes the form 15*8.2)
dy, ■ ■' = ~ 2 an7i* dt J
If (x,,...,^), (y1 »...»yn ) are any two solutions of the respective systems then clearly (5.8.3)
2 (Xjdyj^ + y ^ i )
= 0
and so (58.4)
I
■ C,
a constant. This is the analogue of (5 .4 ),andcould In fact he deduced from It. If we have a base {y^i,y^(yjM,... ,yjn), for the solutions of the adjoint to (1 .3 ) then ly^l ^ 0 . From (5.8.4) follows also (5.8.5)
2y.^
 Cj,
.
a system of equations of the first degree which may be solved for the x^. The solution thus obtained is in terms of n arbitrary constants, the Cy and is the gen eral solution of (5 *8 . 1 ). We may thus state: (5*9) Given a linear homogeneous system of differ ential equations (5 .8 . 1 ) if we have n linearly indenendeit solutions of the ad joint system (5 *8 .2 ), then the com plete solution of the system (5 .8 .1 ) Itself is reduced to the solution of an algebraic system of n linear equa tions in n unknowns. 8 * The linear homogeneous differential equation of order n. Since (1.4) is reducible t.o a system (1 .3 ) its properties may be deduced from those of (1.3)* For con venience in the applications we shall express them di rectly. Por the sake of expediency we introduce the custom ary operator D « ^ • Then (1.4) takes the form
III. LINEAR SYSTEMS
(6.1 )
IPx + a^tOlP"1* + ... +
or with the conventions D° = i,
(6.2)
57 «• Q,
— 1:
2 alc(t)lP "lcx  o.
Writing x  Xj, the equivalent system (5 .8 .1 ) is:
te 1 (6 .3 )

X2 ^
D^n “ ‘V i
’ ®nix2 " ••• " ai*n*
Thia yields in particular: {S.k)
xk » r ^ x .
The domain of (6.1) or of (6 .3 ) i3 of the form I * V f where Iis a component of the interior of the set of points of the real t line at which all the a^(t) are continuous • Let us look at the question of linear dependence of the solutions of (6 .1 ) at first directly, i.e. without referring to (6 .3 ). A set of solutions f§*(t)J is said to be linearly dependent whenever there can be found real scalars not all zero such that (6 . 5 )
I C 1§1( t )  0 .
When this holds we also have (6.6)k
2 C j E ^ U )  0,
for all k. Of course since by (6.1 ) the are lin early dependent upon ..., iP 1 (6.6) need only be considered for k < n. It is clear that the $*(t) will be linearly dependent whenever (6.6)Q, ..., (6.6)n1 have a solution in the not all zero, and hence certainly if
58
DIFFERENTIAL EQUATIONS
matrix (6 .7 )
II I^ 1 II, (k « o,...,ni; 1  l,2 ,...,r)
has more than n columns. Hence there cannot be more than n linearly Independent solutions. Given on the other hand n solutions a n.a.s.c. for their linear de pendence ia that the detenninant called Wronskian:
A (5
V D?1 • >5 n
)
...*n

We have already shown (3 .3 ) that there are n linearly In dependent solutions and no more. A set of n linearly In dependent solutions f1, ..., £n is known as a base for the solutions. Any other $ Is given by a relation (t)  I c ^ t ) . If we think of (t) as a function (t;c1,... ,cn ) then it is described as a general solution of (6. 1 ). Coupling what precedes with the properties ofthe Wronskian we have: (6 .8 ) A n.a.s.c. in order that {f1,...,!11! be a base for the solutions of (6 .1 ) is that the Wronskian M f 1 >•••>fn ) + 0 for some t° € I, and hence for every t € I. (6 .9 ) Given n linearly independent functions £ 1, .... 5 n each n time differentiable on an interval I: t1 < t < t2, they satisfy the differential equation (6*9*1)
^(^*5
)“ 0
and this equation is unique to within a factor in t. If we replace x by A acquires two identical rows and so vanishes. Hence satisfies (6 .9 .1 )* TJnicity will follow if we can prove more generally:
III. (6.10)
LINEAR SYSTEMS
59
If the two gffliftt.lfma
(6.10.1) iftc + a^t)!?11* + ... + a ^ x  o . (6 .1 0 .2 ) iftc + b1(t)lP1 + ... + bn(t)x  0 are satisfied over the Interval I by the aame set of n linearly Independent functions then they are Identical. For otherwise the difference Is of order < n, not identically zero, yet with n linearly independent solu tions which is ruled out. Therefore (6.10) holds and so does (6 .9 ). A(t)  A(t0 ) exp (  5r* a^tjdt)). t0 In fact by differentiation (6.11)
I1 D*1 •
11 * rP ^ 1
I T 2*1 dV
2
from which to (6.11) is but a step. (6 .1 2 ) It is not difficult to see that In what pre cedes the Wronskian matrix plans the role of the previous matrix X. In fact referring to the system (6 .3 ) associ ated with, and equivalent to (6 .1 ), if we set 1?^(t) xJcl(t), then aci1,...,!11) * n **1^) 11 * x To !*(t) there corresponds now the vector solution of (6 .3 ) gnd clearly the correspondence ?*(t) is one to one and preserves linear dependence. As consequence the properties just obtained for (6.1) be obtained directly from those of (6 .3 ). 7 . We will now discuss ad joints for the type
x*(t) ) a could (6.1).
60
differential equations
First the adjoint of (6 .3 ) is " V n ^ 2 “ _y1 + an~l^n
(7.1 )
"^n1 + alyn If we differentiate the (k+1 )st relation k times and set yn = y we find (7.2)
J^V  D ^ ^ y ) + ... + (1 ) ^ 7 = 0 ,
and we now define (6.1 ), (7.2) as ad .joint to one another. To derive the analogue of (5.8.4) we must solve (7.1 ) for the y^. We first have = ^sfe+ii+ ^ and this yields 7k " J o (_1 )h^ (a**hy)Substituting finally in (5 .8 .U) we obtain as the analogue of (5.8.*0: (7 .3 )
I (1 )hr?c"1xr?1(a11_k.hy) = C. h,k If we possess a base fi^j for the solution of the adjoint equation (7«i) to (6 .1 ) then the general solution x(t) of (6 . 1 ) may be obtained by solving the linear (algebraic) system (7.4)
2 ^ h,k
^^^Sikh1* ^ * ^i
^
for x. (7 .5 ) The relation just obtained for adjoints may also be derived directly by means of integration by parts. For any two functions x(t), y(t) with a suitable number of derivatives we have:
III. LINEAR SYSTUB
61
ylftc  DCyl^x)  E y E ^ x I^yl^kx  DO^yl^"11"1*) ^ ' ’y . x  DCD^'Vek)  & 7 • x . From this follows y i£ x
♦(  1
faah
 d K  i ) hr? V i^ "h ' 1x .
Replacing k by nk and y by a^y we find: J . l 7 6>
. 2,...,n)
where n is the number of parameters x^. One may however dissociate the problem altogether from conditions of dif ferentiability by defining a position of equilibrium as an extremum of V. This definition will be ample for our purpose. Now Lagrange formulated the following theorem first proved (later) by Dirichlet: (1.4) Whenever in a certain position of the system V is. miniTmim then equilibrium in that position is stable. The appropriate definition of stability Is: (1.5) Corresponding to any t > 0 there is an ij > o such that if II x  + II x 1 II < >) at the beginning of the movement, it ramalng < £ ever after. This property will now be proved. (1.6) For convenience we assume that the minimum of V occurs at jt  0 and that V(o) ** o. Since V Is con tinuous, there is a f ) 0 such that ln^/(0,f>) «■»/(^) :
74
DIFFERENTIAL EQUATIONS
IIx II < p, we have V(5c) > o,save at 3c «■ o where V(o) o. Consider now any t < p. Since V is positive on the compact set B : 0 x II «■ £, it will have a positive lower bound p. on the set.Similarly T is continuous and posi tive on the compact product of the sets H II ^ £, lit1 II « £, and so it has a positive lower bound V on the pro duct . Take now Z to an initial position (x°, 5?'0)where at all events x° € »/(£), and denote by TQ,V0 the corres ponding values of T,V. Then (1.7)
T  T0 + VQ  V.
Now T0 and T0 + V0 are 2 0 and vanish only when x° *■ x '0 ■* o. Since they are continuous there is an < £• such that IIx°  + II x'° II < ^ — *T0 + VQ and T0 < inf (p»v). By (1.7) then V < /t, T < V, hence I x II + II x* II < 2£. This proves Lagrange's theorem. (1.8) It may be observed that differentiability plays no role whatever in Dirichlet's proof. Indeed^the proof does not even make use of the fact that 5?' ■= it could be a vector unrelated to it. Nor did the finite ness of the dimension of the spaces of x,x' enter into the argument. Assuming now that we are dealing with a system Z of a finite degree of freedom, and that T,V possess all the first partials as to their variables, the motion of the system is governed by Lagrange's equations, direct derivatives from Newton's laws: M ol
d_ ,82_, . 3P dt
av_ 3x± *
Generally T is a quadratic in the and so (1.9) is equivalent to a system with constant field. The sys tem will have the solution ? «■ x' o which is a critical point, and stability in the sense of (1 .5 ) asserts an ’evident property regarding the characteristics passing near the singular point. This property may be formulated
IV. CRITICAL POINTS AND PERIODIC MOTIONS for more general systems, and so we may expect a stab ility notion for differential equations. Our next object will be to give its explicit definition. 2. (2 .1 ) We consider then our usual real system (22)
£  p(5?,t)
under the assumptions of the theorem of existence (II, 4.1), save that it is assumed here that the domain of (2.2) in the sense of (II, 10) is of the form L * R, where L is the real t line and R a connected open subset of the basic vector space V . Let K be a family of tra jectories, rQ : x°(t) a trajectory of K • (2 .3 ) Definition. The trajectory rQ is said to be stable relatively to the family X of trajectories. when ever given any t° and any £ > 0 there is an ij > 0 and a time r such that if x \ t } € Jfand Kx°(t°)  x1(t°)I < tj, then for every t > T we will have ll(x°(t)  x1(t)B < £. The stability is said to be of asymptotic type whenever II x°(t)  x^'tt) I —* o when t —* +oo and this for every reX . In a somewhat less precise way one may say that if at time t° the point of any trajectory r of X passes close enough to the corresponding point of rQ, then for every t > r it will remain quite close to the corres ponding point of rQ. Since the solution 5c(t) is a continuous function of the initial point if we replace t° by t1 and rQ has the property just described relative to t° then it will still have it relative to t1. Thus the property just described is in fact independent of the initial time. (2.4) Definition. The trajectory rQ is said to be unstable if the only class relative to which it is stable consists of rQ itself. (2.4.1) A direct characterization is as follows: given §uay £ and any trajectory x*1(t) such that
76
DIFFERENTIAL EQUATIONS
II (xft0)?1(t°) I < £, then for some t1 > t° we will* have  (3?(t1)x' (t •)  > £. (2 .5 ) Definition. The trajectory rQ is said to be absolutely stable. or more simply stable (of asymptotic type or otheiwise) whenever it is stable (of asymptotic type or otherwise) relative to all classes Xj i*e. when ever it is stable (of asymptotic type or otherwise) rel ative to the class of all trajectories. (2 .6 ) Definition. A trajectory rQ which is neither absolutely stable nor unstable is said to be condition ally stable. Such a trajectory is stable with respect to some partial class X, but not with respect to the class of all trajectories. Examples of each type will be considered later. (2 .7 ) As a special case of the preceding situation a critical point x° would be a point of equilibrium of the system. The corresponding definitions are: (2 .8 ) The critical point M is stable relative to K whenever given t° and any €. > 0 there is an > o* and a r such that if x1(t) € K and x ^ t 0) is in the sphere «/(M9yj) then for t >r, x^t) remains in*/(M,£). If this holds for no family X other than M, the point is unstable: while if it holds for the family of all tra jectories M is stable. If M is neither stable nor un stable it is conditionally stable. Stability is defined as being of asymptotic type whenever  x'(t)  — * 0 when t — ► +0 0 . (2 .9 ) Positive and negative stability. The pre ceding definitions refer to the behavior of the.traject ories for t ) t. To underscore the fact one may prefix the terms "stable, ... * with "positive". Thus one may have critical points which are positively unstable, etc. The same questions may be raised regarding the behavior of the trajectories for t < V, and they will lead to the concepts of negative stability, ....
IV. CRITICAL POINTS AND PERIODIC MOTICKS
77
• (s.io) An example. Consider a real homogeneous system with constant coefficients ( 2. 10. 1)
 Ax
and suppose the characteristic roots of A distinct and none with zero real parts. The transformation of coor dinates for the reduction of A to Its real canonical .form will replace (2.10.1) by a similar system satisfied by the real points
A.t _ A,t Art x ■ (C^e » C^6 » ...» Cr » At Cpe
A , ^2r+ie
.t
A _t 1
^ne
Since —> 0 for t — > +00 when the real part of A la negative, and Ce^—■► +00 for t'—* +00 when the real part la positive we can assert this: Let k be the number of A^ with negative real parts. Making all the constants Cj except those corresponding to the A^ with negative real parts equal to zero, there is obtained a kfold family of characteristics stable as to the origin. The family is clearly maximal. Hence we have 2 conditional stability of asymptotic type for k < n; absolute stability of asymptotic type for k « n; instability for k  0.
78
DIFFERENTIAL EQUATIONS
As we shall see later, this type of behavior Is quite general. 3* (3*1) Uniformly regular transformations. In discussing stability It is sometimes convenient to have transformations which do not affect the stability pro perties at the origin. Such a type Is Introduced here. Consider a mapping T : y = T(x,t) of Vx into which sends the origin of V into the origin of V and * y is topological between two fixed neighborhoods, U of the origin in and V of the origin In within which we will operate throughout. We assume the usual properties, notably that the partial derivatives 3fi exist wherever they are needed and that the Jacobian J
3fl
j ^ o for x € U and all
t >t, where r ia fixed once for all. (3 .2 ) Definition. The mapping T is said to be uni formly regular _ at — the — origin whenever given any ^ p > 0 there is a corresponding such that fl x I ^ p and t > r — ♦ II Tx I ^ p1 and likewise for T*1. We prove: (3 .3 ) A mapping which is. uniformly regular at the origin preserves the stability properties in both direc tions. It is sufficient to prove that stability asymptotic orotherwise in implies the same in Consider first stability. Denote by j/(£), (£) the spheres IIx II < £, IIy I 0 there, is an ij > 0 such that if x € kf(rj) at any time t >tthen x €*/(£) at any ulterior time. Now corresponding to our £ and t* there is an £ 1 such that 0 < £, < I T5? I for II1 1 ^ £ and all t > t. Hence for all t > t the original T1y of any point "j € (£1) lies In */(£). By the assumption of stability In there Is an ■»], such that If y € (^ ) at any time t > t then y
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remains in (£1) ever after. Clearly T~1 )c j^(£). By our restriction on the mapping there is an tj > o such that  y  ^ yjy — I T’V I > >7 for all t > r. Hence f) (tj^ )°j/ (*>])• If x eJ(yf) then Tx € t/1(£1) for every t > r and hence x € ^(£) for every t > t .Thus we have stability ±n~1/x and so (3 .3 ) holds as regards stability. Suppose we have now asymptotic stability in 2£. To prove that we have it also in we must show that if £ — »0 then there is an £^ (£) — * 0 with £ such that !&/(£) t. If this is false then there is an a ) 0 such that Tj/(£)