Contributions to the Theory of Nonlinear Oscillations (AM-36), Volume III 9781400882175

Table of contents :
CONTENTS
Preface
I. A Rotated Vector Approach to the Problem of Stability of Solutions of Pendulum-Type Equations
II. Asymptotically Autonomous Differential Systems
III. Nonlinear Differential Equations Systems
IV. On a Nonlinear Differential Equation Containing a Small Parameter
V. Critical Points at Infinity and Forced Oscillation
VI. On Certain Critical Points of a Differential System in the Plane
VII. On the Total Number of Singular Points and Limit Cycles of a Differential Equations
VIII. Banach Spaces and the Perturbation of Ordinary Differential Equations
IX. A Fixed Point Theorem
X. Perturbation Theorems for Nonlinear Ordinary Differential Equations
XI. A Note on the Existence of Periodic Solutions of Differential Equations
XII. An Invariant Surface Theorem for a Nondegenerate System
XIII. An Application of Periodic Surfaces (Solution of a Small Divisor Problem)
XIV. Repeating Solutions for a Degenerate System
XV. Bounds for Periods of Periodic Solutions
XVI. One-Dimensional Repeating Curves in the Nondegenerate Case

Citation preview

ANNALS OF MATHEMATICS STUDIES

Number 36

ANNALS OF MATHEMATICS STUDIES

1. 3. 6.

Edited by Emil Artin and Marston Morse Algebraic Theory of Numbers, by H W Consistency of the Continuum Hypothesis, by K G The Calculi of Lambda-Conversion, by A C Finite Dimensional Vector Spaces, by P R. H Topics in Topology, by S L Introduction to Nonlinear Mechanics, by N. K and erm an n

eyl

urt

lonzo

od el

h urch

7. aul alm os 10. olo m on e fsc h e tz 11. ryloff N. B o g o l iu b o f f 15. Topological Methods in the Theory of Functions of a Complex Variable,

by

M a r s t o n M o r se

Transcendental Numbers, by C a r l L u d w ig S ie g e l Probleme General de la Stabilite du Mouvement, by M. A. L ia p o u n o f f Fourier Transforms, by S. B o c h n e r and K. C h a n d r a s e k h a r a n Contributions to the Theory of Nonlinear Oscillatioiis, Vol. I, edited by S. L e f s c h e t z 21. Functional Operators, Vol. I, by J oh n v o n N e u m a n n 22. Functional Operators, Vol. II, by J o h n v o n N e u m a n n 23. Existence Theorems in Partial Differential Equations,by D o r o t h y L.

16. 17. 19. 20.

B e r n s t e in

24. Contributions to the Theory of Games, Vol. I, edited by H. W. K u h n and A. W. T u c k e r 25. Contributions to Fourier Analysis, edited by A. Z y g m u n d , W. T r a n s u e , M. M o r s e , A. P. C a l d e r o n , and S. B o c h n e r 26. A Theory of Cross-Spaces, by R o b e r t S c h a t t e n 27. Isoperimetric Inequalities ki Mathematical Physics, by G. P o l y a and G. S ze g o 28. Contributions to the Theory of Games, Vol. II, edited by H. K u h n and A. W. T u c k e r 29. Contributions to the Theory of Nonlinear Oscillations, Vol. II, edited by S. L e f s c h e t z 30. Contributions to the Theory of Riemann Surfaces, edited by L. A h l f o r s

et al.

31. Order-Preserving Maps and Integration Processes, by E d w a r d J. M c S h a n e 32. Curvature and Betti Numbers, by K. Y a n o and S. B o c h n e r 33. Contributions to the Theory of Partial Differential Equations, edited by L. B e r s , S. B o c h n e r , and F. J o h n 34. Automata Studies, edited by C . E. S h a n n o n and J. M c C a r t h y 35. Surface Area, by L a m b e r t o C e s a r i 36. Contributions to the Theory of Nonlinear Oscillations, Vol. Ill, edited by S. L e f s c h e t z 37. Lectures on the Theory of Games, by H a r o l d W. K u h n . In press 38. Linear Inequalities and Related Systems, edited by H. W. K u h n and A. W. T u c k e r . In p re s s 39. Contributions to the Theory of Games, Vol. Ill, edited by M. D r e s h e r and A. W. T u c k e r . In p re s s

CONTRIBUTIONS TO THE THEORY OF NONLINEAR OSCILLATIONS VOLUME I I I SEIFER T G. SEIFERT

F. HAAS F. HAAS

L. MARKUS

G.G.HUFFORD HUFFORD

E. PINNEY PINNEY W. W. T. T. KYNER KYNER V. B. HAAS

S. P. D ILIBERTO

R. E. GOMORY

M. D. D. MARCUS MARCUS M.

S. BAROCIO

P. KOOSIS P. KOOSIS

EDITED BY S. LEFSCHETZ

PRINCETON PRINCETON UNIVERSITY PRESS

1956

Copyright © 1956 , by Princeton University Press London: Geoffrey Cumberlege Oxford University Press All Rights Reserved L. C. Card 50 -2^00

This research was supported in part by the Office of Naval Research. Reproduction, translation, publication, use and disposal in whole or in part by or for the United States Government is permitted.

Printed in the United States of America

PREFACE The papers in this third volume of Contributions fall under two distinct categories. The second beginning with the paper by Hufford, to the end of the collection, consists of contributions by Hufford, Diliberto and students of the latter, essentially on biperiodic systems. This part of the volume offers a certain degree of homogeneity and may be left to speak for itself. The first category is more heterogeneous. In the paper by Seifert there is discussed an equation arising in the study of the oscillations of a synchronous motor about its average velocity. L. Markus considers the equation x = f(x; t) (x, f are n-vectors) where f(x; t)-- ► f(x) as initial and limiting systems.

t -- ► + «> and compares

Pinney attacks the problem of "practical11 determination of os­ cillations in n-th order systems. The paper by Violet Haas deals with the general problem of ap­ proach of a solution of ex + f(t, x, x, e) = 0 as

e -- ► o to a solution of f(t, x, x, o) = 0 .

Gomory's paper is a contribution (the first so far) to the re­ lation between the critical points at infinity of a system x = X(x), (x, X

are 2 -vectors)

and the periodic solutions of x = X(x) + E(t) where

E

is a periodic vector.

He also analyzes the behavior in the large v

vi

PREFACE

of the solutions of x + f(x)x + g (x) = 0 where f and g are polynomials with degree f > degree g, and applies this to periodic solutions of the same system with a periodic right handside. Barocio fully analyzes the critical point at the origin of a system x = X(x, y),

y = Y(x, y)

where X, Y are analytic and have first degree terms whose coefficient matrix, although not zero, does have both characteristic roots zero. Felix Haas studies equations on a two-dimensional manifold with vector not necessarily a gradient, and obtains relations extending those of Morse for the gradient case. These relations involve not only the numbers of critical points but also the numbers of limit-cycles of various types. A number of the papers of this collection have been written under sponsorship by certain governmental agencies. The nature of the agency and the contract number are indicated in each case.

Solomon Lefschetz

CONTENTS Page Preface By Solomon Lefschetz

I.

II. III. IV.

V. VI.

VII.

VIII.

IX. X. x.

XI.

XII.

XIII.

XIV. XV.

XVI.

v

A Rotated Vector Approach to the Problem of Stability of Solutions of Pendulum-Type Equations By George Seifert

1

Asymptotically Autonomous Differential Systems By L. Markus

17

Nonlinear Differential Equations Systems By Edmund Pinney

31

On a Nonlinear Differential Equation Containing a Small Parameter By Violet B. Haas Critical Points at Infinity and Forced Oscillation By Ralph E. Gomory

57 85

On Certain Critical Points of a Differential System in the Plane By Samuel Barocio

127

On the Total Number of Singular Points and Limit Cycles of a Differential Equations By Felix Haas

137

Banach Spaces and the Perturbation of Ordinary Differential Equations By George Hufford

173

A Fixed Point Theorem By "Walter T. Kyner

197

Perturbation Theorems for Nonlinear Ordinary Differential Equations By S. P. Diliberto and G. Hufford

207

A Note on the Existence of Periodic Solutions of Differential Equations By S. P. Diliberto and M. D. Marcus

237

An Invariant Surface Theorem for a Nondegenerate System By Marvin D. Marcus

2^3

An Application of Periodic Surfaces (Solution of a Small Divisor Problem) By Stephen P. Diliberto

257

Repeating Solutions for a Degenerate System By Marvin D. Marcus

261

Bounds for Periods of Periodic Solutions By Stephen P. Diliberto

269

One-Dimensional Repeating Curves in the Nondegenerate Case By Paul Koosis

277

I.

A ROTATED VECTOR APPROACH TO THE PROBLEM OP STABILITY OP SOLUTIONS OF PENDULUM-TYPE EQUATIONS George Seifert

We consider the differential equation

(i 1)

+ f(e, a)— = g(fl) dt

dt where (i)

f(e, a) is defined and has continuous second partial derivatives for a > 0, - °° < 6 < *>, and in this domain

f(e, and (ii)

f(e + 2 jt,

a)

a)

= f(e,

>

0, H

> 0,

a).

g(e) has continuous second derivatives everywhere, g(e) = g(0 + 2 n ), g(e) has zeros for each of which g'(ej_) 4 °> and

J

2 jt

g(e)de > o.

o

This equation, of the so-called pendulum-type, arises in the study of the oscillations of a synchronous motor about its average angular d .s velocity [1]. If it has a solution e(t, a ) such that if 3£-(t, a ) y = y(e, or) is periodic in e with period 2 it, then e(t, a ) will be called a periodic solution of the second kind of Eq. (i). With respect to the performance of the synchronous motor such a solution would correspond to a continuous slipping of the rotor; that is, asynchronous performance. It is known [3 ] that for fixed oe if such a solution exists, it is unique and its time derivative is non-negative, while if no such solution

I.

A ROTATED VECTOR APPROACH TO THE PROBLEM OP STABILITY OP SOLUTIONS OF PENDULUM-TYPE EQUATIONS George Seifert

We consider the differential equation

(i 1)

+ f(e, a)— = g(fl) dt

dt where (i)

f(e, a) is defined and has continuous second partial derivatives for a > 0, - °° < 6 < *>, and in this domain

f(e, and (ii)

f(e + 2 jt,

a)

a)

= f(e,

>

0, H

> 0,

a).

g(e) has continuous second derivatives everywhere, g(e) = g(0 + 2 n ), g(e) has zeros for each of which g'(ej_) 4 °> and

J

2 jt

g(e)de > o.

o

This equation, of the so-called pendulum-type, arises in the study of the oscillations of a synchronous motor about its average angular d .s velocity [1]. If it has a solution e(t, a ) such that if 3£-(t, a ) y = y(e, or) is periodic in e with period 2 it, then e(t, a ) will be called a periodic solution of the second kind of Eq. (i). With respect to the performance of the synchronous motor such a solution would correspond to a continuous slipping of the rotor; that is, asynchronous performance. It is known [3 ] that for fixed oe if such a solution exists, it is unique and its time derivative is non-negative, while if no such solution

PENDULUM-TYPE EQUATIONS

3

closed phase trajectories become applicable. In view of the fact that Amerio’s work [3 ] to which we will frequently refer, employs the usual car­ tesian phase plane, we will suggest the use of cylindrical phase space in an auxiliary capacity only. Part 1:

The General Case

We shall assume in this case that the set P2 is non-void and be­ gin by proving two non-intersection theorems for integral curves of Eq. (2 ) corresponding to distinct values of oc. THEOREM 1.1. Let < a , and y(e, ) and y(e, oc2 ) be solutions of Eq. (2 ) positive on a < e < b such that y(a, a^) > y(a, a?2 ). Then y(e, oc^) > y(e, a2 ) for a < e < b. PROOF. Suppose c to be thesmallest value of e in (a, b) such that y(c, a1 ) = y (c, a2). From Eq. (2 ), y f (c, a1) - y !(c, oc2 ) = f(c, a2 ) f (c, a1 ) > 0; i.e., y f(c, oc} ) >y'(c, of2 ). Since y(e, cr1) and y(e, a2 ) have continuous derivatives for a < e < b, this is impossible. THEOREM 1.2. Let oc1 < c*2 and y(e, ) and y(e, a2) be solutions of Eq.(2 ) positive on a < 0 < b. Then the in­ tegral curves of y(e, ) and y(e, a2 ) cannot be tan­ gent nor intersect more than once in this interval. PROOF. If at 0 = c, a < c < b, we have y 1(c, oc^ ) = y ’(c, ag ) and y(c, or1 ) = y(c, of2 ), an impossibility. Hence the integral curves cannot be tangent. If there are two points of intersection, then at one of these two points, say e = g ^, we must have y'Cc-j, a1) < y ’Cc^ or ), which implies f(c.,, oc^ ) > f(c^ a?2), a contradiction. This completes the proof. The next theorem shows that each integral curve of a P2 function has a neighborhood, each point of which contains an integral curve correspond­ ing to another P2 function, and that inthis neighborhood the P2 curves vary analytically with oc. We shall need the following LEMMA. Each P2 solutionof Eq. (2 )is stable; i.e., given e Q and yQ(0> oc) e P2, there exists an e > 0 such that if ly(eo> oc) - y0(e0, oc) | < e where j ( e , oc) is a solution of Eq. (2 ), then lim |y(e, oc) - J n ( e , or)| = 0. e — ► 00 u PROOF.Let z ( 6, a ) = y(e, oc) - yQ( e , o c ) . Thenfor ||| known equation for this first variation | is

small, the

well-

4

SEIFERT s'(e , a ) = -

g (e)

5 ( e , or)

J 0 ( s , a )

from which we obtain

g(e) = j'q (Q> a)y0(e, oc) + f(e, a)y0 (e* cc), we have that

But since

Hence

1(0, a) ---> o for

e --

THEOREM 1-5* Let J n (d> of ) € P and 0 be fixed; then there exists an €Q > o such that to each e, Ie| < eQ , there corresponds an a = a(e) and a y(0, a(e)) e ?2 for which j ( e Q , cc(e)) = y ( 0 Q ,a Q ) + €, the or(e) being differentiable for |e| < €. PROOF. We shall prove the theorem only for lower neighborhoods. The corresponding proof for upper neighborhoods follows along the same lines and will be omitted. a > ocQ

y(0, y(0, h(0, able

a,e) = J Q (eQ > « 0 ) € > 0. Define h(0, a, e) = J Q (e, ccQ ) a, €) and note that for a - ocQ and e sufficiently small, cc, e) is defined and hence differentiable with respect to each va r i ­ for 6 q < e < e2 = eQ + 2n. In what follows, these restrictions on

a

and

and denote by

the solution of Eq. (2 ) for

Let

y(0, cc, e)

€ are presumed.

We show first that

-J— h ( e 2 , a,

c)> 0.

a ) S iiJ— + f ( e , a ).

y0(e>a0^ Differentiating with respect to differentiation on the left,

n '(e, a, €) =

y

(2 )

By Eq.

-

h'(e, a, e) = — siii---- f(e,

(3)

which

y(e^a>e) a,

■ g (e)

we have, after changing the order of

e)

n (e, a ,

e) + ^

(e, a )

da

PENDULUM-TYPE EQUATIONS where r}(e, a , e) = -J^h(e, a > £)- Multiplying Eq. (3 ) by the usual in­ tegrating factor, and integrating from e to e 20 we obtain

i)(02>

a>

e) exp j^- J

e?r2

G(e, a, e )de

il(0O,

e)

eo (*: e2

=J

If exp [ " J

e^dt

de

eo O

where G(0 , a, €) = g ( e ) / j (0 , oc, e). Since by Theorem i.l y(0 , oc, e ) < j Q( e , a Q) for 0Q oc, e ) = o, we have that t) ( e 2, oc, e ) = -Jjj- h(02, oc, e) > 0 . Now for fixed > ocQ we have h(e2, c^, 0 ) - h(e2,aQ, o) = h(02, a.,, o) = h(e2, I, o)](a1 - a0), where | = £(a.,, QrQ) is such that aQ < | < . Put -J— h(02, I, o) = MQ(|) and note that MQ(|) > o. Since h(02, oc^, e ) is continuous in e, there exists an eQ > 0 such that eQ < M0 (|)(a1 - oc )/2 and for € < e Q, (5)

Mn(l) h(02, oc^, e) > ---- (a1 - ocQ) > e Q > e .

If, now, eQ is sufficiently small, we also have for € < eQ, h(02, ocQ, e ) J 0 ( e 2> « 0 ] " * 0> £) < e > ^ n o t , h(e, a , e) would not tend to o for e -- > oo as is required by the lemma. (On the phase cylinder, the curve of y(0 , oc , e) must move up after one circuit in the positive 0 direction in order that it approach the closed curve of yQ(0> a0) f°r e -- >oo.)Hence for €q sufficiently small corresponding to each e, o < € < €q, there exists an a = a(e) > a Q such that h(02, o;(s), e) = €. This shows that y(e, o c ( e ) , e ) e Pg. That this o c ( e ) is differentiable for o < € < eQ is a consequence of the implicit function theorem. This completes the proof. Prom this last theorem we see that L is open; the next theorem shows that the set L is an interval, not necessarily finite, and thatfor fixed e, y(0 , a) e P2 is a strictly decreasing function of oc for oc e L. THEOREM 1.1*. If y(0 , ocQ) e Pg, then for any oc, o < a < ocQ, there exists a y(0 , oc) > y ( 0 , o?0) such that y(0, a) e P2; i.e., if aQ e L, then the in­ terval 0 < oc < ocQ is contained in L.

6

SEIFERT

PROOF. Fix 0 = gq and oc < aQ, and consider the solution Y0 ( e > oc) of Eq. (2) such that YQ( 0Q, oc) = y(0Q, ceQ) . Since f(0Q, oc) < f(0Q, ocQ) we have Y'0 ( e 0> °c) > y !( e Q, ocQ). Hence, by Theorem 1 .2 , we must have yo^eo + 2tc' ^ > + 2jt> ao} = ? ( e o ’ ao^ = yo^0' From this we con" elude that there exists a y(0, a) € P2 (Cf. [2 ], [3 ]); and since the curves of P2 functions cannot intersect by Theorem 1.2, y(0, or) > y(e, a ) This proves the theorem. This theorem shows that the function or(e) determined in the proof of Theorem 1 . 3 is strictly decreasing; hence has an inverse e(a) which is also strictly increasing and differentiable and may be defined by the equation: e ( a ) = j ( e Q, oc) - j ( o Q> ocQ), for y(e, oc) and y(e, ocQ) each of class P2 and \oc - or01 < SQ, b Q sufficiently small. Consider now the function e*(a) = j ( Q Q> o c ) - j ( e Q, oc^ ) for y(0, oc) and y(0, oc^ ) of class P2; € L is fixed and or e L is the variable. Suppose at oc = ocQ e L, €*(a) is not differentiable. By Theorem 1 . 3 e(oc) = j ( 0 Q, oc) - y(eQ> ocQ) is differentiable with respect to oc at oc = ocQ; but e*(a) = ty(e0> °c) - y(0Q, ocQ)] + [y(0Q, ocQ) y(0Q,)] = e(a) + C where C is independent of a, and we conclude thate*(oO is differentiable on L. We now consider the setL*. Clearly, if L is the interval 0 < oc < °o,then since L C L*,we have L = L*, and everything we have proved for thesets L and P2 holds also for the sets L* and P*. If y e P

Suppose now L is bounded, let ocQ be its least upper bound. then for each 0, lim y(0, or) = y*(0) exists. a - > a c-

Suppose first y*(0) > 0 for all0 . Consider the solution 7 0 ( e , a Q) of Eq. (2 ) such that Yq (Qq > ocq ) = y*(0Q). If J * ( 9 Q + a) yo(0Q + a, ocQ) = e > 0 for some positive a < 2it, then there exists an oc € L so close to ocQ such that if y(0, oc) e P2 and j ( e 0> oc) = yQ, the solution y1(0 , ocQ) for which y1(0Q, ocQ) = j Q will be such that y 1(0Q + a, c*c)- J o ( 0Q + a, ocQ) < ~ , and that y(0Q + a, a ) y 1(0Q + a, ac)< |- . Hencey*(eQ + a) - J q ( oq + a, ocQ) < y(0Q + a, a) y_(0_ 0 0 + a, a_) c < e, a contradiction. Suppose now that for some positive b < 2*, yo(0o + ccQ) - y*(0Q + b) > e1. This is also impossible since we can find a y(0, oc) e P2 with a e L so close to occ that y(0, oc) must intersect Y0 (°> ac) at some point 0 = 0Q + c,.0 < c < b; this is impossible by Theorem 1 .1 . ocQ € L,

Hence we can only conclude that J Q( d > ocQ) = y*(0); which is a contradiction, since L is open. Thus we see that y*(0) > 0 is impossible; i.e.,

i.e., y*(0)

must

PENDULUM-TYPE EQUATIONS

7

have zeros. Since the slope of solutions of Eq. (2 ) on the line y = 0 is 00 or undefined except at the saddle points, the zeros of y*(0) must clearly be found among these saddle points. If we denote one of the zeros of y*(e) by then the solution yo(0> ac) which ocQ) = 0 can be shown to be coincident with y*(e) in a manner similar to that of the previous case. Hence e L*, and y*(e) e P*. Suppose now that for some > ocQ, oc1 e L*. Then the corre­ sponding y(0, ) e P* must also have a zero at the saddle point 9 = 6^. The slope of the integral curve of y(0, ) approaches the value

-f (e 1,a , ) - /f2 (e1,a1 )+it-g' (e. )

m, = -------g------

as 0 -- » from the left. Since this is less than the slope of y*(0 ) = yo(0> &c ) defined above, it follows that for values of 0 < 9 ]f y (0 , a1) must pass above J Q( e > ac )’ Hence there exists a solution y (0 , a ) e P2 whose curve is intersected by y (0 , or1) twice. This is im­ possible by Theorem 1 .2 . Hence an oc.1 > ocn c such that oc.1 e L* is impossible. We can now assert that if L is bounded, then L* simply con­ sists of L plus its least upper bound oc ; also that in this case P* consists of P2 plus a periodic solution of Eq. (2 ) which has zeros. The next theorem gives an expression fory (^0 , o oc) for y(e, oc) € P2. 9 = 9Q,

THEOREM 1 .5 . Let y(0, a) e P2. Then for y(e,of)E(e,of) (6) Sa *

where

/ J0

u'

6 2 = 6^ + 2it,

E( e ,

a

f (0,of)d0

E ( e 2, a )_1

and

a ) = exp

[J

dt

PROOF. If we put t](0 , a ) = y(^> ®), differentiate Eq. (2 ) with respect to oc, and interchange the order of differentiation on the left, we obtain

8

SEIFERT ^'(e, a) = -- i ) ( e , a) - — y (e,a)

( e , a). da

Multiplying this equation by the usual integrating factor, and integrating from e^ to we obtain 2

de 1 - n(e , or) y (e,a) J 0

i)(e , a) exp f f L (7)

g(t) dt L e y 2 (t,a)

e„ Since

ir(ep, a) = T](eo, a),

de.

and since, as before, e

/

K ( t J _ d t = lo g . >. + y2 (t,a y(e0>a)

we obtain, solving Eq. (7) for

r|(eQ, a) = ^

— y ( e o > « ) = ---------- ------

y ( s 0, a )

da

I

f

dt; y(t,a)

y ( S Q, a);

y(e,a)E(e,a) ^

f(e,ct) de

E(e„,a)-1

from which Eq. (6) follows easily. COROLLARY.

Let y(e, a) € Pg, M1(a) = max [g(e)/f(e, a)], M^(a) = max |

f (e, oc) | ,

and E1 = exp

/

f (e, a) de

;

then

(8)

> o lntM. (af)M_(a)E.(a) A y 2(0, a ) > ---- 1--- J --- ]--da E1(or)-1

PROOF. We show first that if y(e, a) e P2, then y(e, a) < M^a). Con­ sider the graph of y = g(e)/f(e, a) in the phase plane. In the region R1 of the phase plane bounded by this graph and in the 9 axis, the phase

PENDULUM-TYPE EQUATIONS

9

trajectories have positive slope; on the graph the slopes are zero, and elsewhere the slopes are negative. Let 0m be a value of Q at which = M-jCa) and suppose y(0 , oc) e P2 is such that y(0m, oc) > U^(oc). Then clearly 7(0m “ 2jt> > which is a contradiction. This shows that y(0 , oc) < M 1 (or). To complete the proof we note that

/

y(e, of)E(e, or) — f ( e , a)de < 2jtE(e2, a)M1(or)M2(of) dor

and that since E(02, a ) > E 1 (oc),

we have

E(e2 ,a)

E -,((*)

E(e2 ,a) - 1

E 1 (or)-!

------------- -------------- .

Using the above inequalities in Eq. (6 ) we obtain (8 ) which completes the proof. The usefulness of inequality (8 ) in obtaining information on values of oc for which oc € L clearly depends on the availability of a lower bound on y2 (0 , ocQ) - y2 (e, oc} ) for some fixed 0 , ocQ < oc^, ocQ and being members of L. Since this lower bound seems rather difficult to obtain, the special case f(0 , oc) = af(e) in which an inequality of type (8 ) from which dependence on y(e, oc) e P2 can be eliminated is of inter­ est. We also note that the equation of the synchronous motor given by Edgerton and Fourmarier [1 ] contains a damping function of the type of this special case.

Part 2 : The Special Case

f(0 , oc) = af(0 )

We first note that the conditions previously imposed on f(0, oc) imply that if f(0 , oc) = orf(e), 'then f(e) has a continuous first deriva­ tive everywhere, is of period 2 k , and has a positive lower bound. The equation of the phase trajectories is now (2 a)

1 (e, a)

We also note that for or sufficiently small, a y(0 , oc) e P2 [2 ]; i.e., the class P2 is in this case non-empty. Let us replace

f(0 , oc)

by

may exist

arf(0 ) in Eq. (6 ) and consider the

10

SEIFERT

numerator of the fraction on the right.

Since

E' (e, oc) = gf (e) E(e, a ) j(e,oc) where E(e, a )

is defined as in Theorem 1 .5 , we have

S2 J

e2 y(e> or)f(e)E(e, a) de = ^

e„o (9)

P [E(e , a ) - i ] y (e ,a) = ---------------- 5— ------------^

y 2 (e, a)E'(e, a) de

J* e°o e n

y 1 (e, a)E(e, a) de.

J eo

But by Eq. (2a),

S'

J

y 2 |(e, a) = 2 tg(e) - cef(e)y(e, a ) ] ;

/2VZ' ( e> «)E(e,

a) de = |r

J

eo

'2

hence

g(e)E(e, a) de

eo e2

2

J

y(e, a)f(e)E(e, a)

Substituting this in Eq. (9 ) and solving for *2

J

y ( e , a)f(e)E(e, a) de

eo we have e„ J

y (e , a)f (e)E(e, a) de

E(e„,a)-i ------- ^ ----------y ^ e , a)

eo (1 0 ) +^ oc

Using Eq. (1 0 ) in Eq. (6) we obtain

J e2 2 J*g(e)E(e, a) de. 6^

PENDULUM- TYPE EQUATIONS

11

We now consider the Integral ^2

0

g(e)E(e, a) de = J

f o

E(e, a) d [ /

g(t) dt |

o

( 12)

A

= E(©2 , a)

J*

A

2

g(e) d 0 -

A

O

g(e) de -

^

2

J

0

g(t) dtde.

E ‘(0 , a)

0o

0

The critical points of the function *0

®(e) = /

g(t) dt

are clearly the zeros e^, 0^ < 0i+1> of g(e) and are independent of the choice of 0.Denote by e2k+i' k = 0, + 1, + 2, ..., the values of 0 at which ®(e) has a relative minimum. We now prove the LEMMA. There exists an odd integer m all n = m + 2k, k = 1, 2, 3, ..., for n < 0 m + 2 it we have

such that for which

0n

J

g(e) de = ®m (0n ) > ° m

PROOF. Assume the lemma false; that is, suppose for each odd integer there exists an odd integer > m for which 0 ^ < 0m + 2* and

m

®

0 ) < °. Starting with m = i = n , we determine an n 1 for which .in . (0v, l < 0 where n^o < en,1 < 0^ n + 2*. Consider the interval \ (n o I\ n-1 /I en > en + 2jtj. Clearly, 0n + 2n is contained in it. Now correspond1

I

1

ing to

n1

.

O

there exists an

n2

in this interval for which

(en

V

] < °*

If % =en + 2lt' we have \ ( \ ) + \ ( \ ) = 4n ( % ) 55 \ ( \ + 2 ") < °' a contradiction. Hence suppose e J 0 + 2*. Next, consider the inter/ \ 2 o val [0 , 0 + 2jt). It must clearly contain 0 + 2it and a point of \

the form n~

2

0n

2

such that

p - i or

2,

I

+ 2*p, p = 1 0 3

or 2.

1

Now corresponding to

n2

is in this interval and $ [0 ) < 0. / \ / 2) 3> t \

we would have

+ *n .(enJ

/3

If \

0 = 0+

+ \ ( en3) = $n(en3) =

\ ( \ + 2*P) < °> whlle ^

en 3 = %

$n (sn j = $n (en

ea°h of these being impossible, we assume

+ 2n)
o, and 2n is the number of zeros of g(e) in e1 < e < e 1 +2*, then the lemma may be formulated as follows: there exists a cyclic permutation of the numbers (c.,, c2, ..., cn ) -- ^(c^, c2!, ..., cn ’) such that k

c^1 >0

for

k = 1, 2,

n.

i= 1 (This was pointed out to the author and proved by Professors L. K. Jackson and H. Ribeiro.) As in the proof of the lemma we may for convenience denote by 01, the em assured us, and since $(e) has a relative maximum at the zero immediately preceding e ^, we have, setting e = e }, that

J

e g(t) dt =

J

e g(t) dt > o

1 for eQ < e < eQ + 2k = e2 where than e1.

e Q is the greatest zero of g(e)

less

This shows that if e Q and e are chosen in the manner indi­ cated above, the last integral in Eq. (1 2 ) is positive and we have

PENDULUM-TYPE EQUATIONS e2

J

e2

a)

g(e)E(e, a )de < E(e2 ,

13

J

S2

J'

g(e)de -

91

9o (1 3 ) 1

61

e2 = E(e2, a)

J

g(e) de

e,

J

g(e)de - (E(e2, a) - 1 ) o

g(e) de. o

Since g(e) < o for e Q < e < - o c f ( e ) , and integrating from

wehave, using Eq. (2 a), to e ^, we obtain

eQ

J'

y(e-,> a) - y ( e Q, a) < - a

y ’(e, oc)
a

e1 f(e)de

J*

+

y ( e 1? a)

>

a

eo for

J

f ( e ) de eo

y (e , a) € P2 Using Inequalities (1 3 ) and (1 ^) in Eq. (1 1 ) we obtain ei

i ^

y 2 ( 0 > a) > aPi 2 + 1

s_ (°) de

[ / O

(1 5 )

E(e2 ,or) g re- ; , a ) - i

J^

] d 0 J

where ei

F, =

J

f ( e ) de. eo Q1

Define E 1 (a) = exp

^

J

f ( e ) deJ o

where M = max[g(Q)/f(e)], and note that as in the proof of the corollary of Theorem 1 .5 , j ( o > oc) < M/a provided y ( e , oc) e P2. Hence

E 1 (a) > E(e2, a),

and we have

SEIFERT

E(e ,a) E,(a) ---- S---- < _J----- =H(a) E(e2,a)-1 E1(a)-i

(16)

Thus (1 5 ) becomes 0 7)

^ - A y 2(v

where H(a)

> aPi2 - 1 [Gi + H