DTIC ADA210562: Homoclinic Orbits for a Class of Hamiltonian Systems

AD-A210 562

^CMS Technical Summary Report #89-36

HOMOCLINIC ORBITS FOR A CLASS
OF HAMILTONIAN SYSTEMS

Paul H. Rabinowitz

UNIVERSITY
OF WISCONSIN

CENTER FOR THE
MATHEMATICAL
SCIENCES

Center for the Mathematical Sciences
University of Wisconsin—Madison
610 Wainut Street
Madison, Wisconsin 53705

June 1989

(Received June 7, 1989)

DTIC

ELECTE

JUL281989

B

Approved for public release
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U. S. Army Research Office
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National Science
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Washington, DC 20550

Office of Naval Research
Department of the Navy
Arlington, VA 22217

UNIVERSITY OF WISCONSIN - MADISON
CENTER FOR THE MATHEMATICAL SCIENCES

HOMOCLINIC ORBITS FOR A CLASSS OF HAMILTONIAN SYSTEMS

Paul H. Rabinowitz

CMS Technical Summary Report ^^89-36
June 1989

ABSTRACT

We establish the existence of a honioclinic solution of the Hamiltonian system

(*) 9-hVi(i,Q) = 0

assuming that the potential V is T periodic in t, grows more rapidly thzm quadratically as
jgf'=-»-oo and satisfies some other technical conditions. The homoclinic solution is obtained
as the limit of subharmonic solutions of (*). The subharmonic solutions are found using a
miniraax argument. ' -

/

AMS (MOS) Subject Classifications: 34C25, 35J60, 58E05, 58F05, 58F22.

Key Words: Hamiltonian system, homoclinic solution, subharmonic, minimax.

Supported in part by the U. S. Army Research Office under Contract No. DAAL03-87-K-
C043, the National Science Foundation under Grant No. MCS-8110556, and the Office of
Naval Research under Grant No. N00014-88-K-0134.

Homoclinic orbits for a class of Hamiltonian systems
Paul H. Rabinowitz

Introduction

There is a large literature on the use of variational methods to prove the existence
of periodic solutions of Hzuniltonian systems. However it is only relatively recently that
these methods have been applied to the existence of homoclinic or heteroclinic orbits of
Hamiltonian systems. See [1-5]. Such orbits have been studied since the time of Poincare
but mainly by perturbation methods.

Our goal in this paper is to prove the existence of homoclinic orbits for the second
order Hamiltonian system:

(HS) q + V,{t,q )=0

where q € R” and V satisfies

(Vi) V(t, 9 ) = —l/ 2 {L{t)q,q) +W{t,q) where L is a continuous T-periodic matrix valued
function and VV € C^(R x R^jR) is T-periodic in f,

(V 2 ) L(t) is positive definite symmetric for all t € [0,r],

(V 3 ) there is a constant /* > 2 such that

0 <(iW{t,q)<{q,W^{t,q))

for all q € R"\{0}, and

(V 4 ) W{t,q) = o{\q\) as g — ♦ 0 uniformly for i€[0,T’].

Note that (Vj) — {V 4 ) imply that q{t) = 0 is a “trivial” homoclinic orbit of (HS). We
will prove:

Theorem 1: If V satisfies (^ 1 ) — (V 4 ), (•^*5') possesses a nontrivial homoclinic solution,
q(t) emanating from 0 such that q € W^’^(R,R’*).

Our study of (ITS) was motivated by a recent paper of Coti-Zelati and Ekelamd [1]
which treated the first order Hamiltonian system.

(2) i = JS.(l,z) J=(,^'’o)

where z € R^". The function H € C^(R x R2'*,R) is T periodic in t with H{t,z] =

lf 2 {Az, z) + R{t, z) where ^4 is a constant symmetric matrix such that JA has no eigenval- -

ues with 0 real part and R is strictly convex, satisfies V 3 (with q € R^"), and |i2(t,z)| < ^

K\z\>* for all z € R^" for some K > 0. The convexity of R leads to a “dual” variational Tial

2

□ □

formulation of ( 2 ) which Coti-Zelati and Ekeland then study. Using a concentration com¬
pactness argument in the sense of P. L. Lions [ 6 ], they are able to apply a variant of the
Mountain Pass Theorem to find a homoclinic orbit of (2).

Recently Hofer and Wysocki [5] have generalized the results of [l| by dropping the
convexity conditions on H but also requiring that \Rz{t,z)\ < for all z €

They use a rather different argument than [l| based on the study of certain first order
elliptic systems that have also been useful in recent work on symplectic geometry. See e.g.

[ 7 ].

Our approach to (HS) differs from both of the above. We will find the homoclinic
solution q as the limit, as fc -+ oo, of 2 kT periodic solutions, Qk- The approximating
solutions are obtained via the Mountain Pass Theorem. Then appropriate estimates for
the critical values, cjt associated with qk and on qk allow us to pass to a limit to get q.
The details will be carried out in § 1 .

§1. Proof of Theorem 2

For each A: € N, let Ek = W 2 ^^^{R,R'‘), the Hilbert space of 2kT periodic functions
on R with values in R'* under the norm

fkT

(/ {i«(oi’+

J-kT

To exploit the form of {HS) and (V 2 ), it is more convenient to work with the equivalent
norm

(3) lklll = Ol«‘)l’ + (i(<)«W.«W)l*

Set

.kT ,

( 4 ) h{q]= [■^\q?-y{t,q)\dt

J-kT ^

= / w{t,q)dt.

^ J-kT

Then € C^(£’jfc,R) and satisfies the Palais-Smale condition. See e.g. [ 8 , Theorem 2.61].
Moreover critical points of Ik in Ek are classical 2 kT periodic solutions of (HS). We will
obtain a critical point of Ik by using a standard version of the Mountain Pass Theorem.
Since the minimax characterization it provides for the critical value is important for what
follows, we state the result precisely. Let Bp{Q) denote an open ball of radixis p about 0.

Proposition 5 [9]: Let E he 3. real Banach space and I £ C^(£^,R) satisfy the Palais-
Smale condition. If further 1(0) =s 0,

3

(/i) There exist constants p,a > 0 such that

I

dB,(0)

and

( 72 ) There exists e 6 JS?\£p(0) such that 7(e) < 0,
then 7 possesses a critical value c > a given by

( 6 )

where

c = inf max 7jfc(p(s))
j€r-€(o.ii

( 7 ) r = {5€C([0,l],i;)lp(0)=0 and p(l) = c}
For our setting, clearly 7 fc( 0 ) = 0 . Moreover by (V 3 ),

(8) for 0<|f|<l

>>»'('. iei>i

It is easy to see that there are constants 0 kilk >0 such that

(9) /^fclklUfi-fcT.JkT] < IklU-i-itr.ikri ^'TfclklU

for all q € Ek- Therefore if [kH* < 7^^, IklU* ^ 1

/ feT rkT /f\

Since fi > 2, (10) shows

fkT

(11) / W^(t,9(t))dt = o(lkllfc) as 9-^0 in Ek

J-kT

4

and

( 12 )

*(«) = i|kllI + o(WII)

as q 0 In Ek‘ Hence Ik satisfies (/i) of Proposition 5.
constants oi, aj > 0 such that

1

(13) h(M < - “M"

^ J-kT

Moreover by (8) again, there are

\ip\^dt + a2

for all /? € R and € Ek\{0}. Now (13) shows (/ 2 ) holds with c = Cjt, a sufficiently large
multiple of any <p € £'fc\{0}. Consequently by Proposition 5, Ik possesses a positive critical
value Ck given by (6) and (7) with E = Ek and P = P*. Let qk denote the corresponding
critical point of I on Ek- Note that 0 since Cfc > 0.

The next step in the proof is to obtain k independent estimates for Ck and qk- Let
<p € Ei\{0} such that

(14) (i) <p{±T) = 0 and (u) Ii{<p) < 0

(By (13), if i} satisfies (i), any sufficiently large multiple (p of V’ satisfies (i) and (ii)). Define

(15) ek{t) = <p{t), \t\ < T

= 0 r < |f| < kT.

Then by (12), cfc € and Ik{ek) = Li(«i) < 0- Note also that gk{s) = scfc 6 Pfc for

all Ar 6 N and /fc(^fc(5)) = Ii{gi{s)). Therefore by (6),

(16) Cfc < max 7i(^i(«)) = M

•€( 0 , 1 )

independently of k.

The estimate (16) leads to a priori bounds for qk- Since /'(gfc) = 0, by (V 2 )>

(17) Cfc = Ikiqk) - ^I'kMqk

J-kT ^

rkT

>(x-i)/ w(t,n)dt.

2 JkT

Hence (4) and (17) yield a k-independent bound for ||gfc|lfc:

/-*r 2

(18) \\qk\\l = 2ck+ W{t,qk)dt < (2 + —^)Af = Mi

J-kT

5

A k-dependent bound on ||9fc||i,«[-fcr,A:Tl n.ow follows from (9). Moreover a better estimate
can be obtained as follows: For q€ Ek and t,T € [—kT^kT],

(19) \q{t)\ < \q{T)\ + \ q{s)ds\.

Integrating (19) over ~ + f] shows

( 20 )

yt+i yt+i yt

k(t)l ^ J j :f(0l‘^^ + / , ly

t—— t—J r

Jt-i

Hence

(21) IklU-l-fcT.AiTl $ ‘^alklU
where az depends on L. Now (18) and (21) imply

(22) ll9*IU<«(-A:r,*r) < = Mz

with Mz independent of k. Finally (HS) provides bounds for qk in C^[—kT,kT\ indepen¬
dent of fc.

The bounds just obtained for qk together with (HS) and (18) show a subsequence
of qk converges in C/J,g(R,R'‘) to a solution q of (HS) satisfying

(23) f [lqp + (Lg,g)l<it <oo

J —oo

It remains to show that g ^ 0 and is a homoclinic solution of (HS). By (23),

(24)

/ llil' + - 0

J |t|>m

as m —*• oo.

Hence by (20), g(t) -+ 0 as t —► ±oo. If

rm+l

(25)

/ -*• 0

J m

as m ±oo, (20) with g replacing g and (24) imply q(t) —^ 0 as t —»• ±oo. To verify (25),
by (HS), (Vi), and (24), it suffices to show

r m+l

( 26 )

-t 0

as m —±c». Since Wq{t,(S) = 0 and it has already been shown that q[t) —^ 0 as t —>• ±oo,
(26) follows.

Lastly we must show that g ^ 0 . Taking the inner product of (HS) with qk and
integrating by parts gives:

(27)

/ kT

(

-kT

{qk,Wq[t,qk))dt.

Set y( 0 ) =0 and for s > 0,

(28)

y(s)

max

teio.ri
l«l< -S

le

Then by (V 3 ) and ( 8 ), it is easy to check that Y € C(R"*‘,R'^), Y'(s) > 0 if s > 0, y is
monotone nondecreasing, and y(s) 00 as s —♦ 00. The definition of Y shows

(29)

--< yill%IU«[-ir,*ri)

for all t € [—kT,kT]. Hence by (27) and (29),

rkT

(30) \\qk\\k<Y{\\qk\\L’>o[-kT.kT]) \qk\^dt

J —kT

< <X3y([[gfc|[L<»(-fer,fcTl)lkA:||fc-

Since |lgfcl|fc > 0,

(31) i^(llg*IU«(-fcr,*ri) >

Consequently the properties of Y imply there is a /? > 0 (and independent of k) such that

(32) ||gfc||L«>[-fcr,Jkri >

Now to complete the proof, observe that by the T-periodicity of L and VT, whenever
p(t) is a 2kT periodic solution of (HS), so is p{t + jT) for all j € Z. Hence by replacing
qk{t) earlier if necessary by qk{t + jT) for some j € [—k,k\ n Z, it can be assumed that the
maximum of |g*(t)| occurs in [0, T]. Therefore if qk{i) —* 0 in along our subsequence,

(33) IkfclU-l-fcT./tT] = ®

contrary to (32). The proof is complete.

7

Remark 34: If V is independent of t, stronger results can be obtained by more direct
arguments as will be shown in a joint paper with K. Tanaka.

References

[1] Coti-Zelati, V. and 1. Ekeland, A variational approach to homoclinic orbits in Hamil¬
tonian systems, preprint

[2] Rabinowitz, P.H., Periodic and heteroclinic orbits for a periodic Hamiltonian system,
to appezu'. Analyse Nonlineaire.

[3] Benci, V. and F. Giannoni, Homoclinic orbits on compact manifolds, preprint.

[4] Tanaka, K., Homoclinic orbits for a singular second order Hamiltonian system, preprint.

[5] Hofer, H. and K. Wysocki, First order elliptic systems and the existence of homoclinic
orbits in Hamiltonian systems, preprint.

[6] Lions, P.L., The concentration-compactness principle in the calculus of variations,
Revista Matematica Iberoamericana, 1, (1985), 145-201.

[7] Floer, A., H. Hofer, and C. Viterbo, The Weinstein conjecture in P x C^, preprint.

[8] Rabinowitz, P. H., Periodic solutions of Hamiltonian systems. Comm. Pure Appl.
Math. 31, (1978), 36-68.

[9] Ambrosetti, A. and P. H. Rabinowitz, Dual variational methods in critical point theory
and applications, J. Functional Anal. 14, (1973), 349-381.

8