Skip to main content

Full text of "DTIC ADA210562: Homoclinic Orbits for a Class of Hamiltonian Systems"

See other formats


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 
Distribution unlimited 


Sponsored by 

U. S. Army Research Office 
P. 0. Box 12211 
Research Triangle Park 
North Carolina 27709 


National Science 
Foundation 

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