Homoclinics and Heteroclinics for a Class of Conservative Singular Hamiltonian Systems

✍ Scribed by Paolo Caldiroli; Louis Jeanjean

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 584 KB
• Volume: 136
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1996.3230

✦ Synopsis

We consider an autonomous Hamiltonian system u +{V(u)=0 where the potential V : R 2 "[!] Ä R has a strict global maximum at the origin and a singularity at some point !{0. Under some compactness conditions on V at infinity and around the singularity ! we study the existence of homoclinic orbits to 0 winding around !. We use a sufficient, and in some sense necessary, geometrical condition (V) on V to prove the existence of infinitely many homoclinics, each one being characterized by a distinct winding number around !. Moreover, under the condition (V) there exists a minimal non contractible periodic orbit uÄ and we establish the existence of a heteroclinic orbit from 0 to uÄ . This connecting orbit is obtained as the limit in the C 1 loc topology of a sequence of homoclinics with a winding number larger and larger.

1997 Academic Press where the potential V has a strict global maximum at the origin and a singularity at some point !{0. More precisely on V we assume that (V1) V # C 1, 1 (R 2 "[!], R) with ! # R 2 "[0]; article no.