This is a further study of the set of homoclinic solutions (i.e., nonzero solutions asymptotic to 0 as ¦x¦→∞) of the reversible Hamiltonian systemu iv +Pu″ +u−u 2=0. The present contribution is in three parts. First, rigorously for P≤ −2, it is proved that there is a unique (up to translation) homoclinic solution of the above system, that solution is even, and on the zero-energy surface its orbit coincides with the transverse intersection of the global stable and unstable manifolds. WhenP=−2 the origin is a node on its local stable and unstable manifolds, and whenP∈(−2,2) it is a focus. Therefore we can infer, rigorously, from the discovery by Devaney of a Smale horseshoe in the dynamics on the zero energy set, there are infinitely many distinct infinite families of homoclinic solutions forP∈(−2, −2+ε) for someε>0. Buffoni has shown globally that there are infinitely many homoclinic solutions for allP∈(−2,0], based on a different approach due to Champneys and Toland. Second, numerically, the development of the set of symmetric homoclinic solutions is monitored asP increases fromP=−2. It is observed that two branches extend fromP=−2 toP=+2 where their amplitudes are found to converge to 0 asP ↗ 2. All other symmetric solution branches are in the form of closed loops with a turning point betweenP=−2 andP=+2. Numerically it is observed that each such turning point is accompanied by, though not coincident with, the bifurcation of a branch of nonsymmetrical homoclinic orbits, which can, in turn, be followed back toP=−2. Finally, heuristic explanations of the numerically observed phenomena are offered in the language of geometric dynamical systems theory. One idea involves a natural ordering of homoclinic orbits on the stable and unstable manifolds, given by the Horseshoe dynamics, and goes some way to accounting for the observed order (in terms ofP-values) of the occurrence of turning points. The near-coincidence of turning and asymmetric bifurcation points is explained in terms of the nontransversality of the intersection of the stable and unstable manifolds in the zero energy set on the one hand, and the nontransversality of the intersection of the same manifolds with the symmetric section in ℝ4 on the other. Some conjectures based on present understanding are recorded.
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Buffoni, B., Champneys, A.R. & Toland, J.F. Bifurcation and coalescence of a plethora of homoclinic orbits for a Hamiltonian system. J Dyn Diff Equat 8, 221–279 (1996). https://doi.org/10.1007/BF02218892
- Homoclinic orbits
- Hamiltonian systems
- dynamical systems
- water waves
- elastic struts