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Bifurcation and coalescence of a plethora of homoclinic orbits for a Hamiltonian system. (English) Zbl 0854.34047

The authors investgate the set of homoclinic solutions of the reversible Hamiltonian system ${u}^{iv}+P{u}^{\text{'}\text{'}}+u-{u}^{2}=0$. It is shown rigorously that for $P\le -2$ there is a unique homoclinic solution of the system, that solution is given, and on the zero energy surface its orbit coincides with the transverse intersection of the global stable and unstable manifold. At $P=-2$ infinitely many families of homoclinic solutions are created which exist for $P\in \left(-2,-2+\epsilon \right)$ for some $\epsilon >0$. Second the development of the set of symmetric solutions is monitored numerically as $P$ increases from $-2$. It is observed that two branches extend from $P=-2$ to $P=2$. All other symmetric branches are in the form of closed loops with a turning point between $P=-2$ and $P=2$.

Numerically, it is observed that close to each turning point a bifurcation of a branch of nonsymmetrical homoclinic orbits occurs, which can be followed back to $P=-2$. Finally, the observed phenomena are explained in the language of geometric dynamical systems theory. It is shown how the different orbits are related to each other and how the highly complicated order of turning points can be related to the intersection of stable and unstable manifolds. This paper demonstrates nicely, how geometric dynamical systems theory can contribute to the understanding of very complicated dynamic phenomena.

Reviewer: A.Steindl (Wien)
##### MSC:
 34C37 Homoclinic and heteroclinic solutions of ODE 37J99 Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems 34C23 Bifurcation (ODE)
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