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Cascades of homoclinic orbits to, and chaos near, a Hamiltonian saddle- center. (English) Zbl 0749.58022

The Hamiltonian two degrees of freedom system under investigation in this paper is modeled by

$H=\frac{1}{2}\omega \left({{p}_{1}}^{2}+{{q}_{1}}^{2}\right)+\frac{1}{2}\lambda \left({{p}_{2}}^{2}-{{q}_{2}}^{2}\right)+\alpha {{q}_{1}}^{3}+\beta {{q}_{1}}^{2}{q}_{2}+\gamma {q}_{1}{{q}_{2}}^{2}+\delta {{q}_{2}}^{3}·$

Assuming $\delta \ne 0$ and $\omega \lambda >0$, rescaling permits to take $\lambda =1$ and $\delta =\frac{1}{3}$. The system has a saddle-centre at the origin and, for $\gamma =0$, a homoclinic solution. Considering $\gamma$ as a small perturbation parameter, the authors first study homoclinic bifurcations on the zero energy surface. A Poincaré map is constructed as the composition of a Shil’nikov-type map and a global map, obtained via an excursion near the homoclinic solution. The reversibility of the system plays a crucial role in many of the subtle arguments and in fact makes the bifurcation problem in the end a codimension one phenomenon. It is shown that for each $n\ge 2$, there is a sequence of values for $\gamma$ (tending to zero), for which there are $n$-homoclinic orbits and that doubling sequences of $2n$-homoclinic values converge to each $n$-homoclinic value. Further, under some generic conditions, the existence of horseshoes is established, implying the existence of sets of $n$-periodic orbits and chaotic orbits. Among the applications, discussed in the final section, we find the Hénon-Heiles Hamiltonian, the orthogonal double pendulum and the plain restricted three-body problem.

Reviewer: W.Sarlet (Gent)
##### MSC:
 37J99 Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems 37G99 Local and nonlocal bifurcation theory 34C25 Periodic solutions of ODE
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