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Zbl 0749.58022
Mielke, A.; Holmes, P.; O'Reilly, O.
Cascades of homoclinic orbits to, and chaos near, a Hamiltonian saddle- center.
(English)
[J] J. Dyn. Differ. Equations 4, No.1, 95-126 (1992). ISSN 1040-7294; ISSN 1572-9222/e

The Hamiltonian two degrees of freedom system under investigation in this paper is modeled by $$H={1\over 2}\omega({p\sb 1}\sp 2+{q\sb 1}\sp 2)+{1\over 2}\lambda({p\sb 2}\sp 2-{q\sb 2}\sp 2)+\alpha {q\sb 1}\sp 3+\beta {q\sb 1}\sp 2q\sb 2+\gamma q\sb 1{q\sb 2}\sp 2+\delta {q\sb 2}\sp 3.$$ Assuming $\delta\ne 0$ and $\omega\lambda > 0$, rescaling permits to take $\lambda=1$ and $\delta={1\over 3}$. 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.
[W.Sarlet (Gent)]
MSC 2000:
*37J99 Finite-dimensional Hamiltonian etc. systems
37G99 Bifurcation theory
34C25 Periodic solutions of ODE

Keywords: homoclinic bifurcations; reversible Hamiltonian systems; saddle-centre; horseshoe

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