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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=1 2ω(p 1 2 +q 1 2 )+1 2λ(p 2 2 -q 2 2 )+αq 1 3 +βq 1 2 q 2 +γq 1 q 2 2 +δq 2 3 ·

Assuming δ0 and ωλ>0, rescaling permits to take λ=1 and δ=1 3. The system has a saddle-centre at the origin and, for γ=0, a homoclinic solution. Considering γ 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 n2, there is a sequence of values for γ (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:
37J99Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
37G99Local and nonlocal bifurcation theory
34C25Periodic solutions of ODE
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