On the multiplicity of periodic orbits and homoclinics near critical energy levels of Hamiltonian systems in $\mathbb{R}^4$
Abstract
We study two-degree-of-freedom Hamiltonian systems. Let us assume that the zero energy level of a real-analytic Hamiltonian function $H:\mathbb{R}^4 \to \mathbb{R}$ contains a saddle-center equilibrium point lying in a strictly convex sphere-like singular subset $S_0\subset H^{-1}(0)$. From previous work [de Paulo-Salomão, Memoirs of the AMS] we know that for any small energy $E>0$, the energy level $H^{-1}(E)$ contains a closed $3$-ball $S_E$ in a neighborhood of $S_0$ admitting a singular foliation called $2-3$ foliation. One of the binding orbits of this singular foliation is the Lyapunoff orbit $P_{2,E}$ contained in the center manifold of the saddle-center. The other binding orbit lies in the interior of $S_E$ and spans a one parameter family of disks transverse to the Hamiltonian vector field. In this article we show that the $2-3$ foliation forces the existence of infinitely many periodic orbits and infinitely many homoclinics to $P_{2,E}$ in $S_E$. Moreover, if the branches of the stable and unstable manifolds of $P_{2,E}$ inside $S_E$ do not coincide then the Hamiltonian flow on $S_E$ has positive topological entropy. We also present applications of these results to some classical Hamiltonian systems.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2017
- DOI:
- 10.48550/arXiv.1712.04720
- arXiv:
- arXiv:1712.04720
- Bibcode:
- 2017arXiv171204720D
- Keywords:
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- Mathematics - Dynamical Systems;
- Mathematics - Symplectic Geometry;
- 37JXX;
- 53DXX
- E-Print:
- 28 pages, 16 figures