Manuel Garzón (ICMAT-CSIC)
The question of whether a Hamiltonian system is typically integrable or chaotic is a central topic in dynamical systems, which traces back to the pioneering works of Poincare in Celestial Mechanics. A satisfactory picture of the typical dynamics of such systems did not emerge until the 1970s, when Markus and Meyer [5] established that a generic (in the Baire category sense) Hamiltonian system on a compact symplectic manifold is neither integrable nor ergodic. On the contrary, the case of natural Hamiltonian systems is much less studied, in spite of its central relevance in mathematical physics. Specifically, a natural Hamiltonian corresponds to the situation in which the symplectic manifold is the cotangent bundle of a manifold M, and the Hamiltonian is given by the sum of a fixed kinetic energy term and a potential field V ∈ C∞(M;R). It is known that a generic potential field on a compact manifold is non-ergodic [4]. Moreover, near the potential
maximum, the system may exhibit positive topological entropy under suitable conditions [1, 2]. Nevertheless, the fundamental question of whether motion at low energy levels is typically integrable or chaotic remains open to date.
This difficulty arises because standard transversality methods are no longer applicable [4], raising the conjecture of whether classical results on generic non-integrability extend to the setting of potential fields. In this talk we shall present the main result of [3] which shows that, on each low energy level, the natural
Hamiltonian system defined by a generic smooth potential V on T2 exhibits an arbitrarily high number of hyperbolic periodic orbits and a positive-measure set of invariant tori. To put this result in perspective, let us recall that hyperbolic periodic orbits are the natural starting point to establish the presence of chaos in dynamical systems.
References
[1] S.V. Bolotin, P.H. Rabinowitz, A variational construction of chaotic trajectories for a Hamiltonian system on a
torus, Boll. Uni. Mat. Ital. 1B (1998) 541–570.
[2] S.V. Bolotin, P.H. Rabinowitz, A variational construction of chaotic trajectories for a reversible Hamiltonian
system, J. Diff. Eqs. 148 (1998) 364–387.
[3] A. Enciso, M. Garz´on, D. Peralta-Salas: Low-energy dynamics in generic potential fields: Hyperbolic periodic
orbits and non-ergodicity. Preprint, 2025
[4] L. Markus, K. R. Meyer. Generic Hamiltonian dynamical systems are not nor ergodic. Proc. Conf. on Nonlinear
Oscillation Kiev, p. 311-332, 1969.
[5] L. Markus, K. R. Meyer: Generic Hamiltonian dynamical systems are neither integrable nor ergodic, Mem. Amer.
Math. Soc. 144, 1974