Low-energy dynamics in generic potential fields: Hyperbolic periodic orbits and non-ergodicity

2025/10/07 by Alberto Enciso, Enciso, Alberto, Manuel Joaquin Castillo Garzón +3
Physics and Astronomy · #Astro and Planetary Science #Dynamical Systems (math.DS) #FOS: Mathematics #Nuclear physics research studies #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2510.06073

openalex publication_date 2025/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03

Abstract

We prove that, on each low energy level, the natural Hamiltonian system defined by a generic smooth potential on T² exhibits an arbitrarily high number of hyperbolic periodic orbits and a positive-measure set of invariant tori. Hence, quasi-periodic motion and hyperbolic behavior typically coexist in the low-energy dynamics of natural Hamiltonian systems with two degrees of freedom.

Citations

• Non-degeneracy of closed orbits for generic potentials
• Non-integrability and chaos for natural Hamiltonian systems with a random potential

Related

• Quasi-periodic motions in generic nearly-integrable mechanical systems
• Non-autonomous KAM theory for lower dimensional invariant tori (I): Normally elliptic and hyperbolic cases
• Non-degeneracy of closed orbits for generic potentials
• Examples of quasi-hyperbolic dynamical systems with slow decay of correlations
• Non-autonomous KAM theory for lower dimensional invariant tori (II): Normally parabolic case