Universality of algebraic laws in Hamiltonian systems

TL;DR

This paper proposes that Hamiltonian systems universally exhibit algebraic laws with exponents 3/2 for recurrence decay and superdiffusion, supported by analytical results and simulations across various systems.

Contribution

It introduces the conjecture of universal algebraic exponents in Hamiltonian systems and provides analytical and simulation evidence for these laws.

Findings

Universal exponents $oldsymbol{eta=3/2}$ and $oldsymbol{\gamma=3/2}$ for unbounded phase space.

Interval $oldsymbol{3/2 ext{ to } 3}$ for bounded phase space with trapping.

Supporting simulations and experiments reinforce the universality of these algebraic laws.

Abstract

Hamiltonian mixed systems with unbounded phase space are typically characterized by two asymptotic algebraic laws: decay of recurrence time statistics ($\gamma$) and superdiffusion ($\beta$). We conjecture the universal exponents $\gamma=\beta=3/2$ for trapping of trajectories to regular islands based on our analytical results for a wide class of area-preserving maps. For Hamiltonian mixed systems with bounded phase space the interval $3/2\leq\gamma_{b}\leq3$ was obtained, given that trapping takes place. A number of simulations and experiments by other authors give additional support to our claims.