Dynamical chaos in the integrable Toda chain induced by time discretization

TL;DR

Numerical time discretization of the integrable Toda chain induces chaos and breaks integrability, with chaos strength depending on the time step, and leads to simulation breakdown due to numerical instabilities.

Contribution

This study reveals how symplectic integrators induce chaos in the Toda chain, an integrable system, and analyzes the mechanisms behind simulation breakdowns.

Findings

Finite Lyapunov time for finite time step $ au$

Chaos strength increases with larger $ au$

Breakdown occurs due to numerical instabilities at large positions and momenta

Abstract

We use the Toda chain model to demonstrate that numerical simulation of integrable Hamiltonian dynamics using time discretization destroys integrability and induces dynamical chaos. Specifically, we integrate this model with various symplectic integrators parametrized by the time step $\tau$ and measure the Lyapunov time $T_{\Lambda}$ (inverse of the largest Lyapunov exponent $\Lambda$). A key observation is that $T_{\Lambda}$ is finite whenever $\tau$ is finite but diverges when $\tau \rightarrow 0$. We compare the Toda chain results with the nonitegrable Fermi-Pasta-Ulam-Tsingou chain dynamics. In addition, we observe a breakdown of the simulations at times $T_B \gg T_{\Lambda}$ due to certain positions and momenta becoming extremely large (``Not a Number''). This phenomenon originates from the periodic driving introduced by symplectic integrators and we also identify the concrete mechanism of the breakdown in the case of the Toda chain.