Dynamical chaos in the integrable Toda chain induced by time discretization

Carlo Danieli; Emil A Yuzbashyan; Boris L Altshuler; Aniket Patra; Sergej Flach

doi:10.1063/5.0171261

Dynamical chaos in the integrable Toda chain induced by time discretization

Chaos. 2024 Mar 1;34(3):033107. doi: 10.1063/5.0171261.

Authors

Carlo Danieli¹, Emil A Yuzbashyan², Boris L Altshuler³, Aniket Patra⁴, Sergej Flach^{4

5}

Affiliations

¹ Physics Department, Sapienza University of Rome, Piazzale Aldo Moro 5, Rome 00185, Italy.
² Department of Physics and Astronomy, Center for Materials Theory, Rutgers University, Piscataway, New Jersey 08854, USA.
³ Physics Department, Columbia University, New York, New York 10027, USA.
⁴ Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon 34126, Republic of Korea.
⁵ Basic Science Program, Korea University of Science and Technology (UST), Daejeon 34113, Republic of Korea.

PMID: 38437872
DOI: 10.1063/5.0171261

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 τ and measure the Lyapunov time TΛ (inverse of the largest Lyapunov exponent Λ). A key observation is that TΛ is finite whenever τ is finite but diverges when τ→0. We compare the Toda chain results with the nonintegrable Fermi-Pasta-Ulam-Tsingou chain dynamics. In addition, we observe a breakdown of the simulations at times TB≫TΛ 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.