Dynamical chaos in the integrable Toda chain induced by time discretization

Area of Science:

• Computational Physics
• Dynamical Systems Theory
• Numerical Analysis

Background:

• Integrable Hamiltonian systems possess conserved quantities and predictable long-term behavior.
• Numerical simulations are essential for studying complex dynamical systems but can introduce artifacts.
• The Toda chain is a classic model of integrable Hamiltonian dynamics.

Purpose of the Study:

• To investigate how time discretization in numerical simulations affects the integrability of Hamiltonian systems.
• To quantify the onset of dynamical chaos induced by numerical methods.
• To analyze the breakdown mechanisms in long-term simulations of the Toda chain.

Main Methods:

• Utilized the Toda chain model for simulations.
• Employed various symplectic integrators with a time step τ.
• Measured the Lyapunov time (TΛ) as the inverse of the largest Lyapunov exponent (Λ).
• Compared results with the nonintegrable Fermi-Pasta-Ulam-Tsingou chain.

Main Results:

• Time discretization with finite τ destroys integrability and induces chaos in the Toda chain.
• Lyapunov time TΛ is finite for finite τ and diverges as τ approaches 0.
• Simulations exhibit breakdown (NaN) at times TB ≫ TΛ due to large positions/momenta.
• Identified periodic driving from symplectic integrators as the cause of simulation breakdown.

Conclusions:

• Numerical integration of integrable Hamiltonian dynamics can lead to the emergence of chaos.
• The choice of time step in numerical integrators critically impacts the preservation of integrability.
• Symplectic integrators, while advantageous, can introduce artifacts leading to simulation instability and breakdown.