Multi-Type Instability Processes of Periodic Orbits in Nonlinear Chains

TL;DR

This paper investigates the instability mechanisms of periodic orbits in nonlinear chains, identifying multiple bifurcation types, a universal scaling law for instability time, and phenomena affecting thermalization in many-body systems.

Contribution

It uncovers the coexistence of three bifurcation types in nonlinear chains and establishes a universal scaling law for instability time, advancing understanding of relaxation dynamics.

Findings

Identified three bifurcation types driving instability.

Discovered a universal scaling law for instability time.

Observed double instability phenomena affecting thermalization.

Abstract

Nonlinear normal modes are periodic orbits that survive in nonlinear many-body Hamiltonian systems, and their instability is crucial for relaxation dynamics. Here, we study the instability process of the $\pi/3$-mode in the Fermi-Pasta-Ulam-Tsingou-$\alpha$ chain with fixed boundary conditions. We find that three types of bifurcations -- period-doubling, tangent, and Hopf -- coexist in this system, each driving instability at specific reduced wave-number $\tilde{k}$. Our analysis reveals a universal scaling law for the instability time $\mathcal{T} \propto (\lambda - \lambda_{\rm c})^{-1/2}$, independent of bifurcation types and models, where the critical perturbation strength $\lambda_{\rm c}$ scales as $\lambda_{\rm c} \propto (\tilde{k} - \tilde{k}_{\rm c})$, with $\tilde{k}_{\rm c}$ varying across bifurcations. We also observe a double instability phenomenon for certain system sizes, meaning that larger perturbations do not always lead to faster thermalization. These results provide new insights into the relaxation and thermalization dynamics in many-body systems.