Multi-Type Instability Processes of Periodic Orbits in Nonlinear Chains

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.