  Nonlinear normal modes are periodic orbits that survive in nonlinear chains,
whose instability plays a crucial role in the dynamics of many-body Hamiltonian
systems toward thermalization. Here we focus on how the stability of nonlinear
modes depends on the perturbation strength and the system size to observe
whether they have the same behavior in different models. To this end, as
illustrating examples, the instability dynamics of the ${N}/{2}$ mode in both
the Fermi-Pasta-Ulam-Tsingou (FPUT) -$\alpha$ and -$\beta$ chains under fixed
boundary conditions are studied systematically. Applying the Floquet theory, we
show that for both models the stability time $T$ as a function of the
perturbation strength $\lambda$ follows the same behavior; i.e.,
$T\propto(\lambda-\lambda_c)^{-\frac{1}{2}}$, where $\lambda_c$ is the
instability threshold. The dependence of $\lambda_c$ on $N$ is also obtained.
The results of $T$ and $\lambda_c$ agree well with those obtained by the direct
molecular dynamics simulations. Finally, the effect of instability dynamics on
the thermalization properties of a system is briefly discussed.
