The goal of this paper is to investigate the stability of a dynamical regime corresponding to the vibrational mode with the shortest wavelength (known as the π-mode) in the Toda lattice with a cubic perturbation of the original potential.

Methods. The study is based on the standard Floquet method. The variational system for the corresponding dynamical regime is decomposed into a set of independent two-dimensional subsystems. This allows us to determine the π-mode stability for a chain with an arbitrary number of particles. The decomposition is carried out both by a general group-theoretic approach and by a new method proposed in this work, which is based on the discrete Fourier transform.

Results. The resulting stability diagrams provide information about the stability of the regime for various oscillation amplitudes and numbers of particles. A correspondence between the perturbed Toda lattice and the Fermi-Pasta-Ulam-Tsingou model is established for large magnitude of the perturbation. For the original (unperturbed) Toda lattice, it is observed that its integrals of motion are functionally dependent in the vicinity of the considered dynamical regime. Therefore, the observed trajectory does not satisfy the conditions of Poincares theorem, which states that the Floquet multipliers of fully integrable systems are equal to one. Despite this fact, the considered regime in the original Toda lattice is shown to be stable for any number of particles and any oscillational amplitude.