ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

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Kedrov A. A., Shcherbinin S. A. Investigation of the stability of the oscillatory zone boundary mode in the perturbed one-dimensional Toda lattice. Izvestiya VUZ. Applied Nonlinear Dynamics, 2026, vol. 34, iss. 2, pp. 206-222. DOI: 10.18500/0869-6632-003202, EDN: NRVWDG

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Language: 
Russian
Heading: 
Bifurcation in Dynamical Systems. Deterministic Chaos. Quantum Chaos.
Article type: 
Article
UDC: 
530.182
DOI: 
10.18500/0869-6632-003202
EDN: 
NRVWDG

Investigation of the stability of the oscillatory zone boundary mode in the perturbed one-dimensional Toda lattice

Autors: 
Kedrov Alexander Alekseevich, Peter the Great St. Petersburg Polytechnic University
Shcherbinin Stepan Aleksandrovich, Peter the Great St. Petersburg Polytechnic University
Abstract: 

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 Poincare’s 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.

Conclusion. We have investigated the stability of the zone boundary mode (π-mode) in the Toda lattice with a cubic perturbation in the interaction potential. The study has been carried out for the decomposed variational system consisting of independent two-dimensional subsystems. The independent subsystems are obtained by the general group-theoretic method. In addition, a new decomposition method is proposed based on the discrete Fourier transform. The proposed approach can be further applied to investigate the stability of any nonlinear regimes possessing temporal and spatial periodicity.

Key words: 
nonlinear dynamics
Toda lattice
stability
zone boundary mode
group-theoretics methods
Acknowledgments: 
The results of the project “FR-2025-75”, carried out within the framework of the Basic Research Program at HSE University in 2025, are presented in this work.
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Received: 
18.09.2025
Accepted: 
19.11.2025
Available online: 
19.11.2025
Published: 
31.03.2026
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