Stability of a One-degree-of-freedom Canonical System in the Case of Zero Quadratic and Cubic Part of a Hamiltonian

Boris S. Bardin^ab, Víctor Lanchares<sup>c

a</sup> Computer Modelling Laboratory, Department of Mechanics and Control of Machines, Mechanical Engineering Research Institute of the Russian Academy of Sciences, M.Kharitonyevskiy per. 4, Moscow, 101990 Russia
^b Department of Mechatronic and Theoretical Mechanics, Faculty of Information Technologies and Applied Mathematics, Moscow Aviation Institute (National Research University), Volokolamskoe sh. 4, Moscow, 125993 Russia
^c Departamento de Matemáticas y Computación, CIME, Universidad de La Rioja, 26006 Logroño, Spain

Abstract: We consider the stability of the equilibrium position of a periodic Hamiltonian system with one degree of freedom. It is supposed that the series expansion of the Hamiltonian function, in a small neighborhood of the equilibrium position, does not include terms of second and third degree. Moreover, we focus on a degenerate case, when fourth-degree terms in the Hamiltonian function are not enough to obtain rigorous conclusions on stability or instability. A complete study of the equilibrium stability in the above degenerate case is performed, giving sufficient conditions for instability and stability in the sense of Lyapunov. The above conditions are expressed in the form of inequalities with respect to the coefficients of the Hamiltonian function, normalized up to sixth-degree terms inclusive.

Keywords: Hamiltonian systems, Lyapunov stability, normal forms, KAM theory, case of degeneracy.

MSC: 34D20, 37C75, 37J40

Received: 04.12.2019
Accepted: 26.03.2020

Language: English

DOI: 10.1134/S1560354720030016

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