This article is cited in 8 papers

On the Stability of Periodic Hamiltonian Systems with One Degree of Freedom in the Case of Degeneracy

Boris S. Bardin^a, Victor Lanchares<sup>b

a</sup> Department of Theoretical Mechanics, Faculty of Applied Mathematics and Physics, Moscow Aviation Institute, Volokolamskoe sh. 4, Moscow, 125993 Russia
^b Departamento de Matemáticas y Computación, CIME, Universidad de La Rioja, 26004 Logroño, Spain

Abstract: We deal with the stability problem of an equilibrium position of a periodic Hamiltonian system with one degree of freedom. We suppose the Hamiltonian is analytic in a small neighborhood of the equilibrium position, and the characteristic exponents of the linearized system have zero real part, i.e., a nonlinear analysis is necessary to study the stability in the sense of Lyapunov. In general, the stability character of the equilibrium depends on nonzero terms of the lowest order $N$ $(N>2)$ in the Hamiltonian normal form, and the stability problem can be solved by using known criteria.
We study the so-called degenerate cases, when terms of order higher than $N$ must be taken into account to solve the stability problem. For such degenerate cases, we establish general conditions for stability and instability. Besides, we apply these results to obtain new stability criteria for the cases of degeneracy, which appear in the presence of first, second, third and fourth order resonances.

Keywords: Hamiltonian systems, Lyapunov stability, stability theory, normal forms, KAM theory, Chetaev's function, resonance.

MSC: 34D20, 37C75, 37J4

Received: 08.09.2015
Accepted: 05.10.2015

Language: English

DOI: 10.1134/S1560354715060015

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