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Regular and Chaotic Dynamics

, Volume 20, Issue 6, pp 627–648 | Cite as

On the stability of periodic Hamiltonian systems with one degree of freedom in the case of degeneracy

  • Boris S. BardinEmail author
  • Victor Lanchares
Article

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 

MSC2010 numbers

34D20 37C75 37J4 

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Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Department of Theoretical Mechanics, Faculty of Applied Mathematics and Physics Moscow Aviation InstituteMoscowRussia
  2. 2.Departamento de Matemáticas y Computación, CIMEUniversidad de La RiojaLogroñoSpain

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