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

  1. Bardin, B.S. 1
  2. Lanchares, V. 2
  1. 1 Department of Theoretical Mechanics, Volokolamskoe sh. 4, Moscow, Russian Federation
  2. 2 Universidad de La Rioja
    info

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

Revista:
Regular and Chaotic Dynamics

ISSN: 1560-3547

Año de publicación: 2015

Volumen: 20

Número: 6

Páginas: 627-648

Tipo: Artículo

DOI: 10.1134/S1560354715060015 SCOPUS: 2-s2.0-84948964011 WoS: WOS:000365809000001 GOOGLE SCHOLAR

Otras publicaciones en: Regular and Chaotic Dynamics

Resumen

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. © 2015, Pleiades Publishing, Ltd.