Stability of equilibrium solutions in the critical case of even-order resonance in periodic Hamiltonian systems with one degree of freedom

• Published in: Celestial mechanics and dynamical astronomy Vol. 116; no. 3; pp. 265 - 277
• Main Authors: Mansilla, José E., Vidal, Claudio
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.07.2013; Springer Nature B.V
• Subjects: Aerospace Technology and Astronautics; Astrophysics and Astroparticles; Classical Mechanics; Dynamical Systems and Ergodic Theory; Geophysics/Geodesy; Original Article; Physics; Physics and Astronomy; Autonomous system; Equilibrium solution; Lyapunov’s Theorem; 34A25; Resonances; 37C75; 34D20; Normal form; Hamiltonian system; Type of Stability; Periodic system
• ISSN: 0923-2958; 1572-9478
• DOI: 10.1007/s10569-013-9488-5

The stability in the Lyapunov sense of an equilibrium position in a periodic Hamiltonian system with one degree of freedom is studied. It is assumed that the equilibrium is stable in the first approximation and that there exists an even resonance of order k , arbitrary. The critical case is considered, i.e., when the system of parameters is such that, in order to draw rigorous conclusions about the stability of the equilibrium position in the Lypaunov sense, terms or order higher than three in the series expansion of the Hamiltonian function must be taken into account. Sufficient conditions are derived for stability and instability.