Resonant Lyapunov families of periodic motions of reversible systems

✍ Scribed by V.N. Tkhai

• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 999 KB
• Volume: 68
• Category: Article
• ISSN: 0021-8928
• DOI: 10.1016/s0021-8928(04)00048-6

✦ Synopsis

Local periodic motions of a reversible system in the neighbourhood of the zero equilibrium position are investigated. In the non-degenerate case, to every pair of pure imaginary roots -+)~j there corresponds a symmetric Lyapunov family Lj, provided there is no resonance )~j + P~k = 0 (p • N). The scenario of the disappearance of the family L/~ as ~ ~ 0 (where e is the resonance detuning) is investigated. It is shown that resonant symmetric Lyapunov families LR~ arise and constructive conditions are obtained for the existence of LRu for both ~ = 0 and ~ # 0. Whenp = 1 the existence of two cycles is observed; the cycles are mutually symmetric about the fixed set of the reversible system and each is distant O('/~) from the origin. For a reversible system written in the form that is standard for oscillation theory, in "amplitude-angle" variables, a general theorem is established according to which symmetric periodic motions exist in the structurally unstable case; the theorem is basic for investigating the families LR~.