Resonant Lyapunov families of periodic motions of reversible systems

Abstract

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 ±λj there corresponds a symmetric Lyapunov family Lj, provided there is no resonance λj + pλk = 0 (p ε N). The scenario of the disappearance of the family Lk as ɛ → 0 (where ɛ is the resonance detuning) is investigated. It is shown that resonant symmetric Lyapunov families LRα arise and constructive conditions are obtained for the existence of LRα for both ɛ = 0 and ɛ ≠ 0. When p = 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(ε) 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 LRα.