Quasi-periodic bifurcations in reversible systems

Abstract

Invariant tori of integrable dynamical systems occur both in the dissipative and in the conservative context, but only in the latter the tori are parametrised by phase space variables. This allows for quasi-periodic bifurcations within a single given system, induced by changes of the normal behaviour of the tori. It turns out that in a non-degenerate reversible system all semi-local bifurcations of co-dimension 1 persist, under small non-integrable perturbations, on large Cantor sets.

Whitney smooth families of invariant tori within the reversible context 2 of KAM theory

“…The meaning of the word "constant" is not made precise explicitly in [25] but it is clear from the text that one has in view vector fields of the form b∂/∂z where b ∈ R 2p is a constant vector (Prof. Hanßmann has confirmed this in a private communication to me), see also [27,28]. Since If ker Q(ω * , 0) ⊂ Fix R, one generically expects a reversible quasi-periodic center-saddle bifurcation in family (3.1) to occur [27]. The Hamiltonian counterpart of this bifurcation scenario has been well studied [52,53].…”

“…It is also possible to study bifurcations of invariant n-tori into invariant (n + d)-tori (1 d p) in G-reversible systems close to Eq. (1.4) (see [27,28] and references therein). Now note that the phase space codimension of an invariant (n + d)-torus T here (0 d p) is equal to m + 2p − d while dim Fix G = m + p. Thus, for involution (1.5) Indeed, nothing prevents one from considering e.g.…”

Non-floquet invariant tori in reversible systems

“…The compatible transformations transform reversible systems into systems reversible with respect to the same involution. During the last 50 years, many authors study the persistence of invariant tori for reversible systems of form (1.1) and obtained many kinds of KAM theorems (see [1,2,3,4,5,6,10,15,16,17,18,21,22,23,24,25,26,27,28,29,30,31,33,35] and the references therein).…”

“…Re 10) for all ξ ∈ C m and some σ > 0. The author proved that all the hyperbolic lower dimensional tori with Diophantine frequency of Hamiltonian systems survive small perturbations.…”

A KAM theorem for the elliptic lower dimensional tori with one normal frequency in reversible systems

“…Reversible systems form a class of special conservative systems with an involution structure. During the last 50 years, many authors study the persistence of invariant tori for reversible systems and obtained many kinds of KAM theorems (see [1,5,6,7,8,9,14,17,19,20,22,24,25,26,27,28,29,30,31,32,33,34,36,38] and the references therein).…”