Partial Preservation of Frequencies and Floquet Exponents of Invariant Tori in the Reversible KAM Context 2

Abstract

We consider the persistence of smooth families of invariant tori in the reversible context 2 of KAM theory under various weak nondegeneracy conditions via Herman's method. The reversible KAM context 2 refers to the situation where the dimension of the fixed point manifold of the reversing involution is less than half the codimension of the invariant torus in question. The nondegeneracy conditions we employ ensure the preservation of any prescribed subsets of the frequencies of the unperturbed tori and of their Floquet exponents (the eigenvalues of the coefficient matrix of the variational equation along the torus).MSC2010 numbers: 70K43, 70H33

Families of invariant tori in KAM theory: Interplay of integer characteristics

“…Quasi-periodic invariant n-tori for n 2 are not isolated in the product M × P of the phase space and the external parameter space, such tori are generically organized into at least one-parameter Whitney smooth families. In all the cases presented in Tables 1 and 2 except for the reversible context 2, the values s = 0 or s = 1 are enough for the generic existence of families of invariant tori in question [15].…”

“…Such conservation laws and symmetry properties constitute what is called the context of KAM theory. The four best explored KAM contexts are the following ones [4,6,7,8,9,10,11,12,13,14,15,16] (n 0 always denotes the dimension of the quasi-periodic invariant tori under consideration, while s is the number of external parameters µ 1 , . .…”

“…Some essential features of generic smooth (for n = 0, 1) and Whitney smooth Cantor-like (for n 2) families of quasi-periodic invariant n-tori in KAM theory [4,8,9,10,11,12,15] are summarized in Tables 1 and 2. To be more precise, in the functional space of all the families V µ of vector fields within the context in question, there is an open subset of families of vector fields Table 1. Key properties of generic families of invariant n-tori T in various KAM contexts.…”

“…The book [5] presents a brilliant semi-popular introduction to the theory. The genericity approach to KAM theory with external parameters (this note is based on) has been developed mainly in the works [4,6,7,8,9,10,11,12,13,14,15] of the Groningen school of dynamical systems.…”