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

Abstract

Abstract-We prove a general theorem on the persistence of Whitney C ∞ -smooth families of invariant tori in the reversible context 2 of KAM theory. This context refers to the situation where dim Fix G < (codim T )/2 where Fix G is the fixed point manifold of the reversing involution G and T is the invariant torus in question. Our result is obtained as a corollary of the theorem by H. W. Broer, M.-C. Ciocci, H. Hanßmann, and A. Vanderbauwhede of 2009 concerning quasi-periodic stability of invariant tori with singular "normal" matrices in reversible systems. MSC2010 numbers: 70K43, 70H33

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

“…The drastic differences between the two reversible contexts and the peculiarities of the reversible context 2 are discussed in detail in our previous articles [41,42,43,44,45]. Here we just demonstrate these differences in the trivial case n = 0 where the invariant tori in question are equilibria and their codimension is the phase space dimension.…”

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

“…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 , . .…”

“…For basic references on Whitney smoothness in KAM theory, see [4,10]. Whitney smoothness of families of invariant tori in the reversible context 2 was proven only in 2016 [14].…”

“…In other words, such families either indicate the existence of some additional (explicit or hidden) symmetries in the systems or exhibit smaller values of the number c of internal parameters (and are "adjacent" to generic families of invariant tori in the same sense as a complicated singularity can be adjacent to a simpler singularity). For instance, the paper [21] is devoted to reducible invariant tori in the reversible (n, a, b, s) context 1 where the multiplicity of zero Floquet exponent of the tori is larger than a − b but the number of internal parameters of the Whitney smooth family of the tori is smaller than a − b + s (see also [14] for a discussion). The Floquet matrices of the invariant tori in [21] are not diagonalizable over C: their Jordan structure involves at least one nilpotent Jordan block of order greater than 1.…”

“…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.…”