A parametrised version of Moser's modifying terms theorem

Abstract

A sharpened version of Moser's 'modifying terms' KAM theorem is derived, and it is shown how this theorem can be used to investigate the persistence of invariant tori in general situations, including those where some of the Floquet exponents of the invariant torus may vanish. The result is 'structural' and works for dissipative, Hamiltonian, reversible and symmetric vector fields. These results are derived for the contexts of real analytic, Gevrey regular, ultradifferentiable and finitely differentiable perturbed vector fields. In the first two cases, the conjugacy constructed in the theorem is shown to be Gevrey smooth in the sense of Whitney on the set of parameters satisfy a "Diophantine" non-resonance condition.

Quasi-periodic bifurcations in reversible systems

“…This follows directly from the conclusion of Theorem 3.1 in [29]. The proof of that result uses (the Main) Theorem 2.3 of [29] which is formulated in terms of an admissible pair ( g , h) of Lie algebras.…”

“…This follows directly from the conclusion of Theorem 3.1 in [29]. The proof of that result uses (the Main) Theorem 2.3 of [29] which is formulated in terms of an admissible pair ( g , h) of Lie algebras. As worked out in Section 3.2 of [9], in the present reversible setting the Lie algebra h of structure-preserving vector fields can be replaced by the vector space h − of reversible vector fields together with the Lie algebra h + of G-equivariant vector fields, while the rôle of the finite-dimensional Lie algebra g < h is taken over by gl − (2, R) together with gl + (2, R).…”

“…See also [20,27]. In this way theorems 1-4 can be deduced from Theorems 2.3 and 3.1 of [29] as well, see [5].…”

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 , . .…”

“…To be more precise, the mapping Φ (1.1) is analytic in x ∈ T n and X ∈ O m (0) for any n 0, is analytic in ν ∈ Ξ for n = 0, 1, and is infinitely differentiable in ν ∈ Ξ for n 2. In fact, Φ is Gevrey regular in ν ∈ Ξ for n 2, see the papers [16,19,20] and references therein. In the C ∞ -category, the mapping Φ (1.1) is of class C ∞ in all its arguments.…”