Invariant tori of locally Hamiltonian systems close to conditionally integrable systems

Abstract

We study the problem of perturbations of quasiperiodic motions in the class of locally Hamiltonian systems. By using methods of the KAM-theory, we prove a theorem on the existence of invariant tori of locally Hamiltonian systems close to conditionally integrable systems. On the basis of this theorem, we investigate the bifurcation of a Cantor set of invariant tori in the case where a Liouville-integrable system is perturbed by a locally Hamiltonian vector field and, simultaneously, the symplectic structure of the phase space is deformed.

Herman's Approach to Quasi-Periodic Perturbations in the Reversible KAM Context 2

“…In KAM theory, there are also results for some "exotic" classes of dynamical systems, for instance, for weakly reversible systems (where the reversing diffeomorphism of the phase space is not assumed to be an involution) [4,30], for locally Hamiltonian vector fields V (defined by the condition that the 1-form i V ω 2 is closed but not necessarily exact, so that the Hamilton function can be multi-valued) [25,26,39], for conformally Hamiltonian vector fields V (defined by the identity d(i V ω 2 ) ≡ ηω 2 with constant η = 0) [12], for generalized Hamiltonian (or Poisson-Hamilton) systems defined on Poisson manifolds [23,24] (see [11,39] for more references), for presymplectic systems (defined in another way on Poisson manifolds where the role of the symplectic form ω 2 is played by a closed degenerate 2-form with constant rank) [1], for b-Hamiltonian vector fields on the so-called b-Poisson (or log-symplectic) manifolds [18], or for equivariant vector fields [45]. Here i V ω 2 is the interior product, or the contraction, of ω 2 with V .…”

“…Herman's method has been used in KAM theory in many other situations. For instance, it was applied to coisotropic [20,25,26] and atropic [26] invariant tori of Hamiltonian [20] and locally Hamiltonian [25,26] systems. In [19,46,47,49], Herman's approach was employed in the case of systems with weak nondegeneracy conditions formulated in terms of the Brouwer topological degree.…”

KAM tori: persistence and smoothness

“…It is also shown to hold for coisotropic tori (m < n, q = 0) in Hamiltonian systems in [33] and in locally Hamiltonian systems in [41]. The paper [43] contains some assertions concerning the atropic context (m < n, q > 0) in the very particular case of a constant matrix F in (1). In all these works, the (locally) Hamiltonian vector fields in question and their dependence on µ are assumed to be analytic while the Cantor families of perturbed invariant tori are proven to be C ∞ in the sense of Whitney (see however the note [64] presenting a general approach to Gevrey-smoothness of families of perturbed invariant tori).…”

“…Coisotropic invariant tori of locally Hamiltonian systems are treated as a subject of KAM theory in [41-43, 50, 51]. Atropic invariant tori of Hamiltonian systems are studied from the KAM viewpoint in [29,30] (see also a discussion in [14,59]) and those of locally Hamiltonian systems in [43]. The atropic context (m < n, q > 0) is to the coisotropic one (m < n, q = 0) as the isotropic context (m = n, q > 0) is to the Lagrangian one (m = n, q = 0).…”

“…The following approach (going back to Moser's 'modifying terms' [45,46]) to constructing invariant tori in dynamical systems is of fundamental character [9,10,13,28,31,43]. Fix some class of systems preserving a certain structure on the phase space (e.g.…”