Persistence of lower-dimensional hyperbolic invariant tori for generalized Hamiltonian systems

Abstract

In this paper we study the persistence of lower dimensional hyperbolic invariant tori for generalized Hamiltonian systems. Here the generalized Hamiltonian systems refer to the systems which may admit a distinct number of action and angle variables. In particular, systems under consideration can be odddimensional. Under Rüssmann-type non-degenerate condition, by introducing a modified linear KAM iterative scheme, we proved that the majority of the lower-dimensional hyperbolic invariant tori persist under small perturbations for generalized Hamiltonian systems.

Generalization of solutions of the Jacobi PDEs associated to time reparametrizations of Poisson systems

“…Accordingly it is J * 12 (y) = 0 everywhere in Ω * . This implies that in order to fulfill the Darboux reduction we only need to perform the time reparametrization dτ = η(y)dt as detailed in (32), which is well defined everywhere because now η(y) = ( {d 1 (x), d 2 (x)} J )| x(y) = (∇ x d 1 (x)) T · J (x) · (∇ x d 2 (x)) x(y) = J * 12 (y)…”

“…see [37] and references therein for an overview and a historical discussion) are ubiquitous in most fields of applied mathematics and related areas such as physics and theoretical biology, for instance in mechanics [9,12,14,32,42], control theory [4], electromagnetism [8,10], plasma physics [40], optics [11,28], population dynamics [23,25,35,36,41], dynamical systems theory [6,7,9,26,32,34], etc. In fact, describing a given dynamical system in terms of a Poisson structure allows the obtainment of a wide range of information which may be in the form of perturbative solutions [8], invariants [10,32,43], bifurcation properties and characterization of chaotic behaviour [11,32,40], efficient numerical integration [29], integrability results [32,34,37], reductions [1,10,11,16], [18]- [21], [23,24], as well as algorithms for stability analysis [3,25,…”

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 .…”

The almost-periodic solutions of the weakly coupled pendulum equations

“…Then, Zehnder in [53,54] has brought a substitute proof of Graff 's conclusion by an implicit function technique. For more results on the evolutions in this direction, one can refer to [55][56][57][58][59][60][61]. By virtue of the KAM theory of this situation, we obtain that there are quasi-periodic solutions for the coupled pendulum equations.…”