Unfoldings of quasi-periodic tori in reversible systems

Abstract

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KAM theory: The legacy of Kolmogorov's 1954 paper

“…Similar results occur when considering reversible systems instead of Hamiltonian ones [50], [53], [64], [16], [22], or systems that are equivariant with respect to certain Lie groups. Moreover, as section 3 already indicates, there also exist kam theorems for the general class of (dissipative) dynamical systems; compare with section 3.…”

Quasi-periodic bifurcations in reversible systems

“…The formulation in [11] similarly already anticipated the (dissipative) frequency-halving bifurcation treated in [1]. Writing (9) ensures that the constant part of any perturbing vector field can be transformed away. In important cases it is possible to first restrict ad N to a subspace B of the space of constant vector fields.…”

“…We follow the proof of theorem 3, but apply at the end the Corollary to the Main Theorem in [9] to obtain quasi-periodic stability of the elliptic or hyperbolic tori.…”

“…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).…”

Quasi-periodic stability of normally resonant tori

“…To define reversibility we consider an involution (i.e. G Following [12][13][14]24] the vector field X is called integrable if it is equivariant with respect to the group action We denote the subspace of infinitesimally reversible linear operators on R 2p by gl − (2p; R) and by gl + (2p; R) the subspace of all R-equivariant linear operators on R 2p , i.e.…”