Families of Quasi-Periodic Motions in Dynamical Systems Depending on Parameters

113

We consider families of dynamical systems having invariant tori that carry quasi-periodic motions. Our interest is the persistence of such tori under small, nearly-integrable perturbations. This persistence problem is studied in the dissipative, the Hamiltonian and the reversible setting, as part of a more general KAM theory for classes of structure preserving dynamical systems. This concerns the parametrized KAM theory as initiated by Moser [J.K. Moser, On the theory of quasiperiodic motions, SIAM Rev. 8 (2) (1966)145-172; J.K. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann. 169 (1967) 136-176] and further developed in [G.B. Huitema, Unfoldings of quasi-periodic tori, PhD thesis, University of Groningen, 1988; H.W. Broer, G.B. Huitema, F. Takens, Unfoldings of quasi-periodic tori, Mem. Amer. Math. Soc. 83 (421) (1990) 1-82; H.W. Broer, G.B. Huitema, Unfoldings of quasi-periodic tori in reversible systems, J. Dynam. Differential Equations 7 (1) (1995) 191-212]. The corresponding nondegeneracy condition involves certain (trans-)versality conditions on the normal linear, leading, part at the invariant tori. We show that as a consequence, a Cantor family of Diophantine tori with positive Hausdorff measure is persistent under nearly-integrable perturbations. This result extends the above references since presently the case of multiple Floquet exponents is included. Our leading example is the normal 1 : −1 resonance, which occurs a lot in applications, both Hamiltonian and reversible. As an illustration of this we briefly describe the Lagrange top coupled to an oscillator.

Section: Normal Linear Stability: Nearly-integrable Reversible Casementioning

confidence: 98%

Normal linear stability of quasi-periodic tori

Broer

Hoo

Naudot

2007

113

“…Let t 8 be as in Lemma 53 and let 0 < ρ * < 1 be sufficiently small such that ρ * t −1 8 and such that following inequality holds: The convergence of the sequence {K m } m 1 follows from the Inverse Approximation Lemma (see, for example, Lemma 2.2 in [4] or Lemma 6.14 in [24]). Indeed, define u m def = K m − K 1 , then the following properties hold:…”

Section: Proofmentioning

confidence: 98%

Analytic smoothing of geometric maps with applications to KAM theory

González-Enríquez¹,

Llave²

2008

We show that finitely differentiable diffeomorphisms which are either symplectic, volume-preserving, or contact can be approximated with analytic diffeomorphisms that are, respectively, symplectic, volumepreserving or contact. We prove that the approximating functions are uniformly bounded on some complex domains and that the rate of convergence, in C r -norms, of the approximation can be estimated in terms of the size of such complex domains and the order of differentiability of the approximated function. As an application to this result, we give a proof of the existence, the local uniqueness and the bootstrap of regularity of KAM tori for finitely differentiable symplectic maps. The symplectic maps considered here are not assumed either to be written in action-angle variables or to be perturbations of integrable systems. Our main assumption is the existence of a finitely differentiable parameterization of a maximal dimensional torus that satisfies a non-degeneracy condition and that is approximately invariant. The symplectic, volumepreserving and contact forms are assumed to be analytic.

“…Based on original works of Melnikov [24], Eliasson [12], Kuksin [18], and Pöschel [25], the KAM method has been extensively developed in finite dimensions concerning the persistence of lower-dimensional tori in Hamiltonian systems (see [2,3,16,23,29,31,32] and references therein). Recently, the KAM method was extended to infinite dimensions in works of Kuksin [19], Wayne [30], and Pöschel [27] in studying quasiperiodic solutions for 1D nonlinear Schrödinger and wave equations with the Dirichlet boundary condition and parametrized potentials.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Quasi-periodic solutions in a nonlinear Schrödinger equation

Geng

2007

In this paper, one-dimensional (1D) nonlinear Schrödinger equation iu t − u xx + mu + |u| 4 u = 0 with the periodic boundary condition is considered. It is proved that for each given constant potential m and each prescribed integer N > 1, the equation admits a Whitney smooth family of small amplitude, time quasi-periodic solutions with N Diophantine frequencies. The proof is based on a partial Birkhoff normal form reduction and an improved KAM method.