Normal linear stability of quasi-periodic tori
- 作者：H.W. Broer ; J. Hoo ; V. Naudot
- 关键词：kam theory ; Nearly-integrable system ; Perturbation theory ; Quasi-periodic stability ; (Uni-)versal unfolding ; Linear centralizer unfolding ; Normal linear part ; Quasi-periodic bifurcations ; Hamiltonian and reversible Hopf bifurcation ; Lagrange top
- 刊名：Journal of Differential Equations
- 出版年：2007
- 期刊代码：74_00220396
- 类别：mt
- 出版时间：15 January 2007
- 卷：232
- 期：2
- 页码：355-418
- 文件大小：776 K
摘要
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 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.