Tracing Kam Tori in Presymplectic Dynamical Systems
نویسندگان
چکیده
We present a KAM theorem for presymplectic dynamical systems. The theorem has a “ a posteriori ” format. We show that given a Diophantine frequency ω and a family of presymplectic mappings, if we find an embedded torus which is approximately invariant with rotation ω such that the torus and the family of mappings satisfy some explicit non-degeneracy condition, then we can find an embedded torus and a value of the parameter close to to the original ones so that the torus is invariant under the map associated to the value of the parameter. Furthermore, we show that the dimension of the parameter space is reduced if we assume that the systems are exact.
منابع مشابه
Kam Theory: the Legacy of Kolmogorov’s 1954 Paper
Kolmogorov-Arnold-Moser (or kam) theory was developed for conservative dynamical systems that are nearly integrable. Integrable systems in their phase space usually contain lots of invariant tori, and kam theory establishes persistence results for such tori, which carry quasi-periodic motions. We sketch this theory, which begins with Kolmogorov’s pioneering work.
متن کاملSurvey on dissipative KAM theory including quasi-periodic bifurcation theory
Kolmogorov-Arnol’d-Moser Theory classically was mainly developed for conservative systems, establishing persistence results for quasi-periodic invariant tori in nearly integrable systems. In this survey we focus on dissipative systems, where similar results hold. In non-conservative settings often parameters are needed for the persistence of invariant tori. When considering families of such dyn...
متن کاملPeriodic Solutions and KAM Tori in a Triaxial Potential
The existence and stability of periodic solutions for an autonomous Hamiltonian system in 1:1:1 resonance depending on two real parameters α and β is established using reduction and averaging theories. The different types of periodic solutions as well as their bifurcation curves are characterized in terms of the parameters. The linear stability of each periodic solution, together with the deter...
متن کاملKAM Tori for 1D Nonlinear Wave Equations with Periodic Boundary Conditions
with periodic boundary conditions are considered; V is a periodic smooth or analytic function and the nonlinearity f is an analytic function vanishing together with its derivative at u = 0. It is proved that for “most” potentials V (x), the above equation admits small-amplitude periodic or quasi-periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dim...
متن کاملGeometric Averaging of Hamiltonian Systems: Periodic Solutions, Stability, and KAM Tori
We investigate the dynamics of various problems defined by Hamiltonian systems of two and three degrees of freedom that have in common that they can be reduced by an axial symmetry. Specifically, the systems are either invariant under rotation about the vertical axis or can be made approximately axially symmetric after an averaging process and the corresponding truncation of higher-order terms....
متن کامل