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Register a new userInvariant Tori of a Two Dimensional Periodic System with the Linear-Cubic Unperturbed Part
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Two classes of time-periodic systems of ordinary differential equations with a small nonnegative parameter, those with fast and slow time, are studied. Right-hand sides of these systems are three times continuously differentiable with respect to phase variables and the parameter, the corresponding unperturbed systems are autonomous, conservative and have nine equilibrium points. For the perturbed systems, which do not depend on the parameter explicitly, we obtain the conditions yielding that the initial system has a certain number of two-dimensional invariant surfaces homeomorphic to a torus for each sufficiently small values of parameter and the formulas of such surfaces. A class of systems with seven invariant surfaces enclosing different configurations of equilibrium points is studied as an example of applications of our method.
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We consider the classical problem of the continuation of periodic orbits surviving to the breaking of invariant lower dimensional resonant tori in nearly integrable Hamiltonian systems. In particular we extend our previous results (presented in CNSNS, 61:198-224, 2018) for full dimensional resonant tori to lower dimensional ones. We develop a constructive normal form scheme that allows to identify and approximate the periodic orbits which continue to exist after the breaking of the resonant torus. A specific feature of our algorithm consists in the possibility of dealing with degenerate periodic orbits. Besides, under suitable hypothesis on the spectrum of the approximate periodic orbit, we obtain information on the linear stability of the periodic orbits feasible of continuation. A pedagogical example involving few degrees of freedom, but connected to the classical topic of discrete solitons in dNLS-lattices, is also provided.
We reconsider the classical problem of the continuation of degenerate periodic orbits in Hamiltonian systems. In particular we focus on periodic orbits that arise from the breaking of a completely resonant maximal torus. We here propose a suitable normal form construction that allows to identify and approximate the periodic orbits which survive to the breaking of the resonant torus. Our algorithm allows to treat the continuation of approximate orbits which are at leading order degenerate, hence not covered by classical averaging methods. We discuss possible future extensions and applications to localized periodic orbits in chains of weakly coupled oscillators.
In this paper, we show the existence of non contractible periodic orbits in Hamiltonian systems defined on $T^*T^n$ separating two Lagrangian tori under certain cone assumption. Our result answers a question of Polterovich in cite{P} in a sharp way. As an application, we find periodic orbits of almost all the homotopy types on a dense set of energy level in Lorentzian type mechanical Hamiltonian systems defined on $T^*T^2$. This solves a problem of Arnold in cite{A}.
We consider the motion of an electron in an atom subjected to a strong linearly polarized laser field. We identify the invariant structures organizing a very specific subset of trajectories, namely recollisions. Recollisions are trajectories which first escape the ionic core (i.e., ionize) and later return to this ionic core, for instance, to transfer the energy gained during the large excursion away from the core to bound electrons. We consider the role played by the directions transverse to the polarization direction in the recollision process. We compute the family of two-dimensional invariant tori associated with a specific hyperbolic-elliptic periodic orbit and their stable and unstable manifolds. We show that these manifolds organize recollisions in phase space.
The KAM (Kolmogorov-Arnold-Moser) theorem guarantees the stability of quasi-periodic invariant tori by perturbation in some Hamiltonian systems. Michel Herman proved a similar result for quasi-periodic motions, with $k$-dimensional involutive manifolds in Hamiltonian systems with $n$ degrees of freedom $n leq k < 2n $. In this paper, we extend this result to the case of a quasi-periodic motion on symplectic tori $k = 2n$.
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