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9 results
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ItemInvariant Tori in Hamiltonian Systems with High Order Proper Degeneracy(Georgia Institute of Technology, 2009) Han, Yuecai ; Li, Yong ; Yi, YingfeiWe study the existence of quasi-periodic, invariant tori in a nearly integrable Hamiltonian system of high order proper degeneracy, i.e., the integrable part of the Hamiltonian involves several time scales and at each time scale the corresponding Hamiltonian depends on only part of the action variables. Such a Hamiltonian system arises frequently in problems of celestial mechanics, for instance, in perturbed Kepler problems like the restricted and non-restricted 3-body problems and spatial lunar problems in which several bodies with very small masses are coupled with two massive bodies and the nearly integrable Hamiltonian systems naturally involve different time scales. Using KAM method, we will show under certain higher order nondegenerate conditions of Bruno-Rüssmann type that the majority of quasi-periodic, invariant tori associated with the integrable part will persist after the non-integrable perturbation. This actually concludes the KAM metric stability for such a properly degenerate Hamiltonian system.
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ItemDegenerate Lower Dimensional Tori in Hamiltonian Systems(Georgia Institute of Technology, 2005) Han, Yuecai ; Li, Yong ; Yi, Yingfei
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ItemNekhoroshev and Kam Stabilities in Generalized Hamiltonian Systems(Georgia Institute of Technology, 2003) Li, Yong ; Yi, YingfeiWe present some Nekhoroshev stability results for nearly integrable, generalized Hamiltonian systems which can be odd dimensional and admit a distinct number of action and angle variables. Using a simultaneous approximation technique due to Lochak, Nekhoroshev stabilities are shown for various cases of quasi-convex generalized Hamiltonian systems along with concrete estimates on stability exponents. Discussions on KAM metric stability of generalized Hamiltonian systems are also made.
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ItemOn Poincaré-Treshchev Tori in Hamiltonian Systems(Georgia Institute of Technology, 2003) Li, Yong ; Yi, YingfeiWe study the persistence of Poincaré-Treshchev tori on a resonant surface of a nearly integrable Hamiltonian system in which the unperturbed Hamiltonian needs not satisfy the Kolmogorov non-degenerate condition. The persistence of the majority of invariant tori associated to g-nondegenerate relative equilibria on the resonant surface will be shown under a Rüssmann like condition.
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ItemPersistence of Invariant Tori in Generalized Hamiltonian Systems(Georgia Institute of Technology, 2001) Li, Yong ; Yi, YingfeiWe present some results of KAM type, comparable to the KAM theory for volume-preserving maps and flows, for generalized Hamiltonian systems which may admit a distinct number of action and angle variables. In particular, systems under consideration can be odd dimensional. Applications to the perturbation of three dimensional steady Euler fluid particle path flows are considered with respect to the existence problem of barriers to fluid transport and mixing.
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ItemPersistence of Invariant Tori on Submanifolds in Hamiltonian Systems(Georgia Institute of Technology, 1999) Chow, Shui-Nee ; Li, Yong ; Yi, YingfeiGeneralizing the degenerate KAM theorem under the Rüssmann non-degeneracy and the isoenergetic KAM theorem, we employ a quasi-linear iterative scheme to study the persistence and frequency preservation of invariant tori on a smooth sub-manifold for a real analytic, nearly integrable Hamiltonian system. Under a nondegenerate condition of Rüssmann type on the sub-manifold, we shall show the following: a) the majority of the unperturbed tori on the sub-manifold will persist; b) the perturbed toral frequencies can be partially preserved according to the maximal degeneracy of the Hessian of the unperturbed system and be fully preserved if the Hessian is nondegenerate; c) the Hamiltonian admits normal forms near the perturbed tori of arbitrarily prescribed high order. Under a sub-isoenergetic nondegenerate condition on an energy surface, we shall show that the majority of unperturbed tori give rise to invariant tori of the perturbed system of the same energy which preserve the ratio of certain components of the respective frequencies.
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ItemPersistence of Lower Dimensional Tori of General Types in Hamiltonian Systems(Georgia Institute of Technology, 1999) Li, Yong ; Yi, YingfeiThe work is a generalization to [40] in which we study the persistence of lower dimensional tori of general type in Hamiltonian systems of general normal forms. By introducing a modified linear KAM iterative scheme to deal with small divisors, we shall prove a persistence result, under a Melnikov type of non-resonance condition, which particularly allows multiple and degenerate normal frequencies of the unperturbed lower dimensional tori.
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ItemA Quasi-Periodic Poincaré's Theorem(Georgia Institute of Technology, 1999) Li, Yong ; Yi, YingfeiWe study the persistence of invariant tori on resonant surfaces of a nearly integrable Hamiltonian system under the usual Kolmogorov non-degenerate condition. By introducing a quasi-linear iterative scheme to deal with small divisors, we generalize the Poincaré theorem on the maximal resonance case (i.e., the periodic case) to the general resonance case (i.e., the quasi-periodic case) by showing the persistence of majority of invariant tori associated to non-degenerate relative equilibria on any resonant surface.
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ItemPersistence of Hyperbolic Tori in Hamiltonian Systems(Georgia Institute of Technology, 1999) Li, Yong ; Yi, YingfeiWe generalize the well-known result of Graff and Zehnder on the persistence of hyperbolic invariant tori in Hamiltonian systems by considering non-Floquet, frequency varying normal forms and allowing the degeneracy of the unperturbed frequencies. The preservation of part or full frequency components associated to the degree of non-degeneracy is considered. As applications, we consider the persistence problem of hyperbolic tori on a submanifold of a nearly integrable Hamiltonian system and the persistence problem of a fixed invariant hyperbolic torus in a non-integrable Hamiltonian system.