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39 results
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ItemMulti-frequency oscillations in biological, electrical, and mechanical systems(Georgia Institute of Technology, 2010-06-05) Yi, Yingfei
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ItemAsymptotic Almost Periodicity of Scalar Parabolic Equations with Almost Periodic Time Dependence(Georgia Institute of Technology, 2009-12-07) Shen, Wenxian ; Yi, Yingfei
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ItemDynamics of Almost Periodic Scalar Parabolic Equations(Georgia Institute of Technology, 2009-12-07) Shen, Wenxian ; Yi, Yingfei
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ItemNonlinear Oscillations and Multiscale Dynamics in a Closed Chemical Reaction System(Georgia Institute of Technology, 2009) Li, Yongfeng ; Qian, Hong ; Yi, YingfeiWe investigate the oscillatory chemical dynamics in a closed isothermal reaction system described by the reversible Lotka-Volterra model. This is a three-dimensional, dissipative, singular perturbation to the conservative Lotka-Volterra model, with the free energy serving as a global Lyapunov function. We will show that there is a natural distinction between oscillatory and non-oscillatory regions in the phase space, that is, while orbits ultimately reach the equilibrium in a non-oscillatory fashion, they exhibit damped, oscillatory behaviors as interesting intermediate dynamics.
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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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ItemOscillations and Multiscale Dynamics in a Closed Chemical Reaction System: Second Law of Thermodynamics and Temporal Complexity(Georgia Institute of Technology, 2008) Li, Yongfeng ; Qian, Hong ; Yi, YingfeiWe investigate the oscillatory reaction dynamics in a closed isothermal chemical system: the reversible Lotka-Volterra model. The Second Law of Thermodynamics dictates that the system ultimately reach an equilibrium. Quasi-stationary oscillations are analyzed while free energy of the system serves as a global Lyapunov function of the dissipative dynamics. A natural distinction between regions near and far from equilibrium in terms of the free energy can be established. The dynamics is analogous to a mechanical system with time-dependent increasing damping. Near equilibrium, no oscillation is possible as dictated by Onsager’s reciprocal symmetry relation. We observe that while free energy decreases in the closed system’s dynamics, it does not follow the steepest descending path.
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ItemQuasi-Periodic Breathers in Hamiltonian Networks of Long-Range Coupling(Georgia Institute of Technology, 2007-07-03) Geng, Jiansheng ; Viveros, Jorge ; Yi, YingfeiThis work is concerned with Hamiltonian networks of weakly and long-range coupled oscillators with either variable or constant on-site frequencies. We derive an infinite dimensional KAM-like theorem by which we establish that, given any N-sites of the lattice, there is a positive measure set of small amplitude, quasi-periodic breathers (solutions of the Hamiltonian network that are quasi-periodic in time and exponentially localized in space) having N-frequencies which are only slightly deformed from the on-site frequencies.
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ItemAlmost Periodically Forced Circle Flows(Georgia Institute of Technology, 2007) Huang, Wen ; Yi, YingfeiWe study general dynamical and topological behaviors of minimal sets in skew-product circle flows in both continuous and discrete settings, with particular attentions paying to almost periodically forced circle flows. When a circle flow is either discrete in time and unforced (i.e., a circle map) or continuous in time but periodically forced, behaviors of minimal sets are completely characterized by classical theory. The general case involving almost periodic forcing is much more complicated due to the presence of multiple forcing frequencies, the topological complexity of the forcing space, and the possible loss of mean motion property. On one hand, we will show that to some extent behaviors of minimal sets in an almost periodically forced circle flow resemble those of Denjoy sets of circle maps in the sense that they can be almost automorphic, Cantorian, and everywhere non-locally connected. But on the other hand, we will show that almost periodic forcing can lead to significant topological and dynamical complexities on minimal sets which exceed the contents of Denjoy theory. For instance, an almost periodically forced circle flow can be positively transitive and its minimal sets can be Li-Yorke chaotic and non-almost automorphic. As an application of our results, we will give a complete classification of minimal sets for the projective bundle flow of an almost periodic, sl(2,R)-valued, continuous or discrete cocycle. Continuous almost periodically forced circle flows are among the simplest non-monotone, multi-frequency dynamical systems. They can be generated from almost periodically forced nonlinear oscillators through integral manifolds reduction in the damped cases and through Mather theory in the damping-free cases. They also naturally arise in 2D almost periodic Floquet theory as well as in climate models. Discrete almost periodically forced circle flows arise in the discretization of nonlinear oscillators and discrete counterparts of linear Schrödinger equations with almost periodic potentials. They have been widely used as models for studying strange, non-chaotic attractors and intermittency phenomena during the transition from order to chaos. Hence the study of these flows is of fundamental importance to the understanding of multifrequency-driven dynamical irregularities and complexities in non-monotone dynamical systems.
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ItemA KAM Theorem for Hamiltonian Networks with Long Ranged Couplings(Georgia Institute of Technology, 2006) Geng, Jiansheng ; Yi, YingfeiWe consider Hamiltonian networks of long ranged and weakly coupled oscillators with variable frequencies.
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ItemFirst Integrals and Normal Forms for Germs of Analytic Vector Fields(Georgia Institute of Technology, 2006) Chen, Jian ; Yi, Yingfei ; Zhang, XiangFor a germ of analytic vector fields, the existence of first integrals, resonance and the convergence of normalization transforming the vector field to a normal form are closely related. In this paper we first provide a link between the number of first integrals and the resonant relations for a quasi-periodic vector field, which generalizes one of the Poincaré’s classical results [18] on autonomous systems and Theorem 5 of [14] on periodic systems. Then in the space of analytic autonomous systems in C[2n] with exactly n resonances and n functionally independent first integrals, our results are related to the convergence and generic divergence of the normalizations. Lastly for a planar Hamiltonian system it is well known that the system has an isochronous center if and only if it can be linearizable in a neighborhood of the center. Using the Euler-Lagrange equation we provide a new approach to its proof.