Persistence of lower-dimensional hyperbolic invariant tori for generalized Hamiltonian systems

In this paper we study the persistence of lower dimensional hyperbolic invariant tori for generalized Hamiltonian systems. Here the generalized Hamiltonian systems refer to the systems which may admit a distinct number of action and angle variables. In particular, systems under consideration can be odd-dimensional. Under Rüssmann-type non-degenerate condition, by introducing a modified linear KAM iterative scheme, we proved that the majority of the lower-dimensional hyperbolic invariant tori persist under small perturbations for generalized Hamiltonian systems.     

Topologically crossing heteroclinic connections to invariant tori

We consider transition tori of Arnold which have topologically crossing heteroclinic connections. We prove the existence of shadowing orbits to a bi-infinite sequence of tori, and of symbolic dynamics near a finite collection of tori. Topologically crossing intersections of stable and unstable manifolds of tori can be found as non-trivial zeroes of certain Melnikov functions. Our treatment relies on an extension of Easton's method of correctly aligned windows due to Zgliczyński.     

Hyperbolicity versus partial-hyperbolicity and the transversality-torsion phenomenon

In this paper, we describe a process to create hyperbolicity in the neighbourhood of a homoclinic orbit to a partially hyperbolic torus for three degrees of freedom Hamiltonian systems: the transversality-torsion phenomenon.     

Energy drift in molecular dynamics simulations

In molecular dynamics, Hamiltonian systems of differential equations are numerically integrated using the Störmer–Verlet method. One feature of these simulations is that there is an unphysical drift in the energy of the system over long integration periods. We study this energy drift, by considering a representative system in which it can be easily observed and studied. We show that if the system is started in a random initial configuration, the error in energy of the numerically computed solution is well modeled as a continuous-time stochastic process: geometric Brownian motion. We discuss what in our model is likely to remain the same or to change if our approach is applied to more realistic molecular dynamics simulations.     

传递子集的性质及其在符号动力系统中的应用

讨论了传递子集的一些性质,并且应用这些性质研究符号动力系统的弱混合子集和传递子集之间的关系,给出了符号动力系统的传递子集是弱混合子集的一个充分条件.     

KAM-Type Theorem on Resonant Surfaces for Nearly Integrable Hamiltonian Systems

Summary. In this paper, we consider analytic perturbations of an integrable Hamiltonian system in a given resonant surface. It is proved that, for most frequencies on the resonant surface, the resonant torus foliated by nonresonant lower dimensional tori is not destroyed completely and that there are some lower dimensional tori which survive the perturbation if the Hamiltonian satisfies a certain nondegenerate condition. The surviving tori might be elliptic, hyperbolic, or of mixed type. This shows that there are many orbits in the resonant zone which are regular as in the case of integrable systems. This behavior might serve as an obstacle to Arnold diffusion. The persistence of hyperbolic lower dimensional tori has been considered by many authors [5], [6], [15], [16], mainly for multiplicity one resonant case. To deal with the mechanisms of the destruction of the resonant tori of higher multiplicity into nonhyperbolic lower dimensional tori, we have to deal with some small coefficient matrices that are the generalization of small divisors. Received December 18, 1997; revised December 30, 1998; accepted June 21, 1999     

Completely degenerate lower-dimensional invariant tori for Hamiltonian system

We study the persistence of lower-dimensional invariant tori for a nearly integrable completely degenerate Hamiltonian system. It is shown that the majority of unperturbed invariant tori can survive from the perturbations which are only assumed the smallness and smoothness.     

Covering relations for multidimensional dynamical systems—II

We use a topological technique based on covering relations to prove the existence of chaotic orbits for certain Hamiltonian systems with several degrees of freedom. This paper relies on an earlier work of Zgliczyński and Gidea.     

Transient and asymptotic dynamics of a prey–predator system with diffusion

In this paper, we study a prey–predator system associated with the classical Lotka–Volterra nonlinearity. We show that the dynamics of the system are controlled by the ODE part. First, we show that the solution becomes spatially homogeneous and is subject to the ODE part as t → ∞ . Next, we take the shadow system to approximate the original system as D → ∞ . The asymptotics of the shadow system are also controlled by those of the ODE. The transient dynamics of the original system approaches to the dynamics of its ODE part with the initial mean as D → ∞ . Although the asymptotic dynamics of the original system are also controlled by the ODE, the time periods of these ODE solutions may be different. Concerning this property, we have that any δ > 0 admits D₀ > 0 such that if , the time period of the ODE, satisfies , then the solution to the original system with D ≥ D₀ cannot approach the stationary state. Copyright © 2012 John Wiley & Sons, Ltd.     

Zonal-meridional decomposition and the Hamiltonian description of planetary fluid dynamics

The basic properties of planetary flows are studied within the framework of the noncanonical Hamiltonian approach formulated by Morrison. A zonal-symmetric decomposition is applied in order to characterize the contributions of the different dynamical terms. Steady states and the Lorenz energy and angular momentum cycles are also written within the Lie-Poisson bracket formalism.     

Bifurcations of First Integrals in the Sokolov Case

We study the phase topology of a new Liouville integrable Hamiltonian system with an additional quartic integral (the Sokolov case).     

Persistence of lower-dimensional tori under the first Melnikov's non-resonance condition

In this paper we prove the persistence of lower-dimensional invariant tori of integrable equations after Hamiltonian perturbations under the first Melnikov's non-resonance condition. The proof is based on an improved KAM machinery which works for the angle variable dependent normal form. By an example, we also show the necessity of the Melnikov's first non-resonance condition for the persistence of lower dimensional tori.     

Generation in Degree Four of Symbolic Blowups of Self-Linked Monomial Space Curves

We study the symbolic blowup ring of a prime ideal defining a monomial curve in the power series ring in 3 variables over a field. We characterize the conditions required to have the symbolic blowup generated in degree 4 when the monomial curve is self-linked.     

哈密顿系统的低维环面的保存性

本文给出了关于哈密顿系统低维环面的一个推广的KAM定理,它适用于同时存在法向频率和双曲法向分量的情况.其证明基于尤建功的一个定理的光滑性表述及法向双曲不变流形理论的应用.文中还给出了另外两种情况下的推广.     

Lyapunov Center Theorem of Infinite Dimensional Hamiltonian Systems

In this paper we reformulate a Lyapunov center theorem of infinite dimensional Hamiltonian systems arising from PDEs. The proof is based on a modified KAM iteration for periodic case.     

Persistence of lower dimensional tori of general types in Hamiltonian systems

This work is a generalization to a result of J. You (1999). 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.

     

Application of the GALI method to localization dynamics in nonlinear systems

We investigate localization phenomena and stability properties of quasiperiodic oscillations in N$N$ degree of freedom Hamiltonian systems and N$N$ coupled symplectic maps. In particular, we study an example of a parametrically driven Hamiltonian lattice with only quartic coupling terms and a system of N$N$ coupled standard maps. We explore their dynamics using the Generalized Alignment Index (GALI), which constitutes a recently developed numerical method for detecting chaotic orbits in many dimensions, estimating the dimensionality of quasiperiodic tori and predicting slow diffusion in a way that is faster and more reliable than many other approaches known to date.     

Containing Symbolic Powers in Regular Rings

We extend some results of Ein–Lazarsfeld-Smith, Hochster–Huneke, and Takagi-Yoshida concerning the containments of symbolic powers of ideals in ordinary powers in regular rings.