Asymptotic Localization of Energy in Nondisordered Oscillator Chains

We study two popular one‐dimensional chains of classical anharmonic oscillators: the rotor chain and a version of the discrete nonlinear Schrödinger chain. We assume that the interaction between neighboring oscillators, controlled by the parameter ? > 0, is small. We rigorously establish that the thermal conductivity of the chains has a nonperturbative origin with respect to the coupling constant ?, and we provide strong evidence that it decays faster than any power law in ? as ? → 0. The weak coupling regime also translates into a high‐temperature regime, suggesting that the conductivity vanishes faster than any power of the inverse temperature. To our knowledge, it is the first time that a clear connection has been established between KAM‐like phenomena and thermal conductivity. © 2015 Wiley Periodicals, Inc.     

Synchronization configurations of two coupled double pendula

We consider the synchronization of two self-excited double pendula hanging from a horizontal beam which can roll on the parallel surface. We show that such pendula can obtain four different robust synchronous configurations. Our approximate analytical analysis allows to derive the synchronization conditions and explains the observed types of synchronizations. We consider the energy balance in the system and show how the energy is transferred between the pendula via the oscillating beam allowing the pendula’ synchronization.     

Weak KAM theorem for Hamilton‐Jacobi equations with Neumann boundary conditions on noncompact manifolds

In this paper, we consider Hamilton–Jacobi equations with homogeneous Neumann boundary condition. We establish some results on noncompact manifold with homogeneous Neumann boundary conditions in view of weak Kolmogorov‐Arnold‐Moser (KAM) theory, which is a generalization of the results obtained by Fathi under the non‐bounded condition. Copyright © 2014 John Wiley & Sons, Ltd.     

Action potential and weak KAM solutions

For convex superlinear lagrangians on a compact manifold M we characterize the Peierls barrier and the weak KAM solutions of the Hamilton-Jacobi equation, as defined by A. Fathi [9], in terms of their values at each static class and the action potential defined by R. Ma né [14]. When the manifold M is non-compact, we construct weak KAM solutions similarly to Busemann functions in riemannian geometry. We construct a compactification of by extending the Aubry set using these Busemann weak KAM solutions and characterize the set of weak KAM solutions using this extended Aubry set. Received: 13 November 2000 / Accepted: 4 December 2000 / Published online: 25 June 2001     

Global weak solutions to the Cauchy problem of compressible Navier–Stokes–Vlasov–Fokker–Planck equations

A fluid–particles system of the compressible Navier‐Stokes equations and Vlasov‐Fokker‐Planck equation (including the case of Vlasov equation) in three‐dimensional space is considered in this paper. The coupling arises from a drag force exerted by the fluid onto the particles. We study a Cauchy problem with large data, and establish the existence of global weak solutions through an approximation scheme, energy estimates, and weak convergence. Copyright © 2016 John Wiley & Sons, Ltd.     

本刊英文版Vol.28(2012),No.11论文摘要

<正>Generalized Maupertuis' Principle with Applications Wei CHENG Abstract We give a rigorous proof of the equivalence of Mane's supercritical potential and the minimal action with respect to an associated Jacobi-Finsler metric.As a consequence, we give an explicit representation of the weak KAM solutions of one-dimensional mechanical systems without the quadratic assumption on the kinetic energy term of the Hamiltonians,and a criterion of the integrability result for such a system of arbitrary degree of freedom by the regularity assumption on Mather'sα-function is discussed.     

Synchronization of the self-excited pendula suspended on the vertically displacing beam

We consider the synchronization of a number of self-excited identical pendula hanging on the same beam which can move vertically. We identify different synchronous configurations and investigate their stability. An approximate analytical analysis of the energy balance allows to derive the synchronization conditions, phase difference between pendula and explains the observed types of synchronizations.     

Synchronization of two flow excited pendula

Two aerodynamically excited pendula are considered as a simple example of two linearly coupled, self sustained mechanical oscillators, modelled by two coupled Van-der-Pol equations. The considered mechanical application admits of a systematic survey of synchronized regimes within the framework of standard nonlinear stability analysis. Using normal form theory and the prevailing direct averaging approach the occurring Hopf bifurcation with two distinct pairs of purely imaginary eigenvalues is studied in the non-resonant case and in the 1:1-resonance corresponding, respectively, to strong and weak coupling. In particular, for the resonant case a graphical approach permits a comprehensive interpretation relating the stable stationary solutions of the averaged system with synchronized regimes and allows an analytical computation of the oscillation amplitudes and the synchronous frequency.     

Generalized Maupertuis’ principle with applications

We give a rigorous proof of the equivalence of Ma's supercritical potential and the minimal action with respect to an associated Jacobi-Finsler metric. As a consequence, we give an explicit representation of the weak KAM solutions of one-dimensional mechanical systems without the quadratic assumption on the kinetic energy term of the Hamiltonians, and a criterion of the integrability result for such a system of arbitrary degree of freedom by the regularity assumption on Mather's α-function is discussed.     

陀螺仪运动的混沌与KAM理论

本文建立了外环轴水平放置的重力对称陀螺仪的运动方程．将非自由陀螺仪转子的质心位置作为扰动，利用Ｍｅｌｎｉｋｏｖ方法和ＫＡＭ理论研究了陀螺仪的运动，研究结果表明：在 螺仪转子的质心与支架中心不重合且充分接近时，或 螺仪能量充分大时，陀螺仪运动的Ｐｏｉｎｃａｒｅ映射既出现Ｓｍａｌｅ马蹄意义下的混沌，又存在ＫＡＭ不变环面和不变闭曲线。     

Minimizing configurations and Hamilton-Jacobi equations of homogeneous N-body problems

For N-body problems with homogeneous potentials we define a special class of central configurations related with the reduction of homotheties in the study of homogeneous weak KAM solutions. For potentials in 1/r ^α with α ∈ (0, 2) we prove the existence of homogeneous weak KAM solutions. We show that such solutions are related to viscosity solutions of another Hamilton-Jacobi equation in the sphere of normal configurations. As an application we prove for the Newtonian three-body problem that there are no smooth homogeneous solutions to the critical Hamilton-Jacobi equation.     

Wigner measures supported on weak KAM tori

In the setting of the Weyl quantization on the flat torus \(\mathbb{T}^n \) , we exhibit a class of wave functions with uniquely associated Wigner probability measure, invariant under the Hamiltonian dynamics and with support contained in weak KAM tori in phase space. These sets are the graphs of Lipschitz-continuous weak KAM solutions of negative type of the stationary Hamilton-Jacobi equation. Such Wigner measures are, in fact, given by the Legendre transform of Mather’s minimal probability measures.     

Numerical simulation and geometrical analysis on the onset of chaos in a system of two coupled pendulums

By means of two geometrical approaches: (i) the potential landscape, and (ii) the Hamiltonian manifold method, we have found the relations that predict the transition to chaos in a two-coupled pendulum system due to a shift from positive to negative curvature. Our scrutiny from numerical simulations show that the splitting of the KAM island at the symmetric mode is another scenario that hints at the incidence of chaotic behaviour. We have uncovered two regimes on the onset of chaos, depending on whether the coupling parameter is greater or less than a critical point. In the first regime, chaos is observed to first occur after the splitting of the KAM island at the symmetric mode and before the switching of the dynamical curvature. On the other hand, an interchange of scenarios arises in the second regime, with chaos now appears after curvature switching and before KAM island splitting at the symmetric mode.     

Ferromagnets with biquadratic exchange coupling energy. Global existence of weak solutions

A global existence theorem is proved for the Landau–Lifshitz–Gilbert equations with biquadratic exchange coupling energy acting on the interfaces of a material composed by two ferromagnetic layers separated by a nonmagnetic one. This energy is not convex. The magnetization M satisfies on the interfaces a coupled non‐linear Neumann boundary condition with cubic growth. We use several regularizations, in particular for the traces of the magnetization at the interfaces, to obtain global weak solutions of the problem with finite energy. Copyright © 2005 John Wiley & Sons, Ltd.     

The complexity of wave propagation of the nonlinear Schrödinger equation with weak periodic external field

A nonlinear Schrödinger equation with external field is obtained for nonlinear optics in a non-homogeneous medium. On the basis of the Melnikov function and KAM theory, the complexity of the wave propagation of the equation is studied in the case of weak periodic external field.     

New Identities for Weak KAM Theory(For Haim Brezis,in continuing admiration)

This paper records for the Hamiltonian H =1/2|p|~2+ W(x) some old and new identities relevant for the PDE/variational approach to weak KAM theory.     

Existence of invariant curves for area-preserving mappings under weaker non-degeneracy conditions

We consider a class of analytic area-preserving mappings C^m-smoothly depending on a parameter. Without imposing on any non-degeneracy assumption, we prove a formal KAM theorem for the mappings, which implies many previous KAM-type results under some non-degeneracy conditions. Moreover, by this formal KAM theorem, we can also obtain some new interesting results under some weaker non-degeneracy conditions. Thus, the formal KAM theorem can be regarded as a general KAM theorem for areapreserving mappings.     

Mather theory for piecewise smooth Lagrangian systems

We establish the Mather theory for a type of piecewise smooth and positive definite Lagrangian systems.It models a mechanical system subject to external impulsive forcing.We show the existence of the minimal measure and the Lipschitz property of Aubry set.In addition,the weak KAM solution to this kind of piecewise smooth Lagrangian is also established.     

General KAM theorems and their applications to invariant tori with prescribed frequencies

In this paper we develop a new KAM technique to prove two general KAM theorems for nearly integrable Hamiltonian systems without assuming any nondegeneracy condition. Many of KAM-type results (including the classical KAM theorem) are special cases of our theorems under some nondegeneracy condition and some smoothness condition. Moreover, we can obtain some interesting results about KAM tori with prescribed frequencies.     

New Identities for Weak KAM Theory

This paper records for the Hamiltonian H =1/2│P│2 + W(x) some old and new identities relevant for the PDE/variational approach to weak KAM theory.