Lipschitz stability of impulsive systems of differential equations

Notions ofLipschitz stability of the zero solution of impulsive systems of differential equations with fixed moments of impulse effect are introduced. Sufficient conditions for various types of uniform Lipschitz stability are obtained and the relations between these notions are investigated. The results obtained are used for the investigation of the uniform Lipschitz stability of the zero solution of linear impulsive systems of differential equations.     

Adaptive chaos control and synchronization in only locally Lipschitz systems

In the existing results on chaos control and synchronization based on the adaptive controlling technique (ACT), a uniform Lipschitz condition on a given dynamical system is always assumed in advance. However, without this uniform Lipschitz condition, the ACT might be failed in both theoretical analysis and in numerical experiment. This Letter shows how to utilize the ACT to get a rigorous control for the system which is not uniformly Lipschitz but only locally Lipschitz, and even for the system which has unbounded trajectories. In fact, the ACT is proved to possess some limitation, which is actually induced by the nonlinear degree of the original system. Consequently, a piecewise ACT is proposed so as to improve the performance of the existing techniques.     

Synchronization of chaotic systems based on adaptive observer design

The synchronizing problem of a chaotic system is investigated based on the observer design. The nonlinear section is assumed to satisfy the Lipschitz condition. Firstly, the normal observer is designed based on the known Lipschitz constant and the results are given in linear matrix inequality (LMI) form. Then a fairly simple adaptive observer is designed with the Lipschitz constant unknown. Simulations on synchronizing the Lorenz system are investigated and the results show the validity and feasibility of our main results.     

Eigenvalue asymptotics for potential type operators on Lipschitz surfaces

For potential-type integral operators on a Lipschitz surface, an asymptotic formula for eigenvalues is proved. The reasoning is based upon the study of the rate of operator convergence as smooth surfaces approximate the Lipschitz surface. Dedicated to Mikhail Agranovich on the occasion of his 75th birthday     

Lipschitz quasistability of impulsive systems of differential equations

A suitable comparison lemma is used to obtain sufficient conditions for uniform Lipschitz quasistability of an arbitrary solution of an impulsive system of differential equations with unfixed moments of impulse effect. The results are applied to finding conditions for uniform Lipschitz quasistability for linear impulsive systems with unfixed moments of impulse effect.     

Static and adaptive feedback control for synchronization of different chaotic oscillators with mutually Lipschitz nonlinearities

This paper addresses the control law design for synchronization of two different chaotic oscillators with mutually Lipschitz nonlinearities. For analysis of the properties of two different nonlinearities, an advanced mutually Lipschitz condition is proposed. This mutually Lipschitz condition is more general than the traditional Lipschitz condition. Unlike the latter, it can be used for the design of a feedback controller for synchronization of chaotic oscillators of different dynamics. It is shown that any two different Lipschitz nonlinearities always satisfy the mutually Lipschitz condition. Applying the mutually Lipschitz condition, a quadratic Lyapunov function and uniformly ultimately bounded stability, easily designable and implementable robust control strategies utilizing algebraic Riccati equation and linear matrix inequalities, are derived for synchronization of two distinct chaotic oscillators. Furthermore, a novel adaptive control scheme for mutually Lipschitz chaotic systems is established by addressing the issue of adaptive cancellation of unknown mismatch between the dynamics of different chaotic systems. The proposed control technique is numerically tested for synchronization of two different chaotic Chua＇s circuits and for obtaining identical behavior between the modified Chua＇s circuit and the R6ssler system.     

Sharp Spectral Asymptotics and Weyl Formula for Elliptic Operators with Non-smooth Coefficients

The aim of this paper is to give the Weyl formula for eigenvalues of self-adjoint elliptic operators, assuming that first-order derivatives of the coefficients are Lipschitz continuous. The approach is based on the asymptotic formula of Hörmander"s type for the spectral function of pseudodifferential operators having Lipschitz continuous Hamiltonian flow and obtained via a regularization procedure of nonsmooth coefficients.     

Adaptive impulsive synchronization of uncertain chaotic systems

The Letter develops an adaptive impulsive scheme that includes a sole restriction criterion to achieve synchronization of chaotic nonlinear systems with unknown parameters. The system is assumed to satisfy the local Lipschitz condition while a Lipschitz constant and the uncertain system parameters are estimated by augmented adaptation equations. Adaptation of all parameters is proven to converge exponentially. The significance of the related control parameters and their margins in the criterion is also discussed in detail. The Lorenz system has been simulated to illustrate the theoretical analysis.     

Weak and strong probabilistic solutions for a stochastic quasilinear parabolic equation with nonstandard growth

In this paper, we investigate a class of stochastic quasilinear parabolic initial boundary value problems with nonstandard growth in the functional setting of generalized Sobolev spaces. The deterministic version of the equation was first introduced and studied by Samokhin in [45] as a generalized model for polytropic filtration. We establish an existence result of weak probabilistic solutions when the forcing terms do not satisfy Lipschitz conditions. Under the Lipschitz property of the forcing terms, we obtain the uniqueness of weak probabilistic solutions. Combining the uniqueness and the famous Yamada–Watanabe result, we prove the existence of a unique strong probabilistic solution of the problem.     

Extended Divergence-Measure Fields and the Euler Equations for Gas Dynamics

 A class of extended vector fields, called extended divergence-measure fields, is analyzed. These fields include vector fields in L ^p and vector-valued Radon measures, whose divergences are Radon measures. Such extended vector fields naturally arise in the study of the behavior of entropy solutions of the Euler equations for gas dynamics and other nonlinear systems of conservation laws. A new notion of normal traces over Lipschitz deformable surfaces is developed under which a generalized Gauss-Green theorem is established even for these extended fields. An explicit formula is obtained to calculate the normal traces over any Lipschitz deformable surface, suitable for applications, by using the neighborhood information of the fields near the surface and the level set function of the Lipschitz deformation surfaces. As an application, we prove the uniqueness and stability of Riemann solutions that may contain vacuum in the class of entropy solutions of the Euler equations for gas dynamics. Received: 7 May 2002 / Accepted: 2 December 2002 Published online: 2 April 2003 Communicated by P. Constantin     

Remarks on strongly elliptic systems in Lipschitz domains

We present some remarks to the general theory of strongly elliptic second-order systems in bounded Lipschitz domains. The most important remarks are related to the use of the ??Weyl decomposition?? of the solution space. In particular, we suggest a simplified approach to the unique choice of the right-hand side of the system and the conormal derivative in the Neumann problem and obtain two-sided a priori estimates for the solutions. We consider the transmission problem for two systems in domains with a common Lipschitz boundary without the assumption that the coefficients do not have jumps on that boundary. We construct examples of strongly elliptic second-order systems for which the Neumann problem is not Fredholm.     

Terminal model of Newtonian dynamics

A new type of dissipation function which does not satisfy the Lipschitz condition at equilibrium states is proposed. Newtonian dynamics supplemented by this dissipation function becomes irreversible and has a well-organized probabilistic structure.     

The classical mechanics of one-dimensional systems of infinitely many particles

We prove a global existence and uniqueness theorem for solutions of the classical equations of motion for a one-dimensional system of infinitely many particles interacting by finite-range two-body forces which satisfy a Lipschitz condition.     

Continuous dependence on a parameter of the solutions of impulsive differential equations in a Banach space

We prove that the solutions of an impulsive differential equation depend continuously on a small parameter under the assumption that the right-hand side of the equation and the impulse operators satisfy conditions of Lipschitz type.     

Existence of almost periodic solutions for SICNNs with time-varying delays

In this Letter, the shunting inhibitory cellular neural networks with time-varying delays are considered. We establish some new results about the existence and stability of almost periodic solutions for SICNNs without the global Lipschitz conditions on the activity functions. An example is given to illustrate the effectiveness of our results.     

Example of indeterminacy in classical dynamics

The case of a particle moving along a nonsmooth constraint under the action of uniform gravity is presented as an example of indeterminancy in a classical situation. The indeterminacy arises from certain initial conditions having nonunique solutions and is due to the failure of the Lipschitz condition at the corresponding points in the phase space of the equation of motion.     

Vector-valued rational forms

We define rational Hermite interpolants to vector-valued functions and show that, in the context of Clifford algebras, the numerator and denominator polynomials belong to a complex extension of the Lipschitz group. We also discuss the problem of constructing an algebraic representation for the generalized inverse of a vector, which is at the heart of the usual development of vector rational approximation.     

The problem of irreversibility in Newtonian dynamics

A new type of dissipation function which does not satisfy the Lipschitz condition at equilibrium states is proposed. It is shown that Newtonian dynamics supplemented by this dissipation function becomes irreversible, i.e., it is not invariant with respect to time inversion. Some effects associated with the approaching of equilibria in infinite time are eliminated. New meanings of chaos and turbulence are discussed.     

Optimal Concentration Inequalities for Dynamical Systems

For dynamical systems modeled by a Young tower with exponential tails, we prove an exponential concentration inequality for all separately Lipschitz observables of n variables. When tails are polynomial, we prove polynomial concentration inequalities. Those inequalities are optimal. We give some applications of such inequalities to specific systems and specific observables.     

两个不相同系统的广义同步化

采用Temam无穷维动力系统的惯性流形理论，证明了两个不相同的系统能实现广义同步化.在两个系统具有吸收集和吸引子的基础上，通过定义在一个函数类上的压缩映射的不动点得到了广义同步化流形，该流形是Lipschitz光滑流形，而且具有不变性与指数吸引性.数值仿真证实了所建立理论的正确性.