Stability of a linear difference system with random parameters

Necessary and sufficient conditions are derived for the mean-square exponential stability of a system described by an n-th order stochastic difference equation.Translated from Matematicheskie Zametki, Vol. 8, No. 6, pp. 753–760, December, 1970.The author wishes to thank R. Z. Khas'minskii for suggesting this problem.     

Stability of a budworm growth model with random perturbations

A budworm growth model perturbed by both white noises and regime switchings is proposed and analyzed. It is proven that there is a threshold. If this threshold is positive, then the model has a unique ergodic stationary distribution; if this threshold is negative, then the zero solution of the model is stable. The results show that both white noises and regime switchings can change the stability of the model greatly. Several numerical simulations based on realistic data are also introduced to illustrate the main results.     

Stability radii for linear time-varying differential-algebraic equations with respect to dynamic perturbations

This paper is concerned with the robust stability for linear time-varying differential-algebraic equations. We consider the systems under the effect of uncertain dynamic perturbations. A formula of the structured stability radius is obtained. The result is an extension of a previous result for time-varying ordinary differential equations proven by Birgit Jacob [B. Jacob, A formula for the stability radius of time-varying systems, J. Differential Equations 142 (1998) 167-187].     

Stability of Hausdorff dimension of Axiom A attractors under random perturbations

This paper proves that the Hausdorff dimension of an Axiom A attractor is stable under random perturbations.     

Investigation of a linear evolution system in the Banach space with random times of perturbations

For a linear evolution system given in the Banach space and characterized by pulse perturbations at random times, we establish conditions for the existence of a unique solution of the Cauchy problem and investigate the stability of the zero solution. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 3, pp. 433–436, March, 1998.     

Survival analysis of a cooperation system with random perturbations in a polluted environment

Taking white noises, Markovian switchings and Lévy jump noises into account, a stochastic cooperation system of two species in a polluted environment is developed and analyzed. Persistence–extinction thresholds are obtained for each population. The results reveal that white noises, Markovian switchings and Lévy jumps have sufficient effect to the persistence and extinction of the species.     

Permanence in a periodic single species system subject to linear/constant impulsive perturbations

In this paper, a single species system with linear/constant impulsive perturbations is established. The linear and constant impulses are realized at fixed moments of time. By using the comparison principle of impulsive differential equations and analysis techniques, sufficient conditions for the permanence of the system with linear and constant impulsive perturbations are obtained, respectively. Numerical simulations are presented to substantiate our analytical results, and show that the impulsive perturbations play a crucial role. Copyright © 2010 John Wiley & Sons, Ltd.     

Stability analysis of positive switched systems via joint linear copositive Lyapunov functions

In this paper, we study the asymptotic stability of continuous-time positive switched linear systems for the case when each subsystem is only stable. By using the so-called “joint linear copositive Lyapunov function” (JLCLF) generalizing the common linear copositive Lyapunov function, we show that the system remains asymptotically stable under appropriate switching if it has a JLCLF. Then, the main result is extended to positive switched linear systems with time delay.     

具有线性脉冲的周期捕食系统的持久性

研究具有HollingIV功能性反应和脉冲的周期捕食食饵系统．找到了影响该系统动力学行为的阈值Ro．证明了当Ro〈1时,该系统的食饵灭绝周期解是局部渐近稳定的;当R0〉1时,该系统的食饵灭绝周期解变得不稳定且食饵将一致持久．     

Stability of solutions to the Cauchy problem for a plane hyperbolic system with time-periodic coefficients

Considering a plane hyperbolic system with time-periodic coefficients, we construct a version of the direct Lyapunov method with the condition on the derivative of the Lyapunov functional along the trajectories of the system which is weakened by use of periodicity of the coefficients. We exhibit an illustrative example.     

On the existence of a stable limit cycle to a piecewise linear system in R2

The present paper is devoted to the existence of limit cycles of planar piecewise linear (PWL) systems with two zones separated by a straight line and singularity of type “focus-focus” and “focus-center.” Our investigation is a supplement to the classification of Freire et al concerning the existence and number of the limit cycles depending on certain parameters. To prove existence of a stable limit cycle in the case “focus-center,” we use a pure geometric approach. In the case “focus-focus,” we prove existence of a special configuration of five parameters leading to the existence of a unique stable limit cycle, whose period can be found by solving a transcendent equation. An estimate of this period is obtained. We apply this theory on a two-dimensional system describing the qualitative behavior of a two-dimensional excitable membrane model.     

Global analysis of the shadow Gierer-Meinhardt system with general linear boundary conditions in a random environment

The global analysis of the shadow Gierer-Meinhardt system with multiplicative white noise and general linear boundary conditions is investigated in this paper. For this reaction-diffusion system, we employ a fixed point argument to prove local existence and uniqueness. Our results on global existence are based on \emph{a priori} estimates of solutions.     

Stability and robustness of quasi-linear impulsive hybrid systems

Both hybrid dynamical systems and impulsive dynamical systems are studied extensively in the literature. However, impulsive hybrid systems are not yet well studied. Nonetheless, many physical systems exhibit both system switching and impulsive jump phenomena. This paper investigates stability and robust stability of a class of quasi-linear impulsive hybrid systems by using the methods of Lyapunov functions and Riccati inequalities. Sufficient conditions for stability and robust stability of those systems are established. Some examples are given to illustrate the applicability of our results.     

The central limit theorem for random perturbations of rotations

Summary.   We prove a functional central limit theorem for stationary random sequences given by the transformations

一类时滞系统的稳定性分析及控制器设计

研究一类具有状态时滞和输入时滞的时变时滞线性系统.首先,通过选取合适的Lyapunov-Krasovskii泛函,应用LMI方法和Lyapunov-Krasovskill稳定性定理对时滞相关的系统进行稳定性分析,并设计了相应的控制器.改进了时变时滞线性系统方面的一些结果.最后用实例验证所得到结果.     

An iterative method for a system of linear complementarity problems with perturbations and interval data

In this paper, we introduce a total step method for solving a system of linear complementarity problems with perturbations and interval data. It is applied to two interval matrices [A] and [B] and two interval vectors [b] and [c]. We prove that the sequence generated by the total step method converges to ([x^∗],[y^∗]) which includes the solution set for the system of linear complementarity problems defined by any fixed A∈[A],B∈[B],b∈[b] and c∈[c]. We also consider a modification of the method and show that, if we start with two interval vectors containing the limits, then the iterates contain the limits. We close our paper with two examples which illustrate our theoretical results.