Invariant Sets and Exact Solutions to Higher-Dimensional Wave Equations

The invariant sets and exact solutions of the （1 ＋ 2）-dimensional wave equations are discussed. It is shown that there exist a class of solutions to the equations which belong to the invariant set E0 = {u ： ux = vxF（u）,uy = vyF（u） }. This approach is also developed to solve （1 ＋ N）-dimensional wave equations.

Finite-time stability with respect to a closed invariant set for a class of discontinuous systems

This paper discusses the problem of finite-time stability with respect to a closed, but not necessarily compact, invariant set for a class of nonlinear systems with discontinuous right-hand sides in the sense of the Filippov solutions. When the Lyapunov function is Lipschitz continuous and regular, the Lyapunov theorem on finite-time stability with respect to a closed invariant set is presented.

Estimating the globally attractive set and positively invariant set of a unified chaotic system

By constructing two suitable generalized Lyapunov functions,we derived a generalized ellipsoidal estimate of the globally attractive set and positively invariant set of the unified chaotic system with the parameters α=1/29 and 1/29<α<2/29,respectively,which extends some related results of Li,et al. [Li DM,Lu JA,Wu XQ,Chen GR,Estimating the global basin of attraction and positively invariant set for the Lorenz system and a unified chaotic system,Journal of Mathematical Analysis and Applications,2006,323(2): 844-853]. The theoretical results obtained in this paper will find wide application in chaos control and synchronization.

THE POSITIVE INVARIANT SET OF THE DIFFERENTIAL EQUATION ON BANACH SPACE

A sufficient condition of the existence of a positive invariant set in the differential equation on Banach space is given, which improves a theorem of Mitio Nagumo.

Exact Solutions and Invariant Sets to General Reaction-Diffusion Equation

In this paper, we introduce new invariant sets, and the invariant sets and exact solutions to general reactiondiffusion equation are discussed. It is shown that there exist a class of exact solutions to the equations that belong to the invariant sets.

Estimating the ultimate bound and positively invariant set for a generalized Lorenz system

A generalized Lyapunov function was employed to investigate the ultimate bound and positively invariant set of a generalized Lorenz system.We derived an ellipsoidal estimate of the ultimate bound and positively invariant set for the generalized Lorenz system,for all the positive values of system parameters a,b,and c.Our results extend the related result of Li,et al.[Li DM,Lu JA,Wu XQ,et al.,Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system,Journal of Mathematical Analysis and Application,2006,323(2):844-653].

Invariant Sets and Exact Solutions to Nonlinear Diffusion Equations with x-Dependent Convection and Absorption

In this paper, we introduce a new invariant set Eo={u：ux=f＇（x）F（u）＋ε[g＇（x）-f＇（x）g（x）]F（u）×exp（-∫^u1/F（z）dz）}where f and g are some smooth functions of x, ε is a constant, and F is a smooth function to be determined. The invariant sets and exact sohltions to nonlinear diffusion equation ut = （ D（u）ux）x ＋ Q（x, u）ux ＋ P（x, u）, are discussed. It is shown that there exist several classes of solutions to the equation that belong to the invariant set Eo.

Stabilizing model predictive control scheme for piecewise affine systems with maximal positively invariant terminal set

An efficient algorithm is proposed for computing the solution to the constrained finite time optimal control （CFTOC） problem for discrete-time piecewise affine （PWA） systems with a quadratic performance index. The maximal positively invariant terminal set, which is feasible and invariant with respect to a feedback control law, is computed as terminal target set and an associated Lyapunov function is chosen as terminal cost. The combination of these two components guarantees constraint satisfaction and closed-loop stability for all time. The proposed algorithm combines a dynamic programming strategy with a multi-parametric quadratic programming solver and basic polyhedral manipulation. A numerical example shows that a larger stabilizable set of states can be obtained by the proposed algorithm than precious work.

La Shalle＇s invariant-set-theory based asymptotic synchronization of duffing system with unknown parameters

A novel La Shalle＇s invariant set theory （LSIST） based adaptive asymptotic synchronization （LSISAAS） method is proposed to asymptotically synchronize Duffing system with unknown parameters which also are considered as system states. The LSISASS strategy depends on the only information, i.e. one state of the master system. According to the LSIST, the LSISASS method can asymptotically synchronize fully the states of the master system and the unknown system parameters as well. Simulation results also validate that the LSISAAS approach can obtain asymptotic synchronization.

A Note on Invariant Sets

The lower bound of Hausdorff dimension of the invariant set is given.

Hybrid Dynamic Variables-Dependent Event-Triggered Fuzzy Model Predictive Control

This article focuses on dynamic event-triggered mechanism(DETM)-based model predictive control(MPC) for T-S fuzzy systems.A hybrid dynamic variables-dependent DETM is carefully devised,which includes a multiplicative dynamic variable and an additive dynamic variable.The addressed DETM-based fuzzy MPC issue is described as a “min-max” optimization problem(OP).To facilitate the co-design of the MPC controller and the weighting matrix of the DETM,an auxiliary OP is proposed based on a new Lyapunov function and a new robust positive invariant(RPI) set that contain the membership functions and the hybrid dynamic variables.A dynamic event-triggered fuzzy MPC algorithm is developed accordingly,whose recursive feasibility is analysed by employing the RPI set.With the designed controller,the involved fuzzy system is ensured to be asymptotically stable.Two examples show that the new DETM and DETM-based MPC algorithm have the advantages of reducing resource consumption while yielding the anticipated performance.

Heteroclinic Cycles in a Class of 3-Dimensional Piecewise Affine Systems

This paper explores the existence of heteroclinic cycles and corresponding chaotic dynamics in a class of 3-dimensional two-zone piecewise affine systems. Moreover, the heteroclinic cycles connect two saddle foci and intersect the switching manifold at two points and the switching manifold is composed of two perpendicular planes.

Better Hausdorff Dimension Estimations of Quadratic and Cubic Functions' Julia Sets

More accurate Hausdorff dimension estimations of Julia sets for two simple functions are given by the methods of composition mapping and invariant set of contraction mapping. For quadratic function fc （ z ） = z^2 ＋ c（c ∈^C）, the range of parameter c is expanded largely and a result on the Hausdorff dimension of its Julia set is gained. Similarly, a better result is obtained for cubic function fc（z） = z^3 ＋ c（c ∈ ^C）.

Existence and Concentration of Sign-Changing Solutions of Quasilinear Choquard Equation

In this paper, we study the following quasilinear equation of choquard type: where A(x,t) is given real functions on RN × R and with N ≥ 3, 1 p N, max{N-2p,1} α N, , and ε > 0 is a small parameter, Iα is the Riesz potential. We establish for small ε the existence of a sequence of sign-changing solutions concentrating near a given local minimum point of the bounded potential function V by using the method of invariant sets of descending flow, perturbation method and truncation technique. .

Invariant sets and solutions to the generalized thin film equation

The invariant sets and the solutions of the 1+2-dimensional generalized thin film equation are discussed. It is shown that there exists a class of solutions to the equations, which are invariant with respect to the set $$E_0 = \{ u:u_x = v_x F(u),u_y = v_y F(u)\} ,$$ where v is a smooth function of variables x, y and F is a smooth function of u. This extends the results of Galaktionov (2001) and for the 1+1-dimensional nonlinear evolution equations.

On the new results of global attractive set and positive invariant set of the Lorenz chaotic system and the applications to chaos control and synchronization

Constructing a family of generalized Lyapunov functions, a new method is proposed to obtain new global attractive set and positive invariant set of the Lorenz chaotic system. The method we proposed greatly simplifies the complex proofs of the two famous estimations presented by the Russian scholar Leonov. Our uniform formula can derive a series of the new estimations. Employing the idea of intersection in set theory, we extract a new Leonov formula-like estimation from the family of the estimations. With our method and the new estimation, one can confirm that there are no equilibrium, periodic solutions, almost periodic motions, wandering motions or other chaotic attractors outside the global attractive set. The Lorenz butterfly-like singular attractors are located in the global attractive set only. This result is applied to the chaos control and chaos synchronization. Some feedback control laws are obtained to guarantee that all the trajectories of the Lorenz systems track a periodic solution, or globally stabilize an unstable (or locally stable but not globally asymptotically stable) equilibrium. Further, some new global exponential chaos synchronization results are presented. Our new method and the new results are expected to be applied in real secure communication systems.

Event-triggered MPC Design for Distributed Systems With Network Communications

This paper deals with the communication problem in the distributed system, considering the limited battery power in the wireless network and redundant transmission among nodes. We design an event-triggered model predictive control(ET-MPC) strategy to reduce the unnecessary communication while promising the system performance. On one hand, for a linear discrete time-invariant system, a triggering condition is derived based on the Lyapunov stability. Here, in order to further reduce the communication rate, we enforce a triggering condition only when the Lyapunov function will exceed its value at the last triggered time, but an average decrease is guaranteed. On the other hand, the feasibility is ensured by minimizing and optimizing the terminal constrained set between the maximal control invariant set and the target terminal set. Finally, we provide a simulation to verify the theoretical results. It's shown that the proposed strategy achieves a good trade-off between the closed-loop system performance and communication rate.

Stability analysis for continuous-time systems with actuator saturation

The aim of this paper is to study the determination of the stability regions for continuous-time systems subject to actuator saturation. Using an affine saturation-dependent Lyapunov function, a new method is proposed to obtain the estimation of the domain of attraction of the closed-loop system. A family of linear matrix inequalities （LMIs） that provides sufficient conditions for the existence of this type of Lyapunov function are presented. The results obtained in this paper can reduce the conservativeness compared with the existing ones. Numerical examples are given to illustrate the effectiveness of the proposed results.

Existence of solutions for fourth-order boundary value problem with parameter

This paper is a sequel to a previous paper （Yang, Y. and Zhang, J. H. Existence of solutions for some fourth-order boundary value problems with parameters. Nonlinear Anal. 69（2）, 1364-1375 （2008）） in which the nontrivial solutions to the fourthorder boundary value problems were studied. In the current work with the same conditions near infinity but different near zero, the positive, negative, and sign-changing solutions are obtained by the critical point theory, retracting property, and invariant sets.

Discrete Conley index and bifurcation points

In this paper, a sufficient condition for the existence of bifurcation points for discrete dynamical systems is presented. The relation between two families of systems is further discussed, and a sufficient condition for determining whether they may have the similar bifurcation points is given.