Yuming Shi

CHAOTIFICATION OF DISCRETE DYNAMICAL SYSTEMS IN BANACH SPACES

This paper is concerned with chaotification of discrete dynamical systems in Banach spaces via feedback control techniques. A criterion of chaos in Banach spaces is first established. This criterion extends and improves the Marotto theorem. Discussions are carried out in general and some special Banach spaces. All the controlled systems are proved to be chaotic in the sense of both Devaney and Li–Yorke. As a consequence, a controlled system described in a finite-dimensional real space studied by Wang and Chen is shown chaotic not only in the sense of Li–Yorke but also in the sense of Devaney. The original system can be driven to be chaotic by using an arbitrarily small-amplitude state feedback control in a certain space. In addition, the Chen–Lai anti-control algorithm via feedback control with mod-operation in a finite-dimensional real space is extended to a certain infinite-dimensional Banach space, and the controlled system is shown chaotic in the sense of Devaney as well as in the sense of both Li–Yorke and Wiggins. Differing from many existing results, it is not here required that the map corresponding to the original system has a fixed point in some cases. An application of the theoretical results to a class of first-order partial difference equations is given with some numerical simulations.

Defect indices and definiteness conditions for a class of discrete linear Hamiltonian systems

Abstract This paper is concerned with a class of discrete linear Hamiltonian systems in finite or infinite intervals. A definiteness condition and its equivalent statements are discussed and three sufficient conditions for the definiteness condition are given. A precise relationship between the defect index of the minimal subspace generated by the system and the number of linearly independent square summable solutions of the system is established. In particular, they are equal if and only if the definiteness condition is satisfied. Finally, two criteria for the limit point case and one criterion for the limit circle case are obtained.

Chaos of time-varying discrete dynamical systems

This paper is concerned with chaos of time-varying (i.e. non-autonomous) discrete systems in metric spaces. Some basic concepts are introduced for general time-varying systems, including periodic point, coupled-expansion for transitive matrix, uniformly topological equiconjugacy, and three definitions of chaos, i.e. chaos in the sense of Devaney and Wiggins, respectively, and in a strong sense of Li–Yorke. An interesting observation is that a finite-dimensional linear time-varying system can be chaotic in the original sense of Li–Yorke, but cannot have chaos in the strong sense of Li–Yorke, nor in the sense of Devaney in a set containing infinitely many points, and nor in the sense of Wiggins in a set starting from which all the orbits are bounded. A criterion of chaos in the original sense of Li–Yorke is established for finite-dimensional linear time-varying systems. Some basic properties of topological conjugacy are discussed. In particular, it is shown that topological conjugacy alone cannot guarantee two topologically conjugate time-varying systems to have the same topological properties in general. In addition, a criterion of chaos induced by strict coupled-expansion for a certain irreducible transitive matrix is established, under which the corresponding nonlinear system is proved chaotic in the strong sense of Li–Yorke. Two illustrative examples are finally provided with computer simulations for illustration.

Introduction to anti-control of discrete chaos: theory and applications.

In this paper, the notion of anti-control of chaos (or chaotification) is introduced, which means to make an originally non-chaotic dynamical system chaotic or enhance the existing chaos of a chaotic system. The main interest in this paper is to employ the classical feedback control techniques. Only the discrete case is discussed in detail, including both finite-dimensional and infinite-dimensional settings.

The limit circle and limit point criteria for second-order linear difference equations☆

Abstract This paper is concerned with the limit circle and limit point criteria of second-order linear difference equations. A sufficient and necessary condition, a sufficient and necessary condition subject to a certain restriction, and several sufficient conditions are established. These results improve and extend some previous results.

CHAOTIFICATION OF DISCRETE DYNAMICAL SYSTEMS GOVERNED BY CONTINUOUS MAPS

This paper is concerned with chaotification of discrete dynamical systems in finite-dimensional real spaces, via feedback control techniques. A chaotification theorem for one-dimensional discrete dynamical systems and a chaotification theorem for general higher-dimensional discrete dynamical systems are established, respectively. The controlled systems are proved to be chaotic in the sense of Devaney. In particular, the maps corresponding to the original systems and designed controllers are only required to satisfy some mild assumptions on two very small disjoint closed subsets in the domains of interest. This condition is weaker than those in the existing relevant literature.

Sensitivity of non-autonomous discrete dynamical systems

Abstract In this paper, the sensitivity for non-autonomous discrete systems is investigated. First of all, two sufficient conditions of sensitivity for general non-autonomous dynamical systems are presented. At the same time, one stronger form of sensitivity, that is, cofinite sensitivity, is introduced for non-autonomous systems. Two sufficient conditions of cofinite sensitivity for general non-autonomous dynamical systems are presented. We generalized the result of sensitivity and strong sensitivity for autonomous discrete systems to general non-autonomous discrete systems, and the conditions in this paper are weaker than the correlated conditions of autonomous discrete systems.

The defect index of singular symmetric linear difference equations with real coefficients

This paper is concerned with the defect index of singular symmetric linear difference equations of order 2n with real coefficients and one singular endpoint. We show that their defect index d satisfies the inequalities n ≤ d ≤ 2n and that all values of d in this range are realized. This parallels the well known result of Glazman for differential equations established about 1950. In addition, several criteria of the limit point and strong limit point cases are established.

Spectral theory of higher-order discrete vector Sturm–Liouville problems☆

Abstract This paper is concerned with spectral problems of higher-order vector difference equations with self-adjoint boundary conditions, where the coefficient of the leading term may be singular. A suitable admissible function space is constructed so that the corresponding difference operator is self-adjoint in it, and the fundamental spectral results are obtained. Rayleighs principles and minimax theorems in two special linear spaces are given. As an application, comparison theorems for eigenvalues of two Sturm–Liouville problems are presented. Especially, the dual orthogonality and multiplicity of eigenvalues are discussed.

COUPLED-EXPANDING MAPS FOR IRREDUCIBLE TRANSITION MATRICES

In this paper, strictly A-coupled-expanding maps in bounded and closed subsets of complete metric spaces are investigated, where A = (aij) is an m × m irreducible transition matrix with one row-sum no less than 2. A map f is said to be strictly A-coupled-expanding in m sets Vi if f(Vi) ⊃ Vj whenever aij = 1 and the distance between any two different sets of these Vi is positive. A new result on the subshift for matrix A is obtained. Based on this result, two criteria of chaos are established, which generalize and relax the conditions of some existing results. These maps are proved to be chaotic either in the sense of both Li–Yorke and Wiggins or in the sense of both Li–Yorke and Devaney. One example is provided to illustrate the theoretical results with a computer simulation for demonstration.