Stability Results for a Class of Difference Systems with Delay

Considering the linear delay difference system x(n+1)=ax(n)+Bx(n-k), where a∈(0,1), B is a p×p real matrix, and k is a positive integer, the stability domain of the null solution is completely characterized in terms of the eigenvalues of the matrix B. It is also shown that the stability domain becomes smaller as the delay increases. These results may be successfully applied in the stability analysis of a large class of nonlinear difference systems, including discrete-time Hopfield neural networks

Chaos in a Discrete Delay Population Model

This paper is concerned with chaos in a discrete delay population model. The map of the model is proved to be chaotic in the sense of both Devaney and Li-Yorke under some conditions, by employing the snap-back repeller theory. Some computer simulations are provided to visualize the theoretical result

Anticontrol of Chaos for a Class of Delay Difference Equations Based on Heteroclinic Cycles Connecting Repellers

This paper is concerned with anticontrol of chaos for a class of delay difference equations via the feedback control technique. The controlled system is first reformulated into a high-dimensional discrete dynamical system. Then, a chaotification theorem based on the heteroclinic cycles connecting repellers for maps is established. The controlled system is proved to be chaotic in the sense of both Devaney and Li-Yorke. An illustrative example is provided with computer simulations

Co-Existence of Chaos and Stable Periodic Orbits in a Simple Discrete Neural Network

Communicated by J. Bélair Summary. We show that a simple discrete network of two identical neurons can demonstrate chaotic behavior near the origin. This is complementary to the results in Wu and Zhang (Disc. Contin. Dynam. Syst. Series B, 4 (2004), 853–865), where it was shown that the same system can have a large capacity of stable periodic orbits in a region away from the origin