Existence of periodic solutions and closed invariant curves in a class of discrete-time cellular neural networks

This paper discusses the dynamical behavior of excitatory-inhibitory discrete-time cellular neural networks (DTCNNs) with piecewise linear output functions. Our analysis shows that such DTCNNs have periodic solutions and closed invariant curves, and all their solutions, except for fixed points, eventually stay on the closed invariant curves. Moreover, these results are also illustrated by examples and figures. These results demonstrate that excitatory-inhibitory DTCNNs can exhibit permanent nonlinear oscillations. Moreover, such DTCNNs with permanent nonlinear oscillations may be chosen arbitrarily to close a DTCNN satisfying the SP-Condition which ensures the complete stability of DTCNNs. Thus, this work indicates that the SP-Condition on complete stability is not robust.