Bifurcations and chaos in two-cell cellular neural networks with periodic inputs

Abstract

This study investigates bifurcations and chaos in two-cell Cellular Neural Networks (CNN) with periodic inputs. Without the inputs, the time periodic solutions are obtained for template A = [r, p, s] with p > 1, r > p - 1 and -s > p - 1. The number of periodic solutions can be proven to be no more than two in exterior regions. The input is b sin 2pit/T with period T > 0 and amplitude b > 0. The typical trajectories Gamma(b, T, A) and their omega-limit set omega(b, T, A) vary with b, T and A are also considered. The asymptotic limit cycles A,,(T, A) with period T of Gamma(b, T, A) are obtained as b --> infinity. When T-0 less than or equal to T-0* (given in (67)), Lambda(infinity) and -Lambda(infinity) can be separated. The onset of chaos can be induced by crises of omega(b, T, A) and -omega(b, T, A) for suitable T and b. The ratio A(b) = aT(b)/a(1)(b), of largest amplitude a(1)(b) except for T-mode and amplitude of the T-mode of the Fast Fourier Transform (FFT) of r(b, T, A), can be used to compare the strength of sustained periodic cycle Lambdao(A) and the inputs. When A(b) much less than 1, Lambda(o)(A) dominates and the attractor omega(b, T, A) is either a quasi-periodic or a periodic. Moreover, the range b of the window of periodic cycles constitutes a devil's staircase. When A(b) similar to 1, finitely many chaotic regions and window regions exist and interweave with each other. In each window, the basic periodic cycle can be identified. A sequence of period-doubling is observed to the left of the basic periodic cycle and a quasi-periodic region is observed to the right of it. For large b, the input dominates, omega(b, T, A) becomes simpler, from quasi-periodic to periodic as b increases.