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

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 2πt/T with period T > 0 and amplitude b > 0. The typical trajectories Γ(b, T, A) and their ω-limit set ω(b, T, A) vary with b, T and A are also considered. The asymptotic limit cycles Λ_∞(T, A) with period T of Γ(b, T, A) are obtained as b → ∞. When T₀ ≤ T₀^* (given in (67)), Λ,_∞ and -Λ_∞ can be separated. The onset of chaos can be induced by crises of ω(b, T, A) and -ω(b, T, A) for suitable T and b. The ratio A(b) = |a_T(b)|/|a₁(b)|, of largest amplitude a₁(b) except for T-mode and amplitude of the T-mode of the Fast Fourier Transform (FFT) of Γ(b, T, A), can be used to compare the strength of sustained periodic cycle Λ₀(A) and the inputs. When A(b) ≪ 1, Λ₀(A) dominates and the attractor ω(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) ∼ 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, ω(b, T, A) becomes simpler, from quasi-periodic to periodic as b increases.