Period-halving Bifurcation of a Neuronal Recurrence Equation

10/17/2011

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by   Renè Ndoundam, et al.

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We study the sequences generated by neuronal recurrence equations of the form x(n) = 1[∑_j=1^h a_j x(n-j)- θ]. From a neuronal recurrence equation of memory size h which describes a cycle of length ρ(m) × lcm(p_0, p_1,..., p_-1+ρ(m)), we construct a set of ρ(m) neuronal recurrence equations whose dynamics describe respectively the transient of length O(ρ(m) × lcm(p_0, ..., p_d)) and the cycle of length O(ρ(m) × lcm(p_d+1, ..., p_-1+ρ(m))) if 0 ≤ d ≤ -2+ρ(m) and 1 if d=ρ(m)-1. This result shows the exponential time of the convergence of neuronal recurrence equation to fixed points and the existence of the period-halving bifurcation.