Necessary and sufficient condition for the global stability of a delayed discrete-time single neuron model

Abstract

We consider the global asymptotic stability of the trivial fixed point of the difference equation x n+1 = mxn − αϕ(x n−1), where (α, m) ∈ R 2 and ϕ is a real function satisfying the discrete Yorke condition: min{0, x} ≤ ϕ(x) ≤ max{0, x} for all x ∈ R. If ϕ is bounded then (α, m) ∈ [|m| − 1, 1] × [−1, 1], (α, m) = (0, −1), (0, 1) is necessary for the global stability of 0. We prove that if ϕ(x) ≡ tanh(x), then this condition is sufficient as well.

Fixed points of a destabilized Kuramoto–Sivashinsky equation

“…Studying this map on the finite dimensional space, where the self-consistent bounds are represented, opens up the possibility for using graph representation techniques [25][26][27][28] to estimate the basin of attraction.…”

Global stability for the three-dimensional logistic map

“…Note that coefficients β ij are independent of c, as in T (r, C) the parameter c appears only in the term c|z| 5 . Actually, that is why we chose the fourth order approximation of the center manifold and the fifth order term c|z| 5 in T .…”

Global stability for the 2-dimensional logistic map

“…For a ≤ 2 the proof of the global stability is a combination of analytical and computer-aided tools. It is based on the method in [9] and [10]. We elaborate the analytical part such that it can be easily applied to similar models.…”