Bifurcation analysis in a neutral differential equation

Abstract

The dynamics of a neural network model in neutral form is investigated. We prove that a sequence of Hopf bifurcations occurs at the origin as the delay increases. The direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are determined by using normal form method and center manifold theory. Global existence of periodic solutions is established using a global Hopf bifurcation result of Krawcewicz et al. and a Bendixson's criterion for higher dimensional ordinary differential equations due to Li and Muldowney.

Positive Almost Periodic Solution for a Model of Hematopoiesis with Infinite Time Delays and a Nonlinear Harvesting Term

“…Neutral functional differential equations arise in many areas of applied mathematics; it has received considerable attention in the recent decades. There have been many remarkable works (see, [1][2][3][4][5]). It is well known that there has been considerable interest in the existence of almost periodic solutions of functional differential equations [3][4][5][6][7][8][9].…”

“…In (2), throughout this paper, the following hypotheses conditions hold: , and (0) > 0 are needed. We write ( 0 , )( ( ; 0 , )) for an admissible solution of (2) with the above initial conditions.…”

Bifurcation Analysis in a Two-Dimensional Neutral Differential Equation

“…One can use the theory to study the existence of Hopf bifurcation to a neutral delay differential equation. In [2], Wang and Wei extend the computation of the properties of Hopf bifurcation, such as the direction of bifurcation and stability of bifurcating periodic solutions, of DDE introduced by Kazarinoff et al Based on combining the global Hopf bifurcation theory of neutral equations due to Krawcewicz et al and the higher dimensional Bendixson's criterion for ordinary differential equations due to Li and Muldowney, the global existence of periodic solutions of neutral differential equations have been studied by Qu et al [3]. But research findings of bifurcation about highdimension neutral differential equations can rarely be found.…”

“…is assumed adequately smooth, for example, ∈ 3 , and satisfies the following condition: (H1) (0) = 0, (0) = 1, and there exists > 0, such that | ( )| ⩽ for all ∈ R, (H2) ( ) > 0 for all ∈ R, ( ) < 0 for all ̸ = 0.…”