Equivariant normal forms for neutral functional differential equations

Abstract

This paper addresses the computation of equivariant normal forms for some Neutral Functional Differential Equations (NFDEs) near equilibria in the presence of symmetry. The analysis is based on the theory previously developed for autonomous retarded Functional Differential Equations (FDEs) and on the existence of center (or other invariant) manifolds. We illustrate our general results by some applications to a detailed case study of additive neurons with delayed feedback.

Hopf Bifurcation Analysis in a Class of Scalar Functional Differential Equations

“…In recent years, a sequence of results about equivariant Hopf bifurcation in neutral functional differential equations [28,29] and functional differential equations of mixed type [30] have been established. In particular, Guo [31] applied the equivariant Hopf bifurcation theorem to study the Hopf bifurcation of a delayed Ginzburg-Landau equation on a two-dimensional disk with the homogeneous Dirichlet boundary condition.…”

Periodic solutions of neutral fractional differential equations

“…It is well known that neutral functional differential equations are used to represent important physical systems. We refer to for a discussion about this aspect of the theory. Similarly, motivated by the fact that abstract neutral functional differential equations (abbreviated, ANFDE) arise in many areas of applied mathematics, this type of equations has received much attention in recent years ().…”

Bifurcation analysis in a neutral differential equation

“…We would like to mention that, as far as we know, there are a few articles on the global existence of periodic solutions for neutral differential equations, we refer to Krawcewicz, Wu and Xia [16,34,36] and Wei and Ruan [31]. Recently, several interesting articles on the stability, bifurcation theory and numerical solutions of neutral differential equations, and the fundamental theory of the neutral type differential equations and inclusions, have been published, we refer to [1,9,10,15,[25][26][27]2] and [11,13,17], respectively. The present paper is the first to study the global existence of periodic solutions of neutral differential equations by combining the global Hopf bifurcation theory of neutral equations due to Krawcewicz, Wu and Xia and the higher dimensional Bendixson's criterion for ordinary differential equations due to Li and Muldowney.…”