Center Manifold Theory for Functional Differential Equations of Mixed Type

Abstract

Abstract. We study the behaviour of solutions to nonlinear autonomous functional differential equations of mixed type in the neighbourhood of an equilibrium. We show that all solutions that remain sufficiently close to an equilibrium can be captured on a finite dimensional invariant center manifold, that inherits the smoothness of the nonlinearity. In addition, we provide a Hopf bifurcation theorem for such equations. We illustrate the application range of our results by discussing an economic life-cycle model that gives rise to functional differential equations of mixed type.

Lin's method and homoclinic bifurcations for functional differential equations of mixed type

“…As a final preparation, we will assume that for every j ∈ J we have obtained the splittings 11) as introduced at the end of Section 5. We write α f S and α f R for the exponential rates associated to the fast spaces S f (ω + , µ) and R f (ω − , µ).…”

“…One of the driving forces in these investigations is the desire to apply the powerful tools that are currently available for ODEs to the infinite dimensional setting of (1.1). The constructions in [10,11] concerning finite dimensional center manifolds, which describe the behaviour of solutions to (1.1) in the vicinity of equilibria and periodic solutions, should be seen in this light.…”

“…In previous work [9,10,11], the absence of a variation-of-constants formula was circumvented by utilizing variants of the Greens function that was constructed by Mallet-Paret for autonomous MFDEs [14]. Continuing in this spirit, we will use the Fredholm theory developed in [14] for nonautonomous MFDEs to construct inverses for inhomogeneous MFDEs on half-lines.…”

Competitive Growth in a Life‐cycle Model: Existence and Dynamics*

“…Hupkes and Verduyn Lunel [14]. However, when the dynamics is non linear, the existence problem studied below, that relies on the dynamics behavior on the initial interval of time is still an open question.…”

Oscillatory Mixed Differential Systems

“…; x n ðtÞ T A R n ; the A i and B j are n-by-n real matrices and the r i and t j are positive real numbers. Systems of this kind can arise in the study on travelling waves in domains with nonlocal interactions initiated in [1], [2], and more recently developed in [3], [4]. Equation (1) can be looked in the more general framework of the mixed functional di¤erential system…”