Related papers: Local bifurcations in differential equations with …
This chapter presents a dynamical systems point of view of the study of systems with delays. The focus is on how advanced tools from bifurcation theory, as implemented for example in the package DDE-BIFTOOL, can be applied to the study of…
The aim of this paper is to provide an effective framework for analysing bifurcations of equilibria in nonlinearly periodically forced delay differential equations. First, we establish the existence of a periodic smooth finite-dimensional…
A recent work by the authors on the existence of a periodic smooth finite-dimensional center manifold near a nonhyperbolic cycle in delay differential equations motivates the derivation of periodic normal forms. In this paper, we prove the…
Discontinuities and delayed terms are encountered in the governing equations of a large class of problems ranging from physics and engineering to medicine and economics. These systems cannot be properly modelled and simulated with standard…
Periodic orbits and associated bifurcations of singularly perturbed state-dependent delay differential equations (DDEs) are studied when the profiles of the periodic orbits contain jump discontinuities in the singular limit. A definition of…
In this paper we perform the parameter-dependent center manifold reduction near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf and transcritical-Hopf bifurcations in delay differential equations (DDEs). This allows us to initialize the…
DDEBIFTOOL is a collection of Matlab routines for numerical bifurcation analysis of systems of delay differential equations with discrete constant and state-dependent delays. The package supports continuation and stability analysis of…
A delayed term in a differential equation reflects the fact that information takes significant time to travel from one place to another within a process being studied. Despite de apparent similarity with ordinary differential equations,…
We consider state-dependent delay equations (SDDE) obtained by adding delays to a planar ordinary differential equation with a limit cycle. These situations appear in models of several physical processes, where small delay effects are…
Delayed processes are ubiquitous in biological systems and are often characterized by delay differential equations (DDEs) and their extension to include stochastic effects. DDEs do not explicitly incorporate intermediate states associated…
Differential equations with state-dependent delays define a semiflow of continuously differentiable solution operators in general only on the associated {\it solution manifold} $X\subset C^1([-h,0],\mathbb{R}^n)$. For systems with discrete…
Differential equations with state-dependent delays define a semiflow of continuously differentiable solution operators in general only on an associated submanifold of the Banach space $C^1([-h,0],\mathbb{R}^n)$. We extend a recent result on…
Partial differential equations with discrete (concentrated) state-dependent delays are studied. The existence and uniqueness of solutions with initial data from a wider linear space is proven first and then a subset of the space of…
This work concerns the dynamics of a certain class of delay differential equations (DDEs) which we refer to as state dependent delay maps. These maps are generated by delay differential equations where the derivative of the current state…
Recent work in [53, 54] by the authors on periodic center manifolds and normal forms for bifurcations of limit cycles in delay differential equations (DDEs) motivates the derivation of explicit computational formulas for the critical normal…
Delays are ubiquitous in applied problems, but often do not arise as the simple constant discrete delays that analysts and numerical analysts like to treat. In this chapter we show how state-dependent delays arise naturally when modeling…
Local bifurcation theory typically deals with the response of a degenerate but isolated equilibrium state or periodic orbit of a dynamical system to perturbations controlled by one or more independent parameters, and characteristically uses…
Neural field models with transmission delay may be cast as abstract delay differential equations (DDE). The theory of dual semigroups (also called sun-star calculus) provides a natural framework for the analysis of a broad class of delay…
In this note we consider local invariant manifolds of functional differential equations representing differential equations with state-dependent delay. Starting with a local center-stable and a local center-unstable manifold of the…
Delay differential equations (DDEs) with large delays play a pivotal role in understanding stability and bifurcations in systems ranging from neural networks to laser dynamics. While prior work has extensively studied DDEs with discrete…