5,057 research outputs found

    Switching to nonhyperbolic cycles from codimension two bifurcations of equilibria of delay differential equations

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    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 continuation of codimension one equilibria and cycle bifurcations emanating from these codimension two bifurcation points. The normal form coefficients are derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative. The obtained expressions give explicit formulas which have been implemented in the freely available numerical software package DDE-BifTool. While our theoretical results are proven to apply more generally, the software implementation and examples focus on DDEs with finitely many discrete delays. Together with the continuation capabilities of DDE-BifTool, this provides a powerful tool to study the dynamics near equilibria of such DDEs. The effectiveness is demonstrated on various models

    Delay Equations and Radiation Damping

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    Starting from delay equations that model field retardation effects, we study the origin of runaway modes that appear in the solutions of the classical equations of motion involving the radiation reaction force. When retardation effects are small, we argue that the physically significant solutions belong to the so-called slow manifold of the system and we identify this invariant manifold with the attractor in the state space of the delay equation. We demonstrate via an example that when retardation effects are no longer small, the motion could exhibit bifurcation phenomena that are not contained in the local equations of motion.Comment: 15 pages, 1 figure, a paragraph added on page 5; 3 references adde

    Hysteresis in Adiabatic Dynamical Systems: an Introduction

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    We give a nontechnical description of the behaviour of dynamical systems governed by two distinct time scales. We discuss in particular memory effects, such as bifurcation delay and hysteresis, and comment the scaling behaviour of hysteresis cycles. These properties are illustrated on a few simple examples.Comment: 28 pages, 10 ps figures, AMS-LaTeX. This is the introduction of my Ph.D. dissertation, available at http://dpwww.epfl.ch/instituts/ipt/berglund/these.htm

    On Local Bifurcations in Neural Field Models with Transmission Delays

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    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 equations, among which DDE. In particular, it may be used advantageously for the investigation of stability and bifurcation of steady states. After introducing the neural field model in its basic functional analytic setting and discussing its spectral properties, we elaborate extensively an example and derive a characteristic equation. Under certain conditions the associated equilibrium may destabilise in a Hopf bifurcation. Furthermore, two Hopf curves may intersect in a double Hopf point in a two-dimensional parameter space. We provide general formulas for the corresponding critical normal form coefficients, evaluate these numerically and interpret the results
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