On Nonoscillation of Mixed Advanced-Delay Differential Equations with Positive and Negative Coefficients

For a mixed (advanced--delay) differential equation with variable delays and coefficients $$\dot{x}(t) \pm a(t)x(g(t)) \mp b(t)x(h(t)) = 0, t\geq t_0$$ where $$a(t)\geq 0, b(t)\geq 0, g(t)\leq t, h(t)\geq t$$ explicit nonoscillation conditions are obtained.Comment: 17 pages; 2 figures; to appear in Computers & Mathematics with Application

Hopf bifurcations in time-delay systems with band-limited feedback

We investigate the steady-state solution and its bifurcations in time-delay systems with band-limited feedback. This is a first step in a rigorous study concerning the effects of AC-coupled components in nonlinear devices with time-delayed feedback. We show that the steady state is globally stable for small feedback gain and that local stability is lost, generically, through a Hopf bifurcation for larger feedback gain. We provide simple criteria that determine whether the Hopf bifurcation is supercritical or subcritical based on the knowledge of the first three terms in the Taylor-expansion of the nonlinearity. Furthermore, the presence of double-Hopf bifurcations of the steady state is shown, which indicates possible quasiperiodic and chaotic dynamics in these systems. As a result of this investigation, we find that AC-coupling introduces fundamental differences to systems of Ikeda-type [Ikeda et al., Physica D 29 (1987) 223-235] already at the level of steady-state bifurcations, e.g. bifurcations exist in which limit cycles are created with periods other than the fundamental ``period-2'' mode found in Ikeda-type systems.Comment: 32 pages, 5 figures, accepted for publication in Physica D: Nonlinear Phenomen

Complex oscillations in the delayed Fitzhugh-Nagumo equation

Motivated by the dynamics of neuronal responses, we analyze the dynamics of the Fitzhugh-Nagumo slow-fast system with delayed self-coupling. This system provides a canonical example of a canard explosion for sufficiently small delays. Beyond this regime, delays significantly enrich the dynamics, leading to mixed-mode oscillations, bursting and chaos. These behaviors emerge from a delay-induced subcritical Bogdanov-Takens instability arising at the fold points of the S-shaped critical manifold. Underlying the transition from canard-induced to delay-induced dynamics is an abrupt switch in the nature of the Hopf bifurcation

Bifurcation analysis of a normal form for excitable media: Are stable dynamical alternans on a ring possible?

We present a bifurcation analysis of a normal form for travelling waves in one-dimensional excitable media. The normal form which has been recently proposed on phenomenological grounds is given in form of a differential delay equation. The normal form exhibits a symmetry preserving Hopf bifurcation which may coalesce with a saddle-node in a Bogdanov-Takens point, and a symmetry breaking spatially inhomogeneous pitchfork bifurcation. We study here the Hopf bifurcation for the propagation of a single pulse in a ring by means of a center manifold reduction, and for a wave train by means of a multiscale analysis leading to a real Ginzburg-Landau equation as the corresponding amplitude equation. Both, the center manifold reduction and the multiscale analysis show that the Hopf bifurcation is always subcritical independent of the parameters. This may have links to cardiac alternans which have so far been believed to be stable oscillations emanating from a supercritical bifurcation. We discuss the implications for cardiac alternans and revisit the instability in some excitable media where the oscillations had been believed to be stable. In particular, we show that our condition for the onset of the Hopf bifurcation coincides with the well known restitution condition for cardiac alternans.Comment: to be published in Chao

A New Formulation of the Initial Value Problem for Nonlocal Theories

There are a number of reasons to entertain the possibility that locality is violated on microscopic scales, for example through the presence of an infinite series of higher derivatives in the fundamental equations of motion. This type of nonlocality leads to improved UV behaviour, novel cosmological dynamics and is a generic prediction of string theory. On the other hand, fundamentally nonlocal models are fraught with complications, including instabilities and complications in setting up the initial value problem. We study the structure of the initial value problem in an interesting class of nonlocal models. We advocate a novel new formulation wherein the Cauchy surface is "smeared out" over the underlying scale of nonlocality, so that the the usual notion of initial data at t=0 is replaced with an "initial function" defined over -M^{-1} \leq t \leq 0 where M is the underlying scale of nonlocality. Focusing on some specific examples from string theory and cosmology, we show that this mathematical re-formulation has surprising implications for the well-known stability problem. For D-brane decay in a linear dilaton background, we are able to show that the unstable directions in phase space cannot be accessed starting from a physically sensible initial function. Previous examples of unstable solutions in this model therefore correspond to unphysical initial conditions, an observation which is obfuscated in the old formulation of the initial value problem. We also discuss implication of this approach for nonlocal cosmological models.Comment: 36 pages, 9 figures. Accepted for publication in Nuclear Physics

Numerical test for hyperbolicity of chaotic dynamics in time-delay systems

We develop a numerical test of hyperbolicity of chaotic dynamics in time-delay systems. The test is based on the angle criterion and includes computation of angle distributions between expanding, contracting and neutral manifolds of trajectories on the attractor. Three examples are tested. For two of them previously predicted hyperbolicity is confirmed. The third one provides an example of a time-delay system with nonhyperbolic chaos.Comment: 7 pages, 5 figure