Tensor Products, Positive Linear Operators, and Delay-Differential Equations

We develop the theory of compound functional differential equations, which are tensor and exterior products of linear functional differential equations. Of particular interest is the equation $\dot x(t)=-\alpha(t)x(t)-\beta(t)x(t-1)$ with a single delay, where the delay coefficient is of one sign, say $\delta\beta(t)\ge 0$ with $\delta\in{-1,1}$. Positivity properties are studied, with the result that if $(-1)^k=\delta$ then the $k$-fold exterior product of the above system generates a linear process which is positive with respect to a certain cone in the phase space. Additionally, if the coefficients $\alpha(t)$ and $\beta(t)$ are periodic of the same period, and $\beta(t)$ satisfies a uniform sign condition, then there is an infinite set of Floquet multipliers which are complete with respect to an associated lap number. Finally, the concept of $u_0$-positivity of the exterior product is investigated when $\beta(t)$ satisfies a uniform sign condition.Comment: 84 page

Snakes: Oriented families of periodic orbits, their sources, sinks, and continuation

AbstractPoincaré observed that for a differential equation x′ = ƒ(x, α) depending on a parameter α, each periodic orbit generally lies in a connected family of orbits in (x, α)-space. In order to investigate certain large connected sets (denoted Q) of orbits containing a given orbit, we introduce two indices: an orbit index φ and a “center” index defined at certain stationary points. We show that genetically there are two types of Hopf bifurcation, those we call “sources” ( = 1) and “sinks” ( = −1). Generically if the set Q is bounded in (x, α)-space, and if there is an upper bound for periods of the orbits in Q, then Q must have as many source Hopf bifurcations as sink Hopf bifurcations and each source is connected to a sink by an oriented one-parameter “snake” of orbits. A “snake” is a maximal path of orbits that contains no orbits whose orbit index is 0. See Fig. 1.1

Traveling Wave Solutions for Systems of ODEs on a Two-Dimensional Spatial Lattice

This is the published version, also available here: http://dx.doi.org/10.1137/S0036139996312703.We consider infinite systems of ODEs on the two-dimensional integer lattice, given by a bistable scalar ODE at each point, with a nearest neighbor coupling between lattice points. For a class of ideal nonlinearities, we obtain traveling wave solutions in each direction $e^{i\theta}$, and we explore the relation between the wave speed c, the angle $\theta$, and the detuning parameter a of the nonlinearity. Of particular interest is the phenomenon of "propagation failure," and we study how the critical value $a=a^*(\theta)$ depends on $\theta$, where $a^*(\theta)$ is defined as the value of the parameter a at which propagation failure (that is, wave speed c=0) occurs. We show that $a^*:\Bbb{R}\to\Bbb{R} is continuous at each point$\theta$where$\tan\theta$is irrational, and is discontinuous where$\tan\theta$ is rational or infinite

Numerical investigation of noise induced changes to the solution behaviour of the discrete FitzHugh-Nagumo equation

In this work we introduce and analyse a stochastic functional equation, which contains both delayed and advanced arguments. This equation results from adding a stochastic term to the discrete FitzHugh-Nagumo equation which arises in mathematical models of nerve conduction. A numerical method is introduced to compute approximate solutions and some numerical experiments are carried out to investigate their dynamical behaviour and compare them with the solutions of the corresponding deterministic equation

Spatial Patterns, Spatial Chaos, And Traveling Waves In Lattice Differential Equations

We survey recent results in the theory of lattice differential equations. Such equations yield continuous-time, usually infinite-dimensional, dynamical systems, which possess a discrete spatial structure modeled on a lattice. The systems we consider, generally over a higher-dimensional lattice such as ZZ D ` IR D , are the simplest nontrivial ones which incorporate both local nonlinear dynamics and short range interactions. Of particular interest are stable equilibria, and the regular patterns, or lack thereof, that are displayed. Traveling wave solutions in such systems are also discussed. 1 Introduction By a lattice differential equation or LDE we mean a system of ordinary differential equations, often of infinite order, in which the state vector u = fu j g j2 is coordinatized by a set , the lattice, which possesses some underlying spatial structure. Typical choices of ` IR D are the D-dimensional integer lattices ZZ D , the hexagonal lattice in the plane, and the crystall..

The Global Structure of Traveling Waves in Spatially Discrete Dynamical Systems

We obtain existence of traveling wave solutions for a class of spatially discrete systems, namely lattice differential equations. Uniqueness of the wave speed c, and uniqueness of the solution with c 6= 0, are also shown. More generally, the global structure of the set of all traveling wave solutions is shown to be a smooth manifold where c 6= 0. Convergence results for solutions are obtained at the singular perturbation limit c ! 0. 1 Introduction We are interested in lattice differential equations, namely infinite systems of ordinary differential equations indexed by points on a spatial lattice, such as the D-dimensional integer lattice Z D . Our focus in this paper is the global structure of the set of traveling wave solutions for such systems. This entails results on existence and uniqueness, and on continuous (or smooth) dependence of traveling waves and their speeds on parameters, as well as some delicate convergence results in the singular perturbation case c ! 0 of the wav..

The Fredholm Alternative for Functional Differential Equations of Mixed Type

We prove a Fredholm alternative theorem for a class of asymptotically hyperbolic linear differential difference equations of mixed type. We also establish the cocycle property and the spectral flow property for such equations, providing an effective means of calculating the Fredholm index. Such systems can arise from equations which describe traveling waves in a spatial lattice