Jérôme Losson

Solution multistability in first-order nonlinear differential delay equations.

The dependence of solution behavior to perturbations of the initial function (IF) in a class of nonlinear differential delay equations (DDEs) is investigated. The structure of basins of attraction of multistable limit cycles is investigated. These basins can possess complex structure at all scales measurable numerically although this is not necessarily the case. Sensitive dependence of the asymptotic solution to perturbations in the initial function is also observed experimentally using a task specific electronic analog computer designed to investigate the dynamics of an integrable first-order DDE.

Phase transitions in networks of chaotic elements with short and long range interactions

Abstract This paper investigates the statistical properties of networks of chaotic elements modeled by coupled map lattices. Transitions separating statistically stable and periodic phases are numerically observed in generic models of excitable media. Similar transitions are studied analytically in lattices of piecewise expanding maps by considering the spectral properties of the Perron-Frobenius operator using the theory of functions of bounded variation in R n.

Coupling induced statistical cycling in two diffusively coupled maps

Abstract This paper discusses the effects of diffusively coupling two identical one dimensional maps. Attention is focused on situations where the local (isolated) maps are statistically stable, but where the coupled system is not. A biologically motivated map and the quadratic map are numerically shown to display this behavior. The piecewise linear tent map is then investigated analytically, and we give a phase diagram of this system which displays the location of nonequilibrium phase transitions. It is conjectured that the diffusive coupling of two chaotic but statistically stable maps (i.e. with asymptotically stable Perron-Frobenius operators) can yield a two-dimensional system which is not statistically stable, whose associated Perron-Frobenius operator is asymptotically periodic.

A Hopf-like equation and perturbation theory for differential delay equations

We extend techniques developed for the study of turbulent fluid flows to the statistical study of the dynamics of differential delay equations. Because the phase spaces of differential delay equations are infinite dimensional, phase-space densities for these systems are functionals. We derive a Hopf-like functional differential equation governing the evolution of these densities. The functional differential equation is reduced to an infinite chain of linear partial differential equations using perturbation theory. A necessary condition for a measure to be invariant under the action of a nonlinear differential delay equation is given. Finally, we show that the evolution equation for the density functional is the Fourier transform of the infinite-dimensional version of the Kramers-Moyal expansion.