Controlling spatiotemporal chaos in oscillatory reaction-diffusion systems by time-delay autosynchronization

Diffusion-induced turbulence in spatially extended oscillatory media near a supercritical Hopf bifurcation can be controlled by applying global time-delay autosynchronization. We consider the complex Ginzburg-Landau equation in the Benjamin-Feir unstable regime and analytically investigate the stability of uniform oscillations depending on the feedback parameters. We show that a noninvasive stabilization of uniform oscillations is not possible in this type of systems. The synchronization diagram in the plane spanned by the feedback parameters is derived. Numerical simulations confirm the analytical results and give additional information on the spatiotemporal dynamics of the system close to complete synchronization.Comment: 19 pages, 10 figures submitted to Physica

Delay induced Turing-like waves for one species reaction-diffusion model on a network

A one species time-delay reaction-diffusion system defined on a complex networks is studied. Travelling waves are predicted to occur as follows a symmetry breaking instability of an homogenous stationary stable solution, subject to an external non homogenous perturbation. These are generalized Turing-like waves that materialize in a single species populations dynamics model, as the unexpected byproduct of the imposed delay in the diffusion part. Sufficient conditions for the onset of the instability are mathematically provided by performing a linear stability analysis adapted to time delayed differential equation. The method here developed exploits the properties of the Lambert W-function. The prediction of the theory are confirmed by direct numerical simulation carried out for a modified version of the classical Fisher model, defined on a Watts-Strogatz networks and with the inclusion of the delay

Stationary localized structures and the effect of the delayed feedback in the Brusselator model

The Brusselator reaction-diffusion model is a paradigm for the understanding of dissipative structures in systems out of equilibrium. In the first part of this paper, we investigate the formation of stationary localized structures in the Brusselator model. By using numerical continuation methods in two spatial dimensions, we establish a bifurcation diagram showing the emergence of localized spots. We characterize the transition from a single spot to an extended pattern in the form of squares. In the second part, we incorporate delayed feedback control and show that delayed feedback can induce a spontaneous motion of both localized and periodic dissipative structures. We characterize this motion by estimating the threshold and the velocity of the moving dissipative structures.Comment: 18 pages, 11 figure

Delay-Controlled Reactions

When the entities undergoing a chemical reaction are not available simultaneously, the classical rate equation of a reaction or, alternatively for the evolution of a population, should be extended by including non-Markovian memory effects. We consider the two cases of an external feedback, realized by fixed functions and an internal feedback originated in a self-organized manner by the relevant concentration itself. Whereas in the first case the fixed points are not changed, although the dynamical process is altered, the second case offers a complete new behaviour, characterized by the existence of a time persistent solution. Due to the feedback the reaction may lead to a finite concentration in the stationary limit even in case of a single-species pair annihilation $A + A \to 0$ process. We argue that the different cases are similar to a coupling of additive or multiplicative noises in stochastic processes.Comment: 18 pages, 3 figure