V. Pérez-Villar

Spiral waves on a 2-D array of nonlinear circuits

Spatio-temporal patterns formed in a 2-D array of Chuas circuits have been studied numerically. It has been found that spiral wave solutions can appear over a large range of parameters and some of their properties have been measured. This demonstrates that spiral wave dynamics can be studied in arrays of discrete electronic circuits, such as a 2-D array of Chuas circuits, where real-time results can be obtained. We also study the influence of small differences in the parameters of the circuits, as is the case in real electronic components, where a 5% device tolerance is typical. >

SPATIOTEMPORAL STRUCTURES IN DISCRETELY-COUPLED ARRAYS OF NONLINEAR CIRCUITS: A REVIEW

Spatiotemporal pattern formation occurring in discretely-coupled nonlinear dynamical systems has been studied numerically. Reaction-diffusion systems can be viewed as an assembly of a large number of identical local subsystems which are coupled to each other by diffusion. Here, the local subsystems are defined by a system of nonlinear ordinary differential equations. While for continuous systems, the characteristic time scale corresponding to the diffusion is slower than that corresponding to the local subsystems, in discretely-coupled systems, both time scales can be of the same order of magnitude. Discrete systems can exhibit behaviors different from those exhibited by their equivalent continuous model: the wave propagation failure phenomenon occurring in nerve-pulse propagation due to transmission blockage is a case in point. In this case, it is found that the wave fails to propagate at or below some critical value of the coupling coefficient. Systems of coupled cells can be found to occur in the trans...

PROPAGATION FAILURE IN LINEAR ARRAYS OF CHUA’S CIRCUITS

Traveling wave fronts are considered for a one-dimensional array of Chua’s circuits. For diffusion coefficients less than some nonzero critical value it has been observed numerically that the traveling fronts fail to propagate. Propagation failure is compared with similar phenomena occurring in pulse propagation in nerves, and in coupled continuously-stirred tank reactors.

Vulnerability in excitable Belousov-Zhabotinsky medium: from 1D to 2D

Abstract Mechanisms for initiating rotating waves in 1D and 2D excitable media were compared and parameters affecting wavefront formation were analyzed. The time delay between two sequentially initiated wavefronts (a conditioning wave followed by a test wave) was varied in order to induce rotating waves, a protocol similar to that utilized in cardiac muscle experiments to reveal vulnerability to rotating wave initiation. We define the vulnerability region, VR, as the range of time delays between conditioning and test waves where the test waves evolves into a rotating wave. The smaller the VR, the more resistant the heart is against origination of dangerous cardiac arrhythmias. Heterogeneity of cardiac muscle is widely recognized as the prerequisite for rotating wave initiation. We have identified the VR in homogeneous 2D excitable media. In the Belousov-Zhabotinsky (BZ) reaction with immobilized catalyst and in the Oregonator model of this reaction, a properly timed test wave gives rise to rotating waves. The VR was increased when the size of the perturbation used for test wave creation was increased or when the threshold for propagation was decreased. Increasing the dimensionality of the medium for 1D to 2D results in diminishing of VR.

Interaction of chaotic rotating waves in coupled rings of chaotic cells

Abstract The interaction of two chaotic rotating waves of the type recently reported by Matias et al. [Europhys. Lett. 37 (1997) 379] is studied experimentally with arrays of non-linear electronic circuits arranged in ring geometries. Unidirectional coupling is assumed for the cell-to-cell coupling within the same ring, but between rings, cells are coupled diffusively. Depending on the relative sense of driving, competition between a rotating chaotic wave and a global synchronized state has been observed. The results are rationalized by means of a linear stability analysis around the uniform synchronized behavior, where the circulant symmetry of the system allows to express the problem as a superposition of a series of Fourier modes.

Spiral breakup induced by an electric current in a Belousov-Zhabotinsky medium.

Sprial breakup in the Belousov-Zhabotinsky reaction has been observed under the influence of an externally applied alternating electric current. The dynamic mechanism of this breakup is explained in the framework of this reaction. The dependence of the critical electric current amplitude on the period of the wave and on the excitability of the medium is analyzed. Spiral breakup is shown to provide a limit of validity of electric-field-induced drift of vortices in excitable media. Experimental results are complemented with numerical simulations provided by two- and three-variable Oregonator models.

CHAOTIC SYNCHRONIZATION OF A ONE-DIMENSIONAL ARRAY OF NONLINEAR ACTIVE SYSTEMS

Linear stability analysis is used to study the synchronization of N coupled chaotic dynamical systems. It is found that the role of the coupling is always to stabilize the system, and then synchronize it. Computer simulations and experimental results of an array of Chuas circuits are carried out. Arrays of identical and slightly different oscillators are considered. In the first case, the oscillators synchronize and sync-phase, i.e., each one repeats exactly the same behavior as the rest of them. When the oscillators are not identical, they can also synchronize but not in phase with each other. The last situation is shown to form structures in the phase space of the dynamical variables. Due to the inevitable component tolerances (±5%), our experiments have so far confirmed our theoretical predictions only for an array of slightly different oscillators.

NONLINEAR WAVES, PATTERNS AND SPATIO-TEMPORAL CHAOS IN CELLULAR NEURAL NETWORKS

Spatio-temporal pattern formation occurring in discretely coupled nonlinear dynamical systems has been studied numerically. In this paper, we review the possibilities of using arrays of discretely coupled nonlinear electronic circuits to study these systems. Spiral wave initiation and Turing pattern formation are some of the examples. Sidewall forcing of Turing patterns is shown to be capable of driving the system into a perfect spatial organization, namely, a rhombic pattern, where no defects occur. The dynamics of the two layers supporting Turing and Hopf modes, respectively, is analysed as a function of the coupling strength between them. The competition between these two modes is shown to increase with the diffusion between layers. As well, the coexistence of low- and high-dimensional spatio-temporal chaos is shown to occur in one-dimensional arrays.

Study of reentry initiation in coupled parallel fibers [cardiology]

Initiation of reentries in coupled parallel fibers is studied as a function of several control parameters intrinsic to those fibers. The influence of inhomogeneities in the fibers leading to drift of the vortices and the interaction between them is also analyzed numerically and experimentally. >

Sidewall forcing of hexagonal Turing patterns: rhombic patterns

Abstract Rhombic arrays were obtained by sidewall forcing during Turing pattern formation in numerical simulations. Locking between the frequency of forcing and the wave length between blobs was obtained in accordance with the Farey sequence. This locking appears as a perfect rhombic array oriented in the direction of the imposed forcing. For a constant forcing in duration and amplitude, the following scheme of bifurcation was observed: parallel stripes ↦ rhombic array ↦ domains of hexagons and rhombi separated by “penta-hepta” defects. Symmetry considerations based on a non-uniform stretching along the x-axis were used to describe these transitions. Unstable “varicose-vein” stripes were observed to evolve during the temporal evolution arrays.