Alexander S. Mikhailov

Complex dynamics of spiral waves and motion of curves

Abstract In this review paper we consider spiral waves in weakly excitable media where they can be described using a simple kinematical model. The model is formulated in terms of the motion of curves with free ends. These curves can grow or contract while moving over a plane. A steadily rotating spiral is a dynamical attractor of such a system. Application of spatial gradients or temporal modulation of the mediums properties induce drift of spirals. When interactions between the curves are taken into account, this results in appearance of complex meandering regimes.

TURING PATTERNS IN NETWORK-ORGANIZED ACTIVATOR-INHIBITOR SYSTEMS

Turing patterns formed by activator-inhibitor systems on networks are considered. The linear stability analysis shows that the Turing instability generally occurs when the inhibitor diffuses sufficiently faster than the activator. Numerical simulations, using a prey-predator model on a scale-free random network, demonstrate that the final, asymptotically reached Turing patterns can be largely different from the critical modes at the onset of instability, and multistability and hysteresis are typically observed. An approximate mean-field theory of nonlinear Turing patterns on the networks is constructed.

From Cells to Societies

1. Introduction.- 2. The Games of Life.- 3. Active Motion.- 4. Ridden by the Noise ...- 5. Dynamics with Delays and Expectations.- 6. Mutual Synchronization.- 7. Dynamical Clustering..- 8. Hierarchical Organization.- 9. Dynamics and Evolution of Networks..- References.

Noise-induced transition from translational to rotational motion of swarms.

We consider a model of active Brownian agents interacting via a harmonic attractive potential in a two-dimensional system in the presence of noise. By numerical simulations, we show that this model possesses a noise-induced transition characterized by the breakdown of translational motion and the onset of swarm rotation as the noise intensity is increased. Statistical properties of swarm dynamics in the weak noise limit are further analytically investigated.

Kinematical theory of spiral waves in excitable media: comparison with numerical simulations

Abstract The kinematical theory of spiral waves is compared with the data of numerical simulations using complete partial differential equations of a reaction-diffusion model. We find good agreement for temporal periods of spiral waves which rotate around minimal holes and in the free medium. No adjustable parameters were used in the calculations.

Mutual synchronization and clustering in randomly coupled chaotic dynamical networks

We introduce and study systems of randomly coupled maps where the relevant parameter is the degree of connectivity in the system. Global (almost-) synchronized states are found (equivalent to the synchronization observed in globally coupled maps) until a certain critical threshold for the connectivity is reached. We further show that not only the average connectivity, but also the architecture of the couplings is responsible for the cluster structure observed. We analyze the different phases of the system and use various correlation measures in order to detect ordered nonsynchronized states. Finally, it is shown that the system displays a dynamical hierarchical clustering which allows the definition of emerging graphs.

NOISE-INDUCED BREAKDOWN OF COHERENT COLLECTIVE MOTION IN SWARMS

We consider swarms formed by populations of self-propelled particles with attractive long-range interactions. These swarms represent multistable dynamical systems and can be found either in coherent traveling states or in an incoherent oscillatory state where translational motion of the entire swarm is absent. Under increasing the noise intensity, the coherent traveling state of the swarms is destroyed and an abrupt transition to the oscillatory state takes place.

Subsurface oxygen formation on the Pt(110) surface : experiment and mathematical modeling

Abstract Experiments using photo emission electron microscopy (PEEM) reveal that regions on a Pt(110) surface covered by chemisorbed O atoms may be converted into a subsurface O-phase, provided that it is preceded by the interaction of CO initiating the 1 × 2 → 1 × 1 transformation of the surface structure. However, the presence of subsurface oxygen also favors lifting of the surface reconstruction. A mathematical model of this process is developed using parameters derived from previous independent experiments and numerical simulations fitting new data to experimental findings.

Controlling turbulence in the complex Ginzburg-Landau equation

Abstract We suggest that diffusion-induced turbulence in distributed dynamical systems near a supercritical Hopf bifurcation can be controlled by means of global delayed feedback. Analytical and numerical investigations of this method for a system described by the complex Ginzburg-Landau equation are performed. Suppression of phase and amplitude turbulence is found inside a window of delay times under increasing the intensity of the control signal.

Evolutionary reconstruction of networks

Can a graph specifying the pattern of connections of a dynamical network be reconstructed from statistical properties of a signal generated by such a system? In this model study, we present a Metropolis algorithm for reconstruction of graphs from their Laplacian spectra. Through a stochastic process of mutations and selection, evolving test networks converge to a reference graph. Applying the method to several examples of random graphs, clustered graphs, and small-world networks, we show that the proposed stochastic evolution allows exact reconstruction of relatively small networks and yields good approximations in the case of large sizes.