From excitability to oscillations: A case study in vasomotion

One consequence of cell-to-cell communication is the appearance of synchronized behavior, where many cells cooperate to generate new dynamical patterns. We present a simple functional model of vasomotion based on the concept of a two-mode oscillator with dual interactions: via relatively slow diffusive coupling that gives rise to wave dynamics and via fast changes in membrane potential that propagate almost instantly over significant distances. The model reproduces the basic calcium dynamics of the vascular smooth muscle cell: calcium waves which upon increased activity of cGMP-sensitive calcium-dependent chloride channels in the plasma membrane may synchronize into whole-cell oscillations which subsequently may spread across a large population of cells. We show how heterogeneity of the system can induce new patterns.     

Frequency locking in spatially extended systems

A variant of the complex Ginzburg-Landau equation is used to investigate the frequency-locking phenomena in spatially extended systems. With appropriate parameter values, a variety of frequency-locked patterns including flats, pi fronts, labyrinths, and 2pi/3 fronts emerge. We show that in spatially extended systems, frequency locking can be enhanced or suppressed by diffusive coupling. Novel patterns such as chaotically bursting domains and target patterns are also observed during the transition to locking.     

Chaotic transients in spatially extended systems

Different transient-chaos related phenomena of spatiotemporal systems are reviewed. Special attention is paid to cases where spatiotemporal chaos appears in the form of chaotic transients only. The asymptotic state is then spatially regular. In systems of completely different origins, ranging from fluid dynamics to chemistry and biology, the average lifetimes of these spatiotemporal transients are found, however, to grow rapidly with the system size, often in an exponential fashion. For sufficiently large spatial extension, the lifetime might turn out to be larger than any physically realizable time. There is increasing numerical and experimental evidence that in many systems such transients mask the real attractors. Attractors may then not be relevant to certain types of spatiotemporal chaos, or turbulence. The observable dynamics is governed typically by a high-dimensional chaotic saddle. We review the origin of exponential scaling of the transient lifetime with the system size, and compare this with a similar scaling with system parameters known in low-dimensional problems. The effect of weak noise on such supertransients is discussed. Different crisis phenomena of spatiotemporal systems are presented and fractal properties of the chaotic saddles underlying high-dimensional supertransients are discussed. The recent discovery according to which turbulence in pipe flows is a very long lasting transient sheds new light on chaotic transients in other spatially extended systems.     

Self-parametric instability in spatially extended systems

We study the stability of almost homoclinic homogeneous limit cycles with respect to spatiotemporal perturbations. It is shown that they are generically unstable. The instability is either the phase instability or a finite wavelength period doubling instability.     

Symmetry and control: spatially extended chaotic systems

In a recent paper [Phys. Rev. E 57 (1998) 1550] it was demonstrated that the symmetries of the evolution equation and the target state have a profound effect on controlling the chaotic behavior. In the present paper we extend these results to the cases of time-periodic target trajectories and inexact symmetries, and apply the developed formalism to the problem of controlling spatiotemporal chaos. We use the example of a lattice dynamical system in arbitrary spatial dimension to show that there exists an intimate relationship between the geometry of an extended system and the geometry of feedback control.     

Synchronization in networks of spatially extended systems

Synchronization processes in networks of spatially extended dynamical systems are analytically and numerically studied. We focus on the relevant case of networks whose elements (or nodes) are spatially extended dynamical systems, with the nodes being connected with each other by scalar signals. The stability of the synchronous spatio-temporal state for a generic network is analytically assessed by means of an extension of the master stability function approach. We find an excellent agreement between the theoretical predictions and the data obtained by means of numerical calculations. The efficiency and reliability of this method is illustrated numerically with networks of beam-plasma chaotic systems (Pierce diodes). We discuss also how the revealed regularities are expected to take place in other relevant physical and biological circumstances.     

Scaling properties of spatially extended chaotic systems

We investigate the scaling properties of Lyapunov eigenvectors and exponents in coupled-map lattices exhibiting space-time chaos. A deep interrelation between spatiotemporal chaos and kinetic roughening of surfaces is postulated. We show that the logarithm of unstable eigenvectors exhibits scale-invariance with roughness exponents that can be predicted by a simple scaling conjecture. We argue that these scaling properties should be generic in spatially homogeneous extended systems with local diffusive-like couplings.     

Generalised equal areas rules for spatially extended systems

It is shown that equal areas rules which exist in such different fields as thermodynamics (Maxwell construction), chemical reaction theory (nonequilibrium phase transitions), and semiconductor physics (Gunn-Hilsum effect; current filamentation; grain boundaries) arise as special cases of a general relation between control parameters and boundary values of a second order ordinary differential equation,. They allow one to extract relevant information about spatial profiles of some physical variable in extended systems directly from the constitutive differential equation without explicitly solving it.     

Chaotic synchronization in coupled spatially extended beam–plasma systems

The appearance of the chaotic synchronization regimes has been discovered for the coupled spatially extended beam–plasma Pierce systems. The coupling was introduced only on the right bound of each subsystem. It has been shown that with coupling increase the spatially extended beam–plasma systems show the transition from asynchronous behavior to the phase synchronization and then to the complete synchronization regime. For the consideration of the chaotic synchronization we used the concept of time-scale synchronization described in work [A.E. Hramov, A.A. Koronovskii, Chaos 14 (3) (2004) 603] and based on the introduction of the continuous set of phases of chaotic signal. In case of unidirectional coupling the generalized synchronization regime has been observed in the spatially extended beam–plasma systems. The generalized synchronization appearance mechanism has been analyzed by means of the offered modified system approach [A.E. Hramov, A.A. Koronovskii, Phys. Rev. E 71 (6) (2005) 067201].     

A study on spatially extended phase objects using speckle photography

We present an application of an improved speckle photography technique for spatially extended phase objects. A contour mapping of a thin lens displaying its phase variation is presented. A theoretical analysis is investigated followed by the experimental presentation. Reasonable interference fringes are obtained and compared with the fringes obtained for hot air. The phase information of the object is extracted using the point-by-point technique.     

Stochastic resonance and energy optimization in spatially extended dynamical systems

We investigate a class of nonlinear wave equations subject to periodic forcing and noise, and address the issue of energy optimization. Numerically, we use a pseudo-spectral method to solve the nonlinear stochastic partial differential equation and compute the energy of the system as a function of the driving amplitude in the presence of noise. In the fairly general setting where the system possesses two coexisting states, one with low and another with high energy, noise can induce intermittent switchings between the two states. A striking finding is that, for fixed noise, the system energy can be optimized by the driving in a form of resonance. The phenomenon can be explained by the Langevin dynamics of particle motion in a double-well potential system with symmetry breaking. The finding can have applications to small-size devices such as microelectromechanical resonators and to waves in fluid and plasma.     

Asymmetric coupling effects in the synchronization of spatially extended chaotic systems

We analyze the effects of asymmetric couplings in setting different synchronization states for a pair of unidimensional fields obeying complex Ginzburg-Landau equations. Novel features such as asymmetry enhanced complete synchronization, limits for the appearance of phase synchronized states, and selection of the final synchronized dynamics are reported and characterized.     

Broken symmetries and directed collective energy transport in spatially extended systems

We study the appearance of directed energy current in homogeneous spatially extended systems coupled to a heat bath in the presence of an external ac field E(t). The systems are described by nonlinear field equations. By making use of a symmetry analysis, we predict the right choice of E(t) and obtain directed energy transport for systems with a nonzero topological charge Q. We demonstrate that the symmetry properties of motion of topological solitons (kinks and antikinks) are equivalent to the ones for the energy current. Numerical simulations confirm the predictions of the symmetry analysis and, moreover, show that the directed energy current drastically increases as the dissipation parameter alpha reduces.     

Characterization of patterned irregularity in locally interacting,spatially extended systems: Ventricular fibrillation

The re-entrant ventricular arrhythmias of monomorphic ventricular tachycardia and fibrillation are produced by abnormal spatio-temporal patterns of propagation in the ventricular myocardium. These behaviors can be described by solutions of reaction-diffusion equation excitable medium models. The direct comparison of such solutions with existing experimental observations is virtually impossible as there are too many factors to be taken into account, including not only the complicated dynamics of the re-entrant waves of excitation in the tissue, but also the way the appearance of these waves on the surface is modified by the inhomogeneity, anisotropy and three-dimensional nature of heart tissue. One way of indirect comparison is to compare characteristics of the complexity of the model and the real data, that are invariant under these modifications of the signal. Karhunen-Loeve decomposition is a standard tool for evaluating the complexity of multidimensional signals. A comparison of the separate and conjoint complexities of the signals on the opposite sides of the preparation can be considered as an indicator how much three-dimensional effects are essential in the preparation behavior. (c) 2001 American Institute of Physics.     

Intermittency near the phase boundary of chaotic synchronization in spatially extended systems

The intermittent behavior of spatially extended systems is investigated using the example of unidirectionally coupled Pierce diodes. It isshown that the same type of intermittency as in finite-scaled systems is characteristic of this system near the boundary of the chaotic phase synchronization regime, i.e., needle-eye type intermittency, which is in fact also equivalent to type I intermittency with noise in the supercritical region.     

One method of calculating Cerenkov losses of spatially extended charged systems

In this article, we shall study the Cerenkov radiation of spatially extended uniformly charged systems (in particular, systems whose charge is distributed over the volume of an ellipsoid of revolution, over the surface of a sphere, and over an infinitely thin ring) and volume currents moving at constant velocity along the optical axis of a uniaxial transparent ferrodielectric.The authors express their appreciation to Professor A. A. Sokolov for his discussion of the results.