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.     

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.     

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.     

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.     

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.     

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 of lyapunov exponents of coupled chaotic systems

We develop a statistical theory of the coupling sensitivity of chaos. The effect was first described by Daido [Prog. Theor. Phys. 72, 853 (1984)]; it appears as a logarithmic singularity in the Lyapunov exponent in coupled chaotic systems at very small couplings. Using a continuous-time stochastic model for the coupled systems we derive a scaling relation for the largest Lyapunov exponent. The singularity is shown to depend on the coupling and the systems' mismatch. Generalizations to the cases of asymmetrical coupling and three interacting oscillators are considered, too. The analytical results are confirmed by numerical simulations.     

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.     

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.     

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].     

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.     

Robust stability of extended delayed-feedback control in one-dimensional chaotic systems

This paper investigates robust stability of the extended delayed-feedback control method [Phys. Rev. E 50 (1994) 3245] from the viewpoint of control theory. The feedback-gain range where control can be achieved for this method is wider than that of the original delayed-feedback control method. We provide a procedure to design the extended delayed-feedback controller with a minimum amount of a priori knowledge for chaotic systems including a parameter uncertainty. Some numerical experiments are given to check the results.     

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.     

Control of chaotic systems using targeting by extended control regions method

In this work, the previously proposed extended control regions (ECR) algorithm for targeting is improved by using individual neural networks for each activation region. The improved version, which exploits the short time predictability of the chaotic system more efficiently, gives better performance with respect to training time and average reaching time while maintaining the advantages of the previous method. Moreover, the simulation results revealed that the meaningful number of activation regions of the controller using improved ECR is nearly linearly related with the prediction horizon of the chaotic system to be targeted, which can be used as a criterion for choosing the number of activation region.     

From oscillations to excitability: A case study in spatially extended systems

This volume is devoted to the presentation of the main contributions to the workshop "From oscillations to excitability: A case study in spatially extended systems," organized by the authors in Nice in June 1993. It gives an overview of the current research on spatiotemporal patterns in a wide range of systems that display self-oscillatory or excitable behavior. It tries to give a better understanding of the transition from the oscillatory to the excitable regime and of its effect on the properties of spiral waves, and to fill the gap between the theories and concepts used to describe both regimes in the so-called "active media."