Properties of generalized synchronization of chaos

A review of recent ideas in the field of generalized synchronization of chaos is presented. This field is concerned with a generalization of the concept of conventional (identical) chaotic synchronization to the case of one-way coupled nonidentical chaotic systems. Generalized synchronization is taken to occur if, ignoring transients, the response system becomes uniquely determined by the current state of the driving system, i. e., all trajectories in the phase space are attracted to a complex synchronization manifold that may have a fractal structure. Different tools for detecting and analyzing the properties of this type of synchronization are discussed

AN ANALYTICAL TREATMENT OF THE DELAYED FEEDBACK CONTROL ALGORITHM AT A SUBCRITICAL HOPF BIFURCATION

We develop an analytical approach for the delayed feedback control algorithm applied to a dynamical system close to a subcritical Hopf bifurcation. A simple nonlinear electronic circuit is considered as a prototypical model of the subcritical Hopf bifurcation. The periodic orbit arising at this bifurcation is torsion free and cannot be controlled by the conventional delayed feedback algorithm. We show the necessity of employing an unstable degree of freedom in a feedback loop as well as a nonlinear coupling between the controlled system and controller. Close to the bifurcation point the system is weakly nonlinear and the problem is treated analytically using the method of averaging

Stability domains for time-delay feedback control with latency

We generalize a known analytical method for determining the stability of periodic orbits controlled by time-delay feedback methods when latencies associated with the generation and injection of the feedback signal cannot be ignored. We discuss the case of extended time-delay autosynchronization (ETDAS) and show that nontrivial qualitative features of the domain of control observed in experiments can be explained by taking into account the effects of both the unstable eigenmode and a single stable eigenmode in the Floquet theory.Comment: 9 pages, 6 figures; Submitted to Physical Review

Application of ultrafast Schottky diodes to high megahertz chaotic oscillators

The considered chaotic oscillator consists of an amplifier, 2nd order LC resonator, Schottky diode and an extra capacitor in parallel to the diode. The diode plays the role of a nonlinear device. Chaotic oscillations are demonstrated numerically and experimentally at low as well as at high megahertz frequencies, up to 250 MHz

On the Mechanism of Time--Delayed Feedback Control

The Pyragas method for controlling chaos is investigated in detail from the experimental as well as theoretical point of view. We show by an analytical stability analysis that the revolution around an unstable periodic orbit governs the success of the control scheme. Our predictions concerning the transient behaviour of the control signal are confirmed by numerical simulations and an electronic circuit experiment.Comment: 4 pages, REVTeX, 4 eps-figures included Phys. Rev. Lett., in press also available at http://athene.fkp.physik.th-darmstadt.de/public/wolfram.htm

Coarse Bifurcation Diagrams via Microscopic Simulators: A State-Feedback Control-Based Approach

The arc-length continuation framework is used for the design of state feedback control laws that enable a microscopic simulator trace its own open-loop coarse bifurcation diagram. The steering of the system along solution branches is achieved through the manipulation of the bifurcation parameter, which becomes our actuator. The design approach is based on the assumption that the eigenvalues of the linearized system can be decomposed into two well separated clusters: one containing eigenvalues with large negative real parts and one containing (possibly unstable) eigenvalues close to the origin

On the Floquet Theory of Delay Differential Equations

We present an analytical approach to deal with nonlinear delay differential equations close to instabilities of time periodic reference states. To this end we start with approximately determining such reference states by extending the Poincar'e Lindstedt and the Shohat expansions which were originally developed for ordinary differential equations. Then we systematically elaborate a linear stability analysis around a time periodic reference state. This allows to approximately calculate the Floquet eigenvalues and their corresponding eigensolutions by using matrix valued continued fractions

Controlling extended systems with spatially filtered, time-delayed feedback

We investigate a control technique for spatially extended systems combining spatial filtering with a previously studied form of time-delay feedback. The scheme is naturally suited to real-time control of optical systems. We apply the control scheme to a model of a transversely extended semiconductor laser in which a desirable, coherent traveling wave state exists, but is a member of a nowhere stable family. Our scheme stabilizes this state, and directs the system towards it from realistic, distant and noisy initial conditions. As confirmed by numerical simulation, a linear stability analysis about the controlled state accurately predicts when the scheme is successful, and illustrates some key features of the control including the individual merit of, and interplay between, the spatial and temporal degrees of freedom in the control.Comment: 9 pages REVTeX including 7 PostScript figures. To appear in Physical Review

Investigation of the complex dynamics and regime control in Pierce diode with the delay feedback

In this paper the dynamics of Pierce diode with overcritical current under the influence of delay feedback is investigated. The system without feedback demonstrates complex behaviour including chaotic regimes. The possibility of oscillation regime control depending on the delay feedback parameter values is shown. Also the paper describes construction of a finite-dimensional model of electron beam behaviour, which is based on the Galerkin approximation by linear modes expansion. The dynamics of the model is close to the one given by the distributed model.Comment: 18 pages, 6 figures, published in Int. J. Electronics. 91, 1 (2004) 1-1