Delayed feedback control of periodic orbits without torsion in nonautonomous chaotic systems: theory and experiment.

We demonstrate theoretically and experimentally that the unstable delayed feedback controller is an efficient tool for stabilizing torsion-free unstable periodic orbits in nonautonomous chaotic systems. To improve the global control performance we introduce a two-step control algorithm. The problem of a linear stability of the system under delayed feedback control is treated analytically. Theoretical results are confirmed by electronic circuit experiments for a forced double-well oscillator.

Delayed feedback control of the Lorenz system: an analytical treatment at a subcritical Hopf bifurcation.

We develop an analytical approach for the delayed feedback control of the Lorenz system close to a subcritical Hopf bifurcation. The periodic orbits arising at this bifurcation have no torsion and cannot be stabilized by a conventional delayed feedback control technique. We utilize a modification based on an unstable delayed feedback controller. The analytical approach employs the center manifold theory and the near identity transformation. We derive the characteristic equation for the Floquet exponents of the controlled orbit in an analytical form and obtain simple expressions for the threshold of stability as well as for an optimal value of the control gain. The analytical results are supported by numerical analysis of the original system of nonlinear differential-difference equations.

Delayed feedback control of forced self-sustained oscillations.

We consider a weakly nonlinear van der Pol oscillator subjected to a periodic force and delayed feedback control. Without control, the oscillator can be synchronized by the periodic force only in a certain domain of parameters. However, outside of this domain the system possesses unstable periodic orbits that can be stabilized by delayed feedback perturbation. The feedback perturbation vanishes if the stabilization is successful and thus the domain of synchronization can be extended with only small control force. We take advantage of the fact that the system is close to a Hopf bifurcation and derive a simplified averaged equation which we are able to treat analytically even in the presence of the delayed feedback. As a result we obtain simple analytical expressions defining the domain of synchronization of the controlled system as well as an optimal value of the control gain. The analytical theory is supported by numerical simulations of the original delay-differential equations.

Delayed feedback control of dynamical systems at a subcritical Hopf bifurcation.

We consider the delayed feedback control of a torsion-free unstable periodic orbit originated in a dynamical system at a subcritical Hopf bifurcation. Close to the bifurcation point the problem is treated analytically using the method of averaging. We discuss the necessity of employing an unstable degree of freedom in the feedback loop as well as a nonlinear coupling between the controlled system and controller. To demonstrate our analytical approach the specific example of a nonlinear electronic circuit is taken as a model of a subcritical Hopf bifurcation.

Adaptive control of unknown unstable steady states of dynamical systems.

A simple adaptive controller based on a low-pass filter to stabilize unstable steady states of dynamical systems is considered. The controller is reference-free; it does not require knowledge of the location of the fixed point in the phase space. A topological limitation similar to that of the delayed feedback controller is discussed. We show that the saddle-type steady states cannot be stabilized by using the conventional low-pass filter. The limitation can be overcome by using an unstable low-pass filter. The use of the controller is demonstrated for several physical models, including the pendulum driven by a constant torque, the Lorenz system, and an electrochemical oscillator. Linear and nonlinear analyses of the models are performed and the problem of the basins of attraction of the stabilized steady states is discussed. The robustness of the controller is demonstrated in experiments and numerical simulations with an electrochemical oscillator, the dissolution of nickel in sulfuric acid; a comparison of the effect of using direct and indirect variables in the control is made. With the use of the controller, all unstable phase-space objects are successfully reconstructed experimentally.

Stabilizing and tracking unknown steady States of dynamical systems.

An adaptive dynamic state feedback controller for stabilizing and tracking unknown steady states of dynamical systems is proposed. We prove that the steady state can never be stabilized if the system and controller in sum have an odd number of real positive eigenvalues. For two-dimensional systems, this topological limitation states that only an unstable focus or node can be stabilized with a stable controller, and stabilization of a saddle requires the presence of an unstable degree of freedom in a feedback loop. The use of the controller to stabilize and track saddle points (as well as unstable foci) is demonstrated both numerically and experimentally with an electrochemical Ni dissolution system.

Analytical properties and optimization of time-delayed feedback control.

Time-delayed feedback control is an efficient method for stabilizing unstable periodic orbits of chaotic systems. If the equations governing the system dynamics are known, the success of the method can be predicted by a linear stability analysis of the desired orbit. Unfortunately, the usual procedures for evaluating the Floquet exponents of such systems are rather intricate. We show that the main stability properties of the system controlled by time-delayed feedback can be simply derived from a leading Floquet exponent defining the system behavior under proportional feedback control. Optimal parameters of the delayed feedback controller can be evaluated without an explicit integration of delay-differential equations. The method is valid for low-dimensional systems whose unstable periodic orbits are originated from a period doubling bifurcation and is demonstrated for the Rössler system and the Duffing oscillator.

Control of chaos via an unstable delayed feedback controller.

Delayed feedback control of chaos is well known as an effective method for stabilizing unstable periodic orbits embedded in chaotic attractors. However, it had been shown that the method works only for a certain class of periodic orbits characterized by a finite torsion. Modification based on an unstable delayed feedback controller is proposed in order to overcome this topological limitation. An efficiency of the modified scheme is demonstrated for an unstable fixed point of a simple dynamic model as well as for an unstable periodic orbit of the Lorenz system.

Dynamics and control of a multimode laser: Reduction of space-dependent rate equations to a low-dimensional system.

We suggest a quantitatively correct procedure for reducing the spatial degrees of freedom of the space-dependent rate equations of a multimode laser that describe the dynamics of the population inversion of the active medium and the mode intensities of the standing waves in the laser cavity. The key idea of that reduction is to take advantage of the small value of the parameter that defines the ratio between the population inversion decay rate and the cavity decay rate. We generalize the reduction procedure for the case of an intracavity frequency doubled laser. Frequency conversion performed by an optically nonlinear crystal placed inside the laser cavity may cause a pronounced instability in the laser performance, leading to chaotic oscillations of the output intensity. Based on the reduced equations, we analyze the dynamical properties of the system as well as the problem of stabilizing the steady state. The numerical analysis is performed considering the specific system of a Nd:YAG (neodymium-doped yttrium aluminum garnet) laser with an intracavity KTP (potassium titanyl phosphate) crystal.

Stabilization of an unstable steady state in intracavity frequency-doubled lasers

We predict theoretically that it is possible to stabilize the steady state in multimode, intracavity doubled, diode pumped Nd:YAG (neodymium-doped yttrium aluminum garnet) lasers using two output signals, namely, the sum intensities of the infrared laser modes polarized in two different orthogonal directions (X and Y) and one feedback input parameter, the pump rate. The stabilization is possible for arbitrarily large numbers of modes polarized in the X and Y direction. Different strategies of stabilization based on proportional feedback, derivative control, and their combination are discussed. The analytical and numerical results of the linear control theory are illustrated with numerical simulations of the underlying nonlinear differential equations. We show that one can maintain the stable steady state of the laser output for an arbitrarily large pump rate by taking advantage of a tracking procedure.