Tracking unstable steady states: Extending the stability regime of a multimode laser system

From chemical systems to systems chemistry: Patterns in space and time

We present a brief, idiosyncratic overview of the past quarter century of progress in nonlinear chemical dynamics and discuss what we view as the most exciting recent developments and some challenges and likely areas of progress in the next 25 years.

A dynamical systems approach to the control of chaotic dynamics in a spatiotemporal jet flow

We present a strategy for control of chaos in open flows and provide its experimental validation in the near field of a transitional jet flow system. The low-dimensional chaotic dynamics studied here results from vortex ring formation and their pairings over a spatially extended region of the flow that was excited by low level periodic forcing of the primary instability. The control method utilizes unstable periodic orbits (UPO) embedded within the chaotic attractor. Since hydrodynamic instabilities in the open flow system are convective, both monitoring and control can be implemented at a few locations, resulting in a simple and effective control algorithm. Experiments were performed in an incompressible, initially laminar, 4 cm diameter circular air jet, at a Reynolds number of 23,000, housed in a low-noise, large anechoic chamber. Distinct trajectory bundles surrounding the dominant UPOs were found from experimentally derived, time-delayed embedding of the chaotic attractor. Velocity traces from a pair of probes placed at the jet flow exit and farther downstream were used to empirically model the UPOs and compute control perturbations to be applied at the jet nozzle lip. Open loop control was used to sustain several nearly periodic states.

Global synchronization in lattices of coupled chaotic systems

Based on the concept of matrix measures, we study global stability of synchronization in networks. Our results apply to quite general connectivity topology. In addition, a rigorous lower bound on the coupling strength for global synchronization of all oscillators is also obtained. Moreover, by merely checking the structure of the vector field of the single oscillator, we shall be able to determine if the system is globally synchronized.

Controlled destruction of chaos in the multistable regime

Autonomous nonlinear systems commonly exhibit simultaneous coexistence, in the phase space, of chaos and stable steady states, created by subcritical Hopf bifurcation. We show that such chaotic instability can be destroyed by small-amplitude modulation of any system parameters. The chaotic attractor undergoes boundary crisis due to a modulation-induced collision with an unstable periodic orbit (UPO). Such a boundary crisis exhibits a new resonance that we refer to as "crisis resonance" in the control parameter space. Crisis resonance implies that crisis occurs at minimum modulation depth due to resonant evolutions of the UPOs and the chaotic attractor. Crisis resonance occurs close to some critical frequency (we refer to it as "crisis resonance frequency") or its multiples. The UPO frequency is a good estimate of the crisis resonance frequency. The small-amplitude parameter modulation destroys chaos in the presence of noise as well. These features are observed theoretically with the paradigm of autonomous systems, namely, Lorenz equations of thermal hydraulics and are in excellent agreement with the experimental results, obtained with an analog circuit of Lorenz equations.

All-optical noninvasive control of unstable steady states in a semiconductor laser

All-optical noninvasive control of a multisection semiconductor laser by means of time-delayed feedback from an external Fabry-Perot cavity is realized experimentally. A theoretical analysis, in both a generic model as well as a device-specific simulation, points out the role of the optical phase. Using phase-dependent feedback we demonstrate stabilization of the continuous-wave laser output and noninvasive suppression of intensity pulsations.

Tracking unstable steady states and periodic orbits of oscillatory and chaotic electrochemical systems using delayed feedback control

Experimental results are presented on successful application of delayed-feedback control algorithms for tracking unstable steady states and periodic orbits of electrochemical dissolution systems. Time-delay autosynchronization and delay optimization with a descent gradient method were applied for stationary states and periodic orbits, respectively. These tracking algorithms are utilized in constructing experimental bifurcation diagrams of the studied electrochemical systems in which Hopf, saddle-node, saddle-loop, and period-doubling bifurcations take place.

Tracking control and synchronization of four-dimensional hyperchaotic Rossler system

In this paper we study the problem of the four-dimensional hyperchaotic Rossler system tracking control. A controller based on the reference signal is designed. It is theoretically proved that the controller can make the error converge to zero exponentially. Numerical results have verified the validity of the controller. The Rossler system cannot only track any reference signal fast, but can synchronize with identical or different chaotic systems.

Laser stabilization with multiple-delay feedback control

Stabilization of chaotic intensity fluctuations of intracavity frequency-doubled solid-state (Nd: YAG) lasers using multiple-delay feedback control (MDFC) is demonstrated by numerical simulations. It is shown that MDFC not only provides stable (cw) output for constant pump rates but also works with slowly varying pump currents, resulting in corresponding (nonchaotic) intensity modulations.

Chaos control using notch filter feedback

A method for stabilizing periodic orbits and steady states of chaotic systems is presented using specifically filtered feedback signals. The efficiency of this control technique is illustrated with simulations (Rössler system, laser model) and a successful experimental application for stabilizing intensity fluctuations of an intracavity frequency-doubled Nd:YAG laser.

Stabilizing unstable steady states using multiple delay feedback control

Feedback control with different and independent delay times is introduced and shown to be an efficient method for stabilizing fixed points (equilibria) of dynamical systems. In comparison to other delay based chaos control methods multiple delay feedback control is superior for controlling steady states and works also for relatively large delay times (sometimes unavoidable in experiments due to system dead times). To demonstrate this approach for stabilizing unstable fixed points we present numerical simulations of Chua's circuit and a successful experimental application for stabilizing a chaotic frequency doubled Nd-doped yttrium aluminum garnet laser.

Tracking fixed-point dynamics in an electrochemical system using delayed-feedback control

We report numerical and experimental results indicating successful tracking of stabilized fixed points solutions in an electrochemical system. By applying a continuous delayed-feedback technique, periodic oscillations are suppressed via stabilization of a steady-state fixed point. Subsequently, using a simple continuation method involving an update term, this stabilized fixed point is tracked through the bifurcation diagram as a system parameter is slowly varied. Under the influence of this tracking protocol, inception of oscillatory dynamics is precluded over large parameter domains and through bifurcations.

Transforming complex multistability to controlled monostability

Multistability, a commonly observed feature among nonlinear systems, could be inconvenient under various circumstances. We demonstrate that a control in the form of slow and weak periodic parameter modulation can be effectively applied to transform a complex multistable system to a controlled monostable one. For the representative of a nonlinear system, we choose the Hénon map as the standard model. The number of coexisting stable states is known to increase as the dissipativity reduces. We show that even in the low dissipative limit, when the number of coexisting states could be arbitrarily large, the periodic parameter modulation can destroy the states coexisting with stable period 1. Thus, the system can be brought from any other branch to period-1 branch, leading to controlled monostability. This method works in the presence of noise as well.

Chaotic function generator: complex dynamics and its control in a loss-modulated Nd:YAG laser

The complex dynamics resulting from electronic feedback of a laser's intensity are explored and characterized. Distinct stable and chaotic regimes can be elicited from the laser by tuning the bias of the feedback loop. An additional branch of the feedback loop, containing a derivative filter, provides access to new kinds of dynamics, including a more gradual transition to chaos. The whole feedback network together allows the laser dynamics to be selected from among a wide range of chaotic wave forms distinguished by statistical or spectral information. In other words, this laser system can be used as a tunable generator of chaotic functions.

Stabilizing unstable discrete systems

A general method for stabilizing unstable discrete systems to a fixed point or high-period orbit is developed analytically and numerically in this paper. It is shown that the method can be equally applied to the systems with one or more positive Lyapunov exponents. Moreover, the method does not require a prior analytical knowledge of the system under investigation, nor any additional control parameters.

Suppression of Q-switch instabilities by feedback control in passively mode-locked lasers

We propose a feedback technique for obtaining stable mode-locked operation in lasers that would normally exhibit Q switching. Using the Haus model with population dynamics, we examine numerically the case in which the intensity is monitored by a slow detector and fed back to the pump power after electronic derivation. This approach allows elimination of Q switching in all situations considered, in particular, in lasers with a long upper-state lifetime.

Characterization and stabilization of the unstable fixed points of a frequency doubled Nd:YAG laser

We demonstrate the successful stabilization of type II chaos of a multimode, intracavity frequency doubled, diode-pumped Nd:YAG (neodymium-doped yttrium aluminum garnet) laser in experiment using an adaptive proportional feedback control. The two orthogonal polarized infrared intensities are fed back to the injection current of the pump diode. The stability properties of the stabilized unstable fixed points are investigated and exploited to explain the performance of our control scheme and to determine suitable measurement signals for the feedback control.

Restricted feedback control of one-dimensional maps

Dynamical control of biological systems is often restricted by the practical constraint of unidirectional parameter perturbations. We show that such a restriction introduces surprising complexity to the stability of one-dimensional map systems and can actually improve controllability. We present experimental cardiac control results that support these analyses. Finally, we develop new control algorithms that exploit the structure of the restricted-control stability zones to automatically adapt the control feedback parameter and thereby achieve improved robustness to noise and drifting system parameters.

Dynamical control of systems near bifurcation points using time series

In order to excite experimentalists to apply a dynamical control method [Zhao et al., Phys. Rev. E 53, 299 (1996); 57, 5358 (1998)], we further introduce a simplified control law in this paper. The law provides a convenient way (in certain circumstance a necessary way) for experimentalists to achieve the system control when the exact position of the desired control objective cannot be known in advance. The validity of the control law is rigidly verified when the system nears a bifurcation point but our numerical examples show that it can be extended to a wide parameter region practically.

Stabilization of unstable fixed points in the dynamics of a laser with feedback

We report theoretical and experimental results on the stabilization of unstable steady states in a CO2 laser with feedback. Periodic and chaotic oscillations have been suppressed by means of a control loop consisting of a high-pass filter, known as washout filter. Although the filter characteristics are determined on the basis of linear analysis, taking into account the problem of robustness with respect to parameter changes, the present control strategy provides a large attractive domain in the phase space.

Stabilization of unstable steady states and periodic orbits in an electrochemical system using delayed-feedback control

We report numerical and experimental results indicating successful stabilization of unstable steady states and periodic orbits in an electrochemical system. Applying a continuous delayed-feedback technique not only periodic and chaotic oscillations are suppressed via stabilization of steady-state solutions but also the chaotic dynamics can be converted to periodic behavior. In all cases the feedback perturbation vanishes as a target state is attained.

Efficient strategy for the occasionally proportional feedback method in controlling chaos

In this work, the generic mechanism of the occasionally proportional feedback (OPF) technique in controlling chaos has been explored extensively. Except for stabilizing the unstable states that are embedded in the chaotic attractors, the OPF method is also found to generate a great number of new states during the control processes. The forms and characteristics of these new states have been addressed. Moreover, we clarify the roles of the parameters in the OPF method and this clarification leads to a practical and systematic approach in adjusting the parameters for control. To demonstrate the validity, an analogous electronic circuit of the resistively shunted Josephson junction oscillator is employed in addition to a numerical illustration of the logistic mapping.

Controlling lasers by use of extended time-delay autosynchronization

A method is described for suppressing chaotic instabilities in lasers by use of a specific form of controlling-chaos feedback. The technique is easy to implement and requires only application of small perturbations to an accessible system parameter or variable.

Tracking controlled chaos: Theoretical foundations and applications

Tracking controlled states over a large range of accessible parameters is a process which allows for the experimental continuation of unstable states in both chaotic and non-chaotic parameter regions of interest. In algorithmic form, tracking allows experimentalists to examine many of the unstable states responsible for much of the observed nonlinear dynamic phenomena. Here we present a theoretical foundation for tracking controlled states from both dynamical systems as well as control theoretic viewpoints. The theory is constructive and shows explicitly how to track a curve of unstable states as a parameter is changed. Applications of the theory to various forms of control currently used in dynamical system experiments are discussed. Examples from both numerical and physical experiments are given to illustrate the wide range of tracking applications. (c) 1997 American Institute of Physics.

Stabilizing Lead-Salt Diode Lasers: Understanding and Controlling Chaotic Frequency Emission

Lead-salt tunable diode lasers (TDLs) are the only devices currently available that can generate tunable monochromatic radiation at arbitrary wavelengths between 3 and 30 micrometers and are particularly useful for high-resolution spectroscopy over a wide range of spectral regimes. Detailed observations of TDLs show that the observed instrumental linewidth is actually a temporal average of many narrow (less than 0.5 megahertz) emission "modes." The time scale characteristic of these "modes," which appear to be of relatively constant intensity, is of the order of a microsecond. The laser's behavior is highly suggestive of a chaotic process, that is, seemingly random excursions of a dynamic variable (frequency) within a bounded range. This report shows experimentally that TDL emissions are indeed chaotic. Furthermore, in a simple and robust fashion, this chaotic behavior has been successfully controlled with the use of recent techniques that take advantage of chaos to produce a narrow band laser output.