Systematic experimental exploration of bifurcations with non-invasive control

We present a general method for systematically investigating the dynamics and bifurcations of a physical nonlinear experiment. In particular, we show how the odd-number limitation inherent in popular non-invasive control schemes, such as (Pyragas) time-delayed or washout-filtered feedback control, can be overcome for tracking equilibria or forced periodic orbits in experiments. To demonstrate the use of our non-invasive control, we trace out experimentally the resonance surface of a periodically forced mechanical nonlinear oscillator near the onset of instability, around two saddle-node bifurcations (folds) and a cusp bifurcation.Comment: revised and extended version (8 pages, 7 figures

Controlling spatiotemporal dynamics with time-delay feedback

We suggest a spatially local feedback mechanism for stabilizing periodic orbits in spatially extended systems. Our method, which is based on a comparison between present and past states of the system, does not require the external generation of an ideal reference state and can suppress both absolute and convective instabilities. As an example, we analyze the complex Ginzburg-Landau equation in one dimension, showing how the time-delay feedback enlarges the stability domain for travelling waves.Comment: 4 pages REVTeX + postscript file with 3 figure

Identification and control of dynamical systems

Practical methods, based upon linear systems theory, are explored for applications to nonlinear phenomena and are extended to a larger class of problems. An algorithm for stabilizing, characterizing, and tracking unstable steady states and periodic orbits in multidimensional dynamical systems is developed and applied to stabilize and characterize an unstable four-cell flame front of the Kuramoto-Sivashinsky equation with six unstable degrees of freedom. A new method is presented for probing chemical reaction mechanisms experimentally with perturbations and measurements of the response. Time series analysis and the methods of linear control theory are used to determine the Jacobian matrix of a reaction at a stable stationary state subjected to random perturbations. The method is demonstrated with time series of a model system, and its performance in the presence of noise is examined. A new theory based on the construction of a multitude of linear models, each serving to represent one small region of the phase space, is presented together. Details of its implementation are presented in predicting chaotic Kuramoto-Sivashinsky wave fronts, demonstrating how it overcomes some of the problems associated with high dimensionality phase spaces. Motivated by the relationship between nonlinear prediction methods and the capabilities of neural systems, we demonstrate the possible role of nonlinear phenomena in the morphogenesis of neural tracts

Systematic experimental exploration of bifurcations with noninvasive control

Copyright © 2013 American Physical SocietyWe present a general method for systematically investigating the dynamics and bifurcations of a physical nonlinear experiment. In particular, we show how the odd-number limitation inherent in popular noninvasive control schemes, such as (Pyragas) time-delayed or washout-filtered feedback control, can be overcome for tracking equilibria or forced periodic orbits in experiments. To demonstrate the use of our noninvasive control, we trace out experimentally the resonance surface of a periodically forced mechanical nonlinear oscillator near the onset of instability, around two saddle-node bifurcations (folds) and a cusp bifurcation.Engineering and Physical Sciences Research Council (EPSRC

A new generalization of delayed feedback control

In this paper, we consider the stabilization problem of unstable periodic orbits of one-dimensional discrete time chaotic systems. We propose a novel generalization of the classical delayed feedback law and present some stability results. These results show that for period 1 all hyperbolic periodic orbits can be stabilized by the proposed method; for higher order periods the proposed scheme possesses some inherent limitations. However, some more improvement over the classical delayed feedback scheme can be achieved with the proposed scheme. The stability proofs also give the possible feedback gains which achieve stabilization. We will also present some simulation results. © 2009 World Scientific Publishing Company

Time-stepping and Krylov methods for large-scale instability problems

With the ever increasing computational power available and the development of high-performances computing, investigating the properties of realistic very large-scale nonlinear dynamical systems has been become reachable. It must be noted however that the memory capabilities of computers increase at a slower rate than their computational capabilities. Consequently, the traditional matrix-forming approaches wherein the Jacobian matrix of the system considered is explicitly assembled become rapidly intractable. Over the past two decades, so-called matrix-free approaches have emerged as an efficient alternative. The aim of this chapter is thus to provide an overview of well-grounded matrix-free methods for fixed points computations and linear stability analyses of very large-scale nonlinear dynamical systems.Comment: To appear in "Computational Modeling of Bifurcations and Instabilities in Fluid Mechanics", eds. A. Gelfgat, Springe

Discrete Time Systems

Discrete-Time Systems comprehend an important and broad research field. The consolidation of digital-based computational means in the present, pushes a technological tool into the field with a tremendous impact in areas like Control, Signal Processing, Communications, System Modelling and related Applications. This book attempts to give a scope in the wide area of Discrete-Time Systems. Their contents are grouped conveniently in sections according to significant areas, namely Filtering, Fixed and Adaptive Control Systems, Stability Problems and Miscellaneous Applications. We think that the contribution of the book enlarges the field of the Discrete-Time Systems with signification in the present state-of-the-art. Despite the vertiginous advance in the field, we also believe that the topics described here allow us also to look through some main tendencies in the next years in the research area