## Introduction to Control of Oscillations and ChaosThis book gives an exposition of the exciting field of control of oscillatory and chaotic systems, which has numerous potential applications in mechanics, laser and chemical technologies, communications, biology and medicine, economics, ecology, etc.A novelty of the book is its systematic application of modern nonlinear and adaptive control theory to the new class of problems. The proposed control design methods are based on the concepts of Lyapunov functions, Poincare maps, speed-gradient and gradient algorithms. The conditions which ensure such control goals as an excitation or suppression of oscillations, synchronization and transformation from chaotic mode to the periodic one or vice versa, are established. The performance and robustness of control systems under disturbances and uncertainties are evaluated.The described methods and algorithms are illustrated by a number of examples, including classical models of oscillatory and chaotic systems: coupled pendula, brusselator, Lorenz, Van der Pol, Duffing, Henon and Chua systems. Practical examples from different fields of science and technology such as communications, growth of thin films, synchronization of chaotic generators based on tunnel diods, stabilization of swings in power systems, increasing predictability of business-cycles are also presented.The book includes many results on nonlinear and adaptive control published previously in Russian and therefore were not known to the West.Researchers, teachers and graduate students in the fields of electrical and mechanical engineering, physics, chemistry, biology, economics will find this book most useful. Applied mathematicians and control engineers from various fields of technology dealing with complex oscillatory systems will also benefit from it. |

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### Contents

Introduction | 5 |

The mathematics of nonlinear control | 29 |

The mathematics of oscillations and chaos | 121 |

Methods of nonlinear and adaptive control | 163 |

Control of oscillatory and chaotic systems | 273 |

loop model | 286 |

Applications | 323 |

What is the message of the book? | 359 |

367 | |

389 | |

### Common terms and phrases

achieved adaptive control applied assume Assumption asymptotically stable attractor bifurcation bounded boundedness chaos chaotic closed loop system Consider the following constant control algorithm control design control goal control law control objective control problem control theory controlled system convergence corresponding defined derivative disturbance eigenvalues ensure equilibrium example exists exponent exponentially finite globally Hamiltonian initial conditions input Jacobi matrix lemma linearizability Lorenz systems Lyapunov exponent Lyapunov function Lyapunov stable matrix minimum phase nonlinear control theory nonlinear system objective function orbit oscillations oscillatory behavior oscillatory systems output feedback overall system passifiable passive pendulum periodic function plant Poincare map positive definite proof pseudogradient radially unbounded reference model right hand side satisfies scalar function semipassive smooth solution x(t solve Speed Gradient Algorithm stability storage function synchronization system x Theorem time-varying systems trajectories uniformly unknown parameters unstable variables vector zero Zhukovskii