Overview

This work is devoted to the nonlocal theory of nonlinear oscillations. The frequency methods of investigating problems of cycle existence in multidimensional analogues of Van der Pol equation, in dynamical systems with cylindrical phase space and dynamical systems satisfying Routh-Hurwitz generalized conditions are systematically presented. To solve these problems methods of Poincare map construction, frequency methods, synthesis of Lyapunov direct methods and bifurcation theory elements are applied. V.M. Popov's method is employed for obtaining frequency criteria, which estimate period of oscillations. Also, an approach to investigate the stability of cycles based on the ideas of Zhukovsky, Borg, Hartmann, and Olech is presented, and the effects appearing when bounded trajectories are unstable are discussed. For chaotic oscillations theorems on localizations of attractors are given. The upper estimates of Hausdorff measure and dimension of attractors generalizing Doudy-Oesterle and Smith theorems are obtained, illustrated by the example of a Lorenz system and its different generalizations. The analytical apparatus developed in the book is applied to the analysis of oscillation of various control systems, pendulum-like systems and those of synchronization.

Full Product Details

Author:   G.A. Leonov ,  I.M. Burkin ,  A.I. Shepeljavyi
Publisher:   Kluwer Academic Publishers
Imprint:   Kluwer Academic Publishers
Edition:   1996 ed.
Volume:   357
Dimensions:   Width: 16.00cm , Height: 2.30cm , Length: 24.00cm
Weight:   0.888kg
ISBN:  

9780792338963

ISBN 10:   0792338960
Pages:   404
Publication Date:   31 December 1995
Audience:   College/higher education ,  Professional and scholarly ,  Postgraduate, Research & Scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   Out of stock  
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Table of Contents

1. Classical two-dimensional oscillating systems and their multidimensional analogues.- §1.1. The van der Pol equation.- §1.2. The equation of oscillations of a pendulum.- §1.3. Oscillations in two-dimensional systems with hysteresis.- §1.4. Lower estimates of the number of cycles of a two-dimensional system.- 2. Frequency criteria for stability and properties of solutions of special matrix inequalities.- §2.1. Frequency criteria for stability and dichotomy.- §2.2. Theorems on solvability and properties of special matrix inequalities.- 3. Multidimensional analogues of the van der Pol equation.- §3.1. Dissipative systems. Frequency criteria for dissipativity.- §3.2. Second-order systems. Frequency realization of the annulus principle.- §3.3. Third-order systems. The torus principle.- §3.4. The main ideas of applying frequency methods for multidimensional systems.- §3.5. The criterion for the existence of a periodic solution in a system with tachometric feedback.- §3.6. The method of transition into the ""space of derivatives"".- §3.7. A positively invariant torus and the function ""quadratic form plus integral of nonlinearity"".- §3.8. The generalized Poincaré–Bendixson principle.- §3.9. A frequency realization of the generalized Poincaré-Bendixson principle.- §3.10. Frequency estimates of the period of a cycle.- 4. Yakubovich auto–oscillation.- §4.1. Frequency criteria for oscillation of systems with one differentiable nonlinearity.- §4.2. Examples of oscillatory systems.- 5. Cycles in systems with cylindrical phase space.- §5.1. The simplest case of application of the nonlocal reduction method for the equation of a synchronous machine.- §5.2. Circular motions and cycles of the second kind in systems with one nonlinearity.- §5.3. The method ofsystems of comparison.- §5.4. Examples.- §5.5. Frequency criteria for the existence of cycles of the second kind in systems with several nonlinearities.- §5.6. Estimation of the period of cycles of the second kind.- 6. The Barbashin-Ezeilo problem.- §6.1. The existence of cycles of the second kind.- §6.2. Bakaev stability. The method of invariant conical grids.- §6.3. The existence of cycles of the first kind in phase systems.- §6.4. A criterion for the existence of nontrivial periodic solutions of a third-order nonlinear system.- 7. Oscillations in systems satisfying generalized Routh-Hurwitz conditions. Aizerman conjecture.- §7.1. The existence of periodic solutions of systems with nonlinearity from a Hurwitzian sector.- §7.2. Necessary conditions for global stability in the critical case of two zero roots.- §7.3. Lemmas on estimates of solutions in the critical case of one zero root.- §7.4. Necessary conditions for absolute stability of nonautonomous systems.- §7.5. The existence of oscillatory and periodic solutions of systems with hysteretic nonlinearities.- 8. Frequency estimates of the Hausdorff dimension of attractors and orbital stability of cycles.- §8.1. Upper estimates of the Hausdorff measure of compact sets under differentiable mappings.- §8.2. Estimate of the Hausdorff dimension of attractors of systems of differential equations.- §8.3. Global asymptotic stability of autonomous systems.- §8.4. Zhukovsky stability of trajectories.- §8.5. A frequency criterion for Poincaré stability of cycles of the second kind.- §8.6. Frequency estimates for the Hausdorff dimension and conditions for global asymptotic stability.

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