Introduction to Nonlinear Dynamics for Physicists

This is a series of lecture notes on nonlinear dynamics for physicists. The level is that of an advanced undergraduate or beginning graduate student. The main aim of these lectures is to present a balance of qualitative and quantitative material. The book also aims to address three main questions about nonlinear dynamics: what is nonlinear dynamics all about and what makes it differ from linear dynamics which permeate all familiar textbooks?; from the physicist's point of view, why should we study nonlinear systems and leave the comfortable territory of linearity?; and how can one progress in the study of nonlinear systems both in the analysis of these systems when we know them, and in learning about new systems from observations of their experimental behaviour? As it would be impossible to answer these questions in the finest detail, this volume nevertheless points the way for the interested reader. Useful problems have also been incorporated as a guide. b0001......Page 1 b0002......Page 2 b0003......Page 3 b0004......Page 4 b0005......Page 5 b0006......Page 6 b0007......Page 7 b0008......Page 8 b0009......Page 9 b0010......Page 10 b0011......Page 11 b0012......Page 12 b0013......Page 13 b0014......Page 14 b0015......Page 15 b0016......Page 16 b0017......Page 17 b0018......Page 18 b0019......Page 19 b0020......Page 20 b0021......Page 21 b0022......Page 22 b0023......Page 23 b0024......Page 24 b0025......Page 25 b0026......Page 26 b0027......Page 27 b0028......Page 28 b0029......Page 29 b0030......Page 30 b0031......Page 31 b0032......Page 32 b0033......Page 33 b0034......Page 34 b0035......Page 35 b0036......Page 36 b0037......Page 37 b0038......Page 38 b0039......Page 39 b0040......Page 40 b0041......Page 41 b0042......Page 42 b0043......Page 43 b0044......Page 44 b0045......Page 45 b0046......Page 46 b0047......Page 47 b0048......Page 48 b0049......Page 49 b0050......Page 50 b0051......Page 51 b0052......Page 52 b0053......Page 53 b0054......Page 54 b0055......Page 55 b0056......Page 56 b0057......Page 57 b0058......Page 58 b0059......Page 59 b0060......Page 60 b0061......Page 61 b0062......Page 62 b0063......Page 63 b0064......Page 64 b0065......Page 65 b0066......Page 66 b0067......Page 67 b0068......Page 68 b0069......Page 69 b0070......Page 70 b0071......Page 71 b0072......Page 72 b0073......Page 73 b0074......Page 74 b0075......Page 75 b0076......Page 76 b0077......Page 77 b0078......Page 78 b0079......Page 79 b0080......Page 80 b0081......Page 81 b0082......Page 82 b0083......Page 83 b0084......Page 84 1. Introduction -- 2. Nonlinear oscillator without dissipation -- 3. Equilibrium states of a nonlinear oscillator with dissipation -- 4. Oscillations in systems with nonlinear dissipation-generators -- 5. The Van der Pol generator -- 6. The Poincaré map -- 7. Slow and fast motions in systems with one degree of freedom -- 8. Forced nonlinear oscillators: linear and nonlinear resonances -- 9. Forced generator: Synchronization -- 10. Competition of modes -- 11. Poincaré indices and bifurcations of equilibrium states -- 12. Resonance interactions between oscillators -- 13. Solitons -- 14. Steady propagation of shock waves -- 15. Formation of shock waves -- 16. Solitons. Shock Waves. Wave Interaction. The spectral approach -- 17. Weak turbulence. Random phase approximation -- 18. Regular patterns in dissipative media -- 19. Deterministic chaos. Qualitative description -- 20. Description of a circuit with chaos. Chaos in maps -- 21. Bifurcations of periodic motions. Period doubling -- 22. Controlled nonlinear oscillator. Intermittency -- 23. Scenarios of the onset of chaos. Chaos through quasi-periodicity -- 24. Characteristics of chaos. Experimental observation of chaos -- 25. Multidimensional chaos. Discrete Ginzburg-Landau model -- 26. Problems to accompany the lectures This series of lectures aims to address three main questions that anyone interested in the study of nonlinear dynamics should ask and ponder over. What is nonlinear dynamics and how does it differ from linear dynamics which permeates all familiar textbooks? Why should the physicist study nonlinear systems and leave the comfortable territory of linearity? How can one progress in the study of nonlinear systems both in the analysis of these systems and in learning about new systems from observing their experimental behavior? While it is impossible to answer these questions in the finest detail, this series of lectures nonetheless successfully points the way for the interested reader. Other useful problems have also been incorporated as a study guide. By presenting both substantial qualitative information about phenomena in nonlinear systems and at the same time sufficient quantitative material, the author hopes that readers would learn how to progress on their own in the study of such similar material hereon. 20 Description of a Circuit with Chaos. Chaos in Maps21 Bifurcations of Periodic Motions. Period Doubling; 22 Controlled Nonlinear Oscillator. Intermittency; 23 Scenarios of the Onset of Chaos. Chaos through Quasi-Periodicity; 24 Characteristics of Chaos. Experimental Observation of Chaos; 24.1 Fractal Dimension; 24.2 Lyapunov Exponents; 25 Multidimensional Chaos. Discrete Ginzburg-Landau Model; 26 Problems to Accompany the Lectures; 26.1 Lecture One; 26.2 Lecture Two; 26.3 Lecture Three; 26.4 Lecture Four; 26.5 Lecture Five; 26.6 Lecture Six; 26.7 Lecture Seven; 26.8 Lecture Eight 7 Slow and Fast Motions in Systems with One Degree of Freedom8 Forced Nonlinear Oscillators: Linear and Nonlinear Resonances; 9 Forced Generator: Synchronization; 10 Competition of Modes; 11 Poincare Indices and Bifurcations of Equilibrium States; 12 Resonance Interactions between Oscillators; 13 Solitons; 14 Steady Propagation of Shock Waves; 15 Formation of Shock Waves; 16 Solitons. Shock Waves. Wave Interaction. The Spectral Approach; 17 Weak Turbulence. Random Phase Approximation; 18 Regular Patterns in Dissipative Media; 19 Deterministic Chaos. Qualitative Description Preface; Contents; INTRODUCTION TO NONLINEAR DYNAMICS FOR PHYSICISTS; 1 Introduction; 2 Nonlinear Oscillator without Dissipation; 1. A ball in a trough; 2. Our equations of motion can be easily integrated:; 3. The integral curves on which the direction of motion is defined are referred to as phase trajectories.; 4. Analytical description of motion on a separatrix loop; 3 Equilibrium States of a Nonlinear Oscillator with Dissipation; 4 Oscillations in Systems with Nonlinear Dissipation-Generators; 5 The Van der Pol Generator; 6 The Poincare Map 26.9 Lecture Nine26.10 Lecture Ten; 26.11 Lecture Eleven; 26.12 Lecture Twelve; 26.13 Lecture Thirteen; 26.14 Lecture Fourteen; 26.15 Lecture Fifteen; 26.16 Lecture Sixteen; 26.17 Lecture Seventeen; 26.18 Lecture Eighteen; 26.19 Lecture Nineteen; 26.20 Lecture Twenty; 26.21 Lecture Twenty-one; 26.22 Lecture Twenty-two; 26.23 Lecture Twenty-three; 26.24 Lecture Twenty-four; 26.25 Lecture Twenty-five