Strange Nonchaotic Attractors: Dynamics Between Order And Chaos In Quasiperiodically Forced Systems Dynamics between Order and Chaos in Quasiperiodically Forced Systems

This book is the first monograph devoted exclusively to strange nonchaotic attractors (SNA), recently discovered objects with a special kind of dynamical behavior between order and chaos in dissipative nonlinear systems under quasiperiodic driving. A historical review of the discovery and study of SNA, mathematical and physically-motivated examples, and a review of known experimental studies of SNA are presented. The main focus is on the theoretical analysis of strange nonchaotic behavior by means of different tools of nonlinear dynamics and statistical physics (bifurcation analysis, Lyapunov exponents, correlations and spectra, renormalization group). The relations of the subject to other fields of physics such as quantum chaos and solid state physics are also discussed. Book jacket Contents......Page 10 Preface......Page 8 1.1 Periodicity and quasiperiodicity......Page 13 1.2 Robustness of quasiperiodic motions......Page 15 1.3 Strange nonchaotic attractors......Page 17 1.4 What is in the book......Page 18 2.1 Differential equations and maps......Page 21 2.2 Quasiperiodically forced one-dimensional maps......Page 23 2.2.1 GOPY model (modulated pitchfork map)......Page 24 2.2.2 Forced circle map......Page 26 2.2.3 Skew shift......Page 27 2.2.4 Forced logistic map......Page 28 2.3 Quasiperiodically forced high-dimensional maps......Page 32 2.4 Quasiperiodically forced continuous-time systems......Page 33 2.4.1 Forced overdamped pendulum......Page 34 2.4.2 Forced Duffing oscillator......Page 35 2.5 Experiments......Page 38 2.6 Bibliographic notes......Page 39 3 Rational approximations......Page 41 3.1 Properties of rational approximations of irrationals......Page 42 3.2 Rational approximations to quasiperiodic forcing......Page 44 3.3.2 Rational approximations to an SNA: An example......Page 45 3.3.3 Rational approximations to an SNA: General consideration......Page 48 3.3.4 Different examples......Page 51 3.4 Bibliographic notes......Page 54 4.1 Theoretical consideration......Page 57 4.2.1 Discrete time mappings......Page 62 4.2.2 Continuous time systems......Page 66 4.3 Bibliographic notes......Page 67 5.1 Fractal properties of SNA......Page 69 5.2.1 Power spectra of regular and irregular motions......Page 72 5.2.2 Spectral properties of fractal tori......Page 75 5.2.3 Singular continuous spectrum in an SNA......Page 77 5.2.4 Theoretical description of the singular continuous spectrum......Page 81 5.3 Bibliographic notes......Page 85 6 Bifurcations in quasiperiodically forced systems and transitions to SNA......Page 87 6.1 Smooth and non-smooth bifurcations......Page 88 6.2 Bifurcations in the quasiperiodically forced logistic map......Page 89 6.2.1 Torus doubling......Page 91 6.2.2 Non-smooth tori collision beyond period-doubling......Page 93 6.2.3 Fractalization of the torus......Page 95 6.2.4 Interior crisis......Page 97 6.2.5 Boundary crisis......Page 103 6.2.6 Basin boundary bifurcation......Page 107 6.3 Bifurcations in the quasiperiodically forced circle map......Page 114 6.3.2 Non-smooth collision of a stable and an unstable torus......Page 116 6.3.3 Phase-locking regions under quasiperiodic forcing......Page 119 6.3.4 Non-smooth pitchfork bifurcation......Page 123 6.4 Loss of transverse stability: blowout transition to SNA......Page 125 6.5 Intermittency......Page 131 6.6 Bibliographic notes......Page 139 7.1 Introduction: The main idea of the renormalization group analysis......Page 143 7.2 The basic functional equations for the golden-mean renormalization scheme......Page 146 7.3 A review of critical points......Page 148 7.3.2 Critical point of the blowout birth of SNA......Page 149 7.3.3 Critical points of torus doubling terminal and torus collision terminal......Page 151 7.3.4 Critical point of torus fractalization......Page 153 7.4 RG analysis of the classic GM critical point......Page 154 7.5 RG analysis of the blowout birth of SNA......Page 157 7.6 RG analysis of the TDT critical point......Page 166 7.7 RG analysis of the TCT critical point......Page 177 7.8 RG analysis of the TF critical point......Page 190 7.9 Critical behavior in realistic systems......Page 199 7.10 Conclusion......Page 205 7.11 Bibliographic notes......Page 207 Bibliography......Page 209 Index......Page 223 Contents 10 Preface 8 1 Introduction 13 1.1 Periodicity and quasiperiodicity 13 1.2 Robustness of quasiperiodic motions 15 1.3 Strange nonchaotic attractors 17 1.4 What is in the book 18 2 Models 21 2.1 Differential equations and maps 21 2.2 Quasiperiodically forced one-dimensional maps 23 2.2.1 GOPY model (modulated pitchfork map) 24 2.2.2 Forced circle map 26 2.2.3 Skew shift 27 2.2.4 Forced logistic map 28 2.2.5 Harper model 32 2.3 Quasiperiodically forced high-dimensional maps 32 2.4 Quasiperiodically forced continuous-time systems 33 2.4.1 Forced overdamped pendulum 34 2.4.2 Forced Duffing oscillator 35 2.5 Experiments 38 2.6 Bibliographic notes 39 3 Rational approximations 41 3.1 Properties of rational approximations of irrationals 42 3.2 Rational approximations to quasiperiodic forcing 44 3.3 Checking strangeness of SNA through rational approximations 45 3.3.1 Rational approximations to a smooth attractor 45 3.3.2 Rational approximations to an SNA: An example 45 3.3.3 Rational approximations to an SNA: General consideration 48 3.3.4 Different examples 51 3.4 Bibliographic notes 54 4 Stability and Instability 57 4.1 Theoretical consideration 57 4.2 Numerical examples 62 4.2.1 Discrete time mappings 62 4.2.2 Continuous time systems 66 4.3 Bibliographic notes 67 5 Fractal and statistical properties 69 5.1 Fractal properties of SNA 69 5.2 Correlations and spectra of SNA 72 5.2.1 Power spectra of regular and irregular motions 72 5.2.2 Spectral properties of fractal tori 75 5.2.3 Singular continuous spectrum in an SNA 77 5.2.4 Theoretical description of the singular continuous spectrum 81 5.3 Bibliographic notes 85 6 Bifurcations in quasiperiodically forced systems and transitions to SNA 87 6.1 Smooth and non-smooth bifurcations 88 6.2 Bifurcations in the quasiperiodically forced logistic map 89 6.2.1 Torus doubling 91 6.2.2 Non-smooth tori collision beyond period-doubling 93 6.2.3 Fractalization of the torus 95 6.2.4 Interior crisis 97 6.2.5 Boundary crisis 103 6.2.6 Basin boundary bifurcation 107 6.3 Bifurcations in the quasiperiodically forced circle map 114 6.3.1 Smooth saddle-node bifurcation of tori 116 6.3.2 Non-smooth collision of a stable and an unstable torus 116 6.3.3 Phase-locking regions under quasiperiodic forcing 119 6.3.4 Non-smooth pitchfork bifurcation 123 6.4 Loss of transverse stability: blowout transition to SNA 125 6.5 Intermittency 131 6.6 Bibliographic notes 139 7 Renormalization group approach to the onset of SNA in maps with the golden-mean quasiperiodic driving 143 7.1 Introduction: The main idea of the renormalization group analysis 143 7.2 The basic functional equations for the golden-mean renormalization scheme 146 7.3 A review of critical points 148 7.3.1 Classic GM point 149 7.3.2 Critical point of the blowout birth of SNA 149 7.3.3 Critical points of torus doubling terminal and torus collision terminal 151 7.3.4 Critical point of torus fractalization 153 7.4 RG analysis of the classic GM critical point 154 7.5 RG analysis of the blowout birth of SNA 157 7.6 RG analysis of the TDT critical point 166 7.7 RG analysis of the TCT critical point 177 7.8 RG analysis of the TF critical point 190 7.9 Critical behavior in realistic systems 199 7.10 Conclusion 205 7.11 Bibliographic notes 207 Bibliography 209 Index 223