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 1. Introduction. 1.1. Periodicity and quasiperiodicity. 1.2. Robustness of quasiperiodic motions. 1.3. Strange nonchaotic attractors. 1.4. What is in the book -- 2. Models. 2.1. Differential equations and maps. 2.2. Quasiperiodically forced one-dimensional maps. 2.3. Quasiperiodically forced high-dimensional maps. 2.4. Quasiperiodically forced continuous-time systems. 2.5. Experiments. 2.6. Bibliographic notes -- 3. Rational approximations. 3.1. Properties of rational approximations of irrationals. 3.2. Rational approximations to quasiperiodic forcing. 3.3. Checking strangeness of SNA through rational approximations. 3.4. Bibliographic notes -- 4. Stability and instability. 4.1. Theoretical consideration. 4.2. Numerical examples. 4.3. Bibliographic notes -- 5. Fractal and statistical properties. 5.1. Fractal properties of SNA. 5.2. Correlations and spectra of SNA. 5.3. Bibliographic notes -- 6. Bifurcations in quasiperiodically forced systems and transitions to SNA. 6.1. Smooth and non-smooth bifurcations. 6.2. Bifurcations in the quasiperiodically forced logistic map. 6.3. Bifurcations in the quasiperiodically forced circle map. 6.4. Loss of transverse stability: blowout transition to SNA. 6.5. Intermittency. 6.6. Bibliographic notes -- 7. Renormalization group approach to the onset of SNA in maps with the golden-mean quasiperiodic driving. 7.1. Introduction: the main idea of the renormalization group analysis. 7.2. The basic functional equations for the golden-mean renormalization scheme. 7.3. A review of critical points. 7.4. RG analysis of the classic GM critical point. 7.5. RG analysis of the blowout birth of SNA. 7.6. RG analysis of the TDT critical point. 7.7. RG analysis of the TCT critical point. 7.8. RG analysis of the TF critical point. 7.9. Critical behavior in realistic systems. 7.10. Conclusion. 7.11. Bibliographic notes