Preface | p. vii |
Introduction | p. 1 |
Periodicity and quasiperiodicity | p. 1 |
Robustness of quasiperiodic motions | p. 3 |
Strange nonchaotic attractors | p. 5 |
What is in the book | p. 6 |
Models | p. 9 |
Differential equations and maps | p. 9 |
Quasiperiodically forced one-dimensional maps | p. 11 |
GOPY model (modulated pitchfork map) | p. 12 |
Forced circle map | p. 14 |
Skew shift | p. 15 |
Forced logistic map | p. 16 |
Harper model | p. 20 |
Quasiperiodically forced high-dimensional maps | p. 20 |
Quasiperiodically forced continuous-time systems | p. 21 |
Forced overdamped pendulum | p. 22 |
Forced Duffing oscillator | p. 23 |
Experiments | p. 26 |
Bibliographic notes | p. 27 |
Rational approximations | p. 29 |
Properties of rational approximations of irrationals | p. 30 |
Rational approximations to quasiperiodic forcing | p. 32 |
Checking strangeness of SNA through rational approximations | p. 33 |
Rational approximations to a smooth attractor | p. 33 |
Rational approximations to an SNA: An example | p. 33 |
Rational approximations to an SNA: General consideration | p. 36 |
Different examples | p. 39 |
Bibliographic notes | p. 42 |
Stability and Instability | p. 45 |
Theoretical consideration | p. 45 |
Numerical examples | p. 50 |
Discrete time mappings | p. 50 |
Continuous time systems | p. 54 |
Bibliographic notes | p. 55 |
Fractal and statistical properties | p. 57 |
Fractal properties of SNA | p. 57 |
Correlations and spectra of SNA | p. 60 |
Power spectra of regular and irregular motions | p. 60 |
Spectral properties of fractal tori | p. 63 |
Singular continuous spectrum in an SNA | p. 65 |
Theoretical description of the singular continuous spectrum | p. 69 |
Bibliographic notes | p. 73 |
Bifurcations in quasiperiodically forced systems and transitions to SNA | p. 75 |
Smooth and non-smooth bifurcations | p. 76 |
Bifurcations in the quasiperiodically forced logistic map | p. 77 |
Torus doubling | p. 79 |
Non-smooth tori collision beyond period-doubling | p. 81 |
Fractalization of the torus | p. 83 |
Interior crisis | p. 85 |
Boundary crisis | p. 91 |
Basin boundary bifurcation | p. 95 |
Bifurcations in the quasiperiodically forced circle map | p. 102 |
Smooth saddle-node bifurcation of tori | p. 104 |
Non-smooth collision of a stable and an unstable torus | p. 104 |
Phase-locking regions under quasiperiodic forcing | p. 107 |
Non-smooth pitchfork bifurcation | p. 111 |
Loss of transverse stability: blowout transition to SNA | p. 113 |
Intermittency | p. 119 |
Bibliographic notes | p. 127 |
Renormalization group approach to the onset of SNA in maps with the golden-mean quasiperiodic driving | p. 131 |
Introduction: The main idea of the renormalization group analysis | p. 131 |
The basic functional equations for the golden-mean renormalization scheme | p. 134 |
A review of critical points | p. 136 |
Classic GM point | p. 137 |
Critical point of the blowout birth of SNA | p. 137 |
Critical points of torus doubling terminal and torus collision terminal | p. 139 |
Critical point of torus fractalization | p. 141 |
RG analysis of the classic GM critical point | p. 142 |
RG analysis of the blowout birth of SNA | p. 145 |
RG analysis of the TDT critical point | p. 154 |
RG analysis of the TCT critical point | p. 165 |
RG analysis of the TF critical point | p. 178 |
Critical behavior in realistic systems | p. 187 |
Conclusion | p. 193 |
Bibliographic notes | p. 195 |
Bibliography | p. 197 |
Index | p. 211 |
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