## Chaos, Noise and FractalsThe study of nonlinear dynamical systems has been gathering momentum since the late 1950s. It now constitutes one of the major research areas of modern theoretical physics. The twin themes of fractals and chaos, which are linked by attracting sets in chaotic systems that are fractal in structure, are currently generating a great deal of excitement. The degree of structure robustness in the presence of stochastic and quantum noise is thus a topic of interest. Chaos, Noise and Fractals discusses the role of fractals in quantum mechanics, the influence of phase noise in chaos and driven optical systems, and the arithmetic of chaos. The book represents a balanced overview of the field and is a worthy addition to the reading lists of researchers and students interested in any of the varied, and sometimes bizarre, aspects of this intriguing subject. |

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### Contents

Singular System Analysis with Application to Dynamical Systems | 15 |

A Review of Progress in the Kicked Rotator Problem | 28 |

Fractals in Quantum Mechanics? | 47 |

Cantori and Quantum Mechanics | 76 |

Chaos in the Micromaser | 102 |

Chaos in a Driven Quantum Spin System | 117 |

### Common terms and phrases

amplitude analysis Anosov diffeomorphisms approximation Arecchi asymptotic atoms attractor average behaviour cantori Casati cavity classically chaotic coherent computed corresponding curve density matrix deterministic diffusion dimension dynamics eigenvalues energy entropy equation ergodic evolution exponent exponential field Figure finite fluctuations fractal frequency function Gaussian Hamiltonian hysteresis infinite dimensional map instability integrable inversion iterates Kerr l6 wave laser Lett linear linearised Lorenz equations lower branch Lyapunov exponent matrix elements modulation Moloney nonlinear obtained operator optical bistability oscillations parameter periodic orbits perturbation phase noise phase space Phys physical plane wave positive-P probability distribution problem quantization quantum chaos quantum mechanics quantum optics quantum system random matrix random matrix theory region resonator ring Sarkar saturable scale SDEs semiclassical shown in Fig shows singular value solitary wave solution spatial spectra stable statistical stochastic theory torus trajectory transverse truncation variables vector wave fixed point wavetrain Wigner