## Dynamical Systems Approach to TurbulenceIn recent decades, turbulence has evolved into a very active field of theoretical physics. The origin of this development is the approach to turbulence from the point of view of deterministic dynamical systems, and this book shows how concepts developed for low dimensional chaotic systems are applied to turbulent states. This book centers around a number of important simplified models for turbulent behavior in systems ranging from fluid motion (classical turbulence) to chemical reactions and interfaces in disordered systems. The theory of fractals and multifractals now plays a major role in turbulence research, and turbulent states are being studied as important dynamical states of matter occurring also in systems outside the realm of hydrodynamics. The book contains simplified models of turbulent behavior, notably shell models, coupled map lattices, amplitude equations and interface models. |

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### Contents

Chapter 1 Turbulence and dynamical systems | 1 |

Chapter 2 Phenomenology of hydrodynamic turbulence | 21 |

Chapter 3 Reduced models for hydrodynamic turbulence | 48 |

Chapter 4 Turbulence and coupled map lattices | 91 |

Chapter 5 Turbulence in the complex GinzburgLandau equation | 138 |

Chapter 6 Predictability in highdimensional systems | 183 |

Chapter 7 Dynamics of interfaces | 211 |

Chapter 8 Lagrangian chaos | 244 |

Appendix B Hamiltonian systems | 294 |

Appendix C Characteristic and generalized Lyapunov exponents | 301 |

Appendix D Convective instabilities and linear front propagation | 309 |

Appendix E Generalized fractal dimensions and multifractals | 315 |

Appendix F Multiaffine fields | 320 |

Appendix G Reduction to a finitedimensional dynamical system | 325 |

Appendix H Directed percolation | 329 |

References | 332 |

### Common terms and phrases

advected attractor average behaviour Bohr boundary conditions boundary layer chaotic coeﬂicients computed convective corresponding coupled map lattice deﬁned degrees of freedom density diffusion dimension dimensional directed percolation discussed dynamical systems energy dissipation enstrophy Eulerian evolution exponential Figure ﬁnd ﬁnite ﬁrst ﬁxed point ﬂow ﬂuctuations fractal fractal dimension gaussian GOY model gradients growth Hamiltonian systems inertial range inﬁnite initial conditions interface intermittency introduced invariant Kolmogorov Kuramoto—Sivashinsky equation Lagrangian Lagrangian chaos laminar length scales linear Lyapunov exponent maximum Lyapunov exponent motion multifractal Navier—Stokes equations obtained parameter perturbation phase space Phys positive Lyapunov exponents power law predictability probability distribution random relevant Reynolds number saddle point scaling invariance separatrices shell model shown in Fig shows Sneppen solution spatial spectrum stable stable manifold statistical structure functions theory trajectories turbulence typical unstable variables velocity ﬁeld viscosity vortex vortices Vulpiani wave numbers zero