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| Introduction | |
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| Motivation | |
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| Nonlinear Continuous Systems | |
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| Linearized stability analysis | |
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| Rayleigh-Taylor instability | |
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| Kelvin-Helmholtz instability | |
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| Rayleigh-Benard problem: basic formulation | |
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| Boussinesq approximation and thermal expansion | |
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| Linearized perturbation equations | |
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| Boundary conditions | |
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| Rayleigh-Benard problem: linearized theory of convective instability | |
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| Proof that solutions are not oscillatory | |
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| Vorticity and the eigenvalue equation | |
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| Free-free boundary conditions: exact solution | |
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| Rigid-rigid boundary conditions: sketch of exact solution | |
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| Rayleigh-Benard problem: expansion in Fourier modes | |
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| Lorenz equations: direct derivation for simple physical configuration | |
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| Discrete Dynamical Systems | |
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| Example of a nonlinear oscillator | |
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| Duffing oscillator: general form | |
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| Duffing oscillator: perturbed simple harmonic oscillator | |
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| Phase-space dynamics and fixed points | |
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| Action-angle variables | |
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| Simple harmonic oscillator | |
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| Pendulum | |
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| Linearized stability analysis | |
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| Lorenz model | |
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| Stationary solutions and fixed points | |
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| Linearized stability analysis | |
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| Periodic oscillatory solutions | |
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| Chaotic solutions | |
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| Two theorems on phase-space convergence of solutions | |
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| Model finite-difference equation: logistic map | |
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| Logistic map | |
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| Power spectrum | |
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| Numerical analysis | |
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| Some analytic results | |
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| Liouville's theorem revisited | |
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| Action-angle variables revisited | |
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| Separable, periodic hamiltonian systems | |
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| Phase plots and motion on tori | |
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| Perturbation of periodic hamiltonian systems | |
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| Hamilton-Jacobi theory revisited | |
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| Direct perturbation analysis of the anharmonic oscillator | |
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| Model one-body problem with time-periodic perturbation | |
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| Resonant disruption of phase space | |
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| Overlap of disrupted regions | |
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| Coupled separable periodic hamiltonian systems | |
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| Two weakly coupled degrees of freedom | |
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| Resonant disruption of phase space | |
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| The Kolmogorov-Arnold-Moser (KAM) theorem | |
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| Problems | |
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| Problems | |
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| References | |
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| Index | |