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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 | |