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Introduction | |
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Solving differential equations | |
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Existence and uniqueness theorems | |
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Phase space and flows | |
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Limit sets and trajectories | |
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Exercises 1 | |
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Stability | |
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Definitions of stability | |
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Liapounov functions | |
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Strong linear stability | |
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Orbital stability | |
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Bounding functions | |
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Non-autonomous equations | |
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Exercises 2 | |
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Linear Differential Equations | |
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Autonomous linear differential equations | |
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Normal forms | |
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Invariant manifolds | |
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Geometry of phase space | |
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Floquet Theory | |
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Exercises 3 | |
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Linearization and Hyperbolicity | |
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Poincare's Linearization Theorem | |
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Hyperbolic stationary points and the stable manifold theorem | |
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Persistence of hyperbolic stationary points | |
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Structural stability | |
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Nonlinear sinks | |
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The proof of the stable manifold theorem | |
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Exercises 4 | |
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Two Dimensional Dynamics | |
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Linear systems in R[superscript 2] | |
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The effect of nonlinear terms | |
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Rabbits and sheep | |
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Trivial linearization | |
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The Poincare index | |
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Dulac's criterion | |
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Pike and eels | |
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The Poincare-Bendixson Theorem | |
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Decay of large amplitudes | |
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Exercises 5 | |
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Periodic Orbits | |
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Linear and nonlinear maps | |
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Return maps | |
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Floquet Theory revisited | |
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Periodically forced differential equations | |
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Normal forms for maps in R[superscript 2] | |
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Exercises 6 | |
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Perturbation Theory | |
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Asymptotic expansions | |
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The method of multiple scales | |
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Multiple scales for forced oscillators: complex notation | |
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Higher order terms | |
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Interlude | |
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Nearly Hamiltonian systems | |
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Resonance in the Mathieu equation | |
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Parametric excitation in Mathieu equations | |
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Frequency locking | |
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Hysteresis | |
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Asynchronous quenching | |
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Subharmonic resonance | |
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Relaxation oscillators | |
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Exercises 7 | |
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Bifurcation Theory I: Stationary Points | |
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Centre manifolds | |
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Local bifurcations | |
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The saddlenode bifurcation | |
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The transcritical bifurcation | |
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The pitchfork bifurcation | |
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An example | |
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The Implicit Function Theorem | |
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The Hopf bifurcation | |
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Calculation of the stability coefficient, ReA(0) | |
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A canard | |
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Exercises 8 | |
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Bifurcation Theory II: Periodic Orbits and Maps | |
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A simple eigenvalue of +1 | |
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Period-doubling bifurcations | |
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The Hopf bifurcation for maps | |
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Arnol'd (resonant) tongues | |
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Exercises 9 | |
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Bifurcational Miscellany | |
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Unfolding degenerate singularities | |
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Imperfection theory | |
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Isolas | |
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Periodic orbits in Lotka-Volterra models | |
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Subharmonic resonance revisited | |
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Chaos | |
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Characterizing chaos | |
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Period three implies chaos | |
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Unimodal maps I: an overview | |
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Tent maps | |
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Unimodal maps II: the non-chaotic case | |
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Quantitative universality and scaling | |
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Intermittency | |
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Partitions, graphs and Sharkovskii's Theorem | |
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Exercises 11 | |
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Global Bifurcation Theory | |
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An example | |
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Homoclinic orbits and the saddle index | |
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Planar homoclinic bifurcations | |
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Homoclinic bifurcations to a saddle-focus | |
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Lorenz-like equations | |
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Cascades of homoclinic bifurcations | |
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Exercises 12 | |
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Notes and further reading | |
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Bibliography | |
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Index | |