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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 | |