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Stability, Instability and Chaos An Introduction to the Theory of Nonlinear Differential Equations

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ISBN-10: 0521425662

ISBN-13: 9780521425667

Edition: 1994

Authors: Paul Glendinning, D. G. Crighton, M. J. Ablowitz, S. H. Davis, E. J. Hinch

List price: $82.99
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Description:

By providing an introduction to nonlinear differential equations, Dr. Glendinning aims to equip the student with the mathematical know-how needed to appreciate stability theory and bifurcations. His approach is readable and covers material both old and new to undergraduate courses. Included are treatments of the Poincar-Bendixson theorem, the Hopf bifurcation and chaotic systems.
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Book details

List price: $82.99
Copyright year: 1994
Publisher: Cambridge University Press
Publication date: 11/25/1994
Binding: Paperback
Pages: 404
Size: 6.25" wide x 9.25" long x 1.00" tall
Weight: 1.188
Language: English

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