Nonlinear Ordinary Differential Equations An Introduction to Dynamical Systems,9780198565628

Nonlinear Ordinary Differential Equations An Introduction to Dynamical Systems

by ;
Edition: 3rd
ISBN13: 9780198565628
ISBN10: 0198565623
Format: Paperback
Pub. Date: 1999-10-21
Publisher(s): Oxford University Press
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Summary

Nonlinear ordinary differential equations was first published in 1977 and has since become a standard text in the teaching of the subject. It takes a qualitative approach, and is designed for advanced undergraduate and graduate students of dynamical systems in mathematics or mathematics-related subjects. The text of this third edition has been completely revised to bring it into line with current teaching, including an expansion of the material on bifurcations and chaos. The book is directed towards practical applications of the theory, with several hundred examples and problems covering a wide variety of applications. Prerequisites are kept to a minimum, with appendices containing the necessary mathematical theory new to this edition. From reviews of the first edition: "The book can profitably be used in a senior undergraduate course. It is well written, well motivated and contains some recent developments of interest which have not been readily accessible before at this level." V. Lakshmikantham in Mathematical Reviews From reviews of the second edition: "The subject has wide applications in physical, biological, and social sciences which continuously supply new problems of practical and theoretical importance. The book does a good job in motivating the reader in such pursuits, and presents the subject in a simple but elegant style."--P. K. Kythe in Applied Mechanics Reviews

Table of Contents

Second-order differential equations in the phase plane
Phase diagram for the pedulum equation
1(4)
Autonomous equations in the phase plane
5(9)
Mechanical analogy for the conservative system x = f(x)
14(8)
The damped linear oscillator
22(4)
Nonlinear damping: limit cycles
26(7)
Some applications
33(6)
Parameter-dependent conservative systems
39(3)
Graphical representation of solutions
42(9)
Problems
43(8)
Plane autonomous systems and linearization
The general phase plane
51(4)
Some population models
55(4)
Linear approximation at equilibrium points
59(1)
The general solution of linear autonomous plane systems
60(6)
The phase paths of linear autonomous plane systems
66(9)
Scaling in the phase diagram for a linear autonomous system
75(1)
Constructing a phase diagram
76(2)
Hamiltonian systems
78(14)
Problems
82(10)
Geometrical aspects of plane autonomous systems
The index of a point
92(8)
The index at infinity
100(3)
The phase diagram at infinity
103(5)
Limit cycles and other closed paths
108(3)
Computation of the phase diagram
111(3)
Homoclinic and heteroclinic paths
114(17)
Problems
117(14)
Periodic solutions; averaging methods
An energy-balance method for limit cycle
131(5)
Amplitude and frequency estimates: polar coordinates
136(5)
An averaging method for spiral phase paths
141(3)
Periodic solutions: harmonic balance
144(2)
The equivalent linear equation by harmonic balance
146(10)
Problems
149(7)
Perturbation methods
Nonautonomous systems: forced oscillations
156(4)
Outline of the direct method for the undamped case; Duffing's equation
160(2)
Forced oscillations far from resonance
162(2)
Forced oscillations near resonance with weak excitation
164(3)
The amplitude equation for the undamped pendulum
167(3)
The amplitude equation for a damped pendulum
170(1)
Soft and hard springs
171(3)
Amplitude-phase perturbation for the pendulum equation
174(2)
Periodic solutions of autonomous equations (Lindstedt's method)
176(2)
Forced oscillation of a self-excited equation
178(3)
The perturbation method and Fourier series
181(2)
Homoclinic bifurcation: an example
183(8)
Problems
186(5)
Singular perturbation methods
Non-uniform approximations to functions on an interval
191(2)
Coordinate perturbation
193(6)
Lighthill's method
199(2)
Time-scaling for series solutions of autonomous equations
201(7)
The multiple-scale technique applied to saddle points and nodes
208(9)
Matching approximations on an interval
217(5)
A matching technique for differential equations
222(15)
Problems
229(8)
Forced oscillations: harmonic and subharmonic response, stability, and entrainment
General forced periodic solutions
237(2)
Harmonic solutions, transients, and stability for Duffing's equation
239(6)
The jump phenomenon
245(3)
Harmonic oscillations, stability, and transients for the forced van der Pol equation
248(5)
Frequency entrainment for the van der Pol equation
253(4)
Subharmonics of Duffing's equation by perturbation
257(5)
Stability and transients for subharmonics of Duffing's equation
262(15)
Problems
266(11)
Stability
Poincare stability (stability of paths)
277(5)
Paths and solution curves for general systems
282(2)
Stability of time solutions: Liapunov stability
284(5)
Liapunov stability of plane autonomous linear systems
289(3)
Structure of the solutions of n-dimensional linear systems
292(5)
Structure of n-dimensional inhomogeneous linear systems
297(3)
Stability and boundedness for linear systems
300(1)
Stability of linear systems with constant coefficients
301(5)
Linear approximation at equilibrium points for first-order systems in n variables
306(3)
Stability of a class of nonautonomous linear systems in n dimensions
309(6)
Stability of the zero solutions of nearly linear systems
315(7)
Problems
317(5)
Determination of stability by solution perturbation
The stability of forced oscillations by solution perturbation
322(3)
Equations with periodic coefficients (Floquet theory)
325(7)
Mathieu's equation arising from a Duffing equation
332(4)
Transition curves for Mathieu's equation by perturbation
336(2)
Mathieu's damped equation arising from a Duffing equation
338(10)
Problems
341(7)
Liapunov methods for determining stability of the zero solution
Introducing the Liapunov method
348(1)
Topographic systems and the Poincare-Bendixson theorem
349(4)
Liapunov stability of the zero solution
353(4)
Asymptotic stability of the zero solution
357(3)
Extending weak Liapunov functions to asymptotic stability
360(3)
A more general theory for autonomous systems
363(4)
A test for instability of the zero solution: n dimensions
367(2)
Stability and the linear approximation in two dimensions
369(7)
Exponential function of a matrix
376(2)
Stability and the linear approximation for nth order autonomous systems
378(6)
Special systems
384(13)
Problems
388(9)
The existence of periodic solutions
The Poincare--Bendixson theorem and periodic solutions
397(7)
A theorem on the existence of a centre
404(4)
A theorem on the existence of a limit cycle
408(6)
Van der Pol's equation with large parameter
414(6)
Problems
417(3)
Bifurcations and manifolds
Examples of simple bifurcations
420(2)
The fold and the cusp
422(4)
Structural stability
426(3)
Further types of bifurcation
429(8)
Hopf bifurcations
437(2)
Higher-order systems: manifolds
439(6)
Linear approximation: centre manifolds
445(12)
Problems
452(5)
Poincare sequences, homoclinic bifurcation, and chaos
Poincare sequences
457(10)
Homoclinic paths, strange attractors and chaos
467(4)
The Duffing oscillator
471(9)
The logistic difference equation
480(4)
Homoclinic bifurcation for forced systems
484(8)
The horseshoe map
492(1)
Melnikov's method for detecting homoclinic bifurcation
493(6)
Power spectra
499(1)
Some characteristic features of chaotic oscillations
500(30)
Problems
502(15)
Hints and answers to the problems
517(13)
Appendices
A Existence and uniqueness theorems
530(2)
B Topographic systems
532(3)
C Norms for vectors and matrices
535(1)
D A contour integral
536(3)
References and further reading 539(3)
Index 542

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