Nonlinear Ordinary Differential Equations An Introduction to Dynamical Systems

by Jordan, D. W.; Smith, P.

• ISBN13: 9780198565635
• ISBN10: 0198565631
• Edition: 3rd
• Format: Hardcover
• Copyright: 1999-10-21
• Publisher: Oxford University Press

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