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nonlinear ordinary differential equations, jordan, smith [pdf]

Nonlinear Ordinary Differential Equations (4E) by D. W. Jordan, P. Smith
(An introduction for Scientists and Engineers)

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Nonlinear Ordinary Differential Equations (4E) (An introduction for Scientists and Engineers) written by D. W. Jordan, P. Smith , Keele University. Published by Oxford University Press , Year: 2007, ISBN: 978–0–19–920824–1.
The book developed from courses on nonlinear differential equations given over many years in the Mathematics Department of Keele University. It presents an introduction to dynamical systems in the context of ordinary differential equations, and is intended for students of mathematics, engineering and the sciences, and workers in these areas who are mainly interested in the more direct applications of the subject. The level is about that of final-year undergraduate, or master’s degree courses in the UK. It has been found that selected material from Chapters 1 to 5, and 8, 10, and 11 can be covered in a one-semester course by students having a background of techniques in differential equations and linear algebra. The book is designed to accommodate courses of varying emphasis, the chapters forming fairly self-contained groups from which a coherent selection can be made without using significant parts of the argument.
General solutions of nonlinear differential equations are rarely obtainable, though particular solutions can be calculated one at a time by standard numerical techniques. However, this book deals with qualitative methods that reveal the novel phenomena arising from nonlinear equations, and produce good numerical estimates of parameters connected with such general features as stability, periodicity and chaotic behaviour without the need to solve the equations. We illustrate the reliability of such methods by graphical or numerical comparison with numerical solutions. For this purpose the Mathematica™software was used to calculate particular exact solutions; this was also of great assistance in the construction of perturbation series, trigonometric identities, and for other algebraic manipulation. However, experience with such software is not necessary for the reader.

Nonlinear Ordinary Differential Equations (4E) (An introduction for Scientists and Engineers) written by D. W. Jordan, P. Smith cover the following topics.

  • 1. Second-order differential equations in the phase plane
    2. Plane autonomous systems and linearization
    3. Geometrical aspects of plane autonomous systems
    4. Periodic solutions; averaging methods
    5. Perturbation methods
    6. Singular perturbation methods
    7. Forced oscillations: harmonic and subharmonic response, stability, and entrainment
    8. Stability
    9. Stability by solution perturbation: Mathieu’s equation
    10. Liapunov methods for determining stability of the zero solution
    11. The existence of periodic solutions
    12. Bifurcations and manifolds
    13. Poincaré sequences, homoclinic bifurcation, and chaos
    Answers to the exercises
    Appendices A Existence and uniqueness theorems
    Appendices B Topographic systems
    Appendices C Norms for vectors and matrices
    Appendices D A contour integral
    Appendices E Useful results
    References and further reading

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