Stability, instability and chaos: an introduction to the theory of nonlinear differential equations

Paul Glendinning

Cambridge texts in appled mathematics

Cambridge University Press 1994



This is a textbook originating from lecture notes to undergraduate and beginning graduate courses. This has produced valuable features. The book covers dynamical system theory at a good interesting pace with many specific examples for students to attempt. Useful algorithmic criteria for existence and stability of equilibrium solutions and periodic solutions are described. It is however difficult to discern a theoretical structure and there is a dearth of detailed discussion of significant applications.

After an introductory chapter the behaviour of solutions near an equilibrium state is studied. Normal forms are derived for linear constant systems and Floquet theory for linear periodic systems is presented. Nonlinear systems near hyperbolic equilibrium states are shown to be similar to the linear approximating system in behaviour. An unusual feature is that it is only after this discussion, including the stable manifold theorem, that the usual two dimensional phase portrait types are discussed. Periodic solutions of limit cycle type are then investigated. Perturbation methods and the quantitative information they can provide form a chapter. Bifurcation theory for systems with one significant parameter is then developed. There is a discussion of chaos in one dimensional maps. Interesting topics in the text are Canardes, resonance with Arnold tongues and a discussion of homoclinic orbits and their importance in bifurcation processes. Printing is adequate but the diagram ( fig 5.8 pp 109) is incorrect and might confuse students. Numerical methods and symbol manipulation languages are not mentioned. Co-dimension of a bifurcation, rotation of vector fields and the concept of degree are not mentioned.

An unusual definition is taken for stability. Usually the concept is defined for an invariant set, here (page 27) it is for a general point. It may help readers to compare the treatment with that in (page 16) of the text [1]. In my opinion the treatment presented using variational equations as in [2] is the most illuminating. The book claims (page 91) that the idea of structural stability was introduced after 1960. The book [3] reveals that these ideas were being investigated in 1937. This latter text also has a wealth of detailed examples showing how abstract theory illuminates real applications. The texts [4],[5] may act as supplements to this treatment to show numerical methods and a wider theoretical approach.

[1] Dynamics and Bifurcations. J. Hale and H. Kocak . Springer-Verlag 1991.
[2] Ordinary Differential Equations. H.K. Wilson. Addison Wesley 1971.
[3] Theory of Oscillators. A.A. Andronov, A.A. Vitt, and S.E. Khaikin.. Pergamon 1966.
[4] Differential Equations and Dynamical Systems. L. Perko. Springer-Verlag 1991
[5] Differential Equations . J.H. Hubbard and B.H. West. Springer-Verlag 1991

David Knapp. e-mail: AMT6DK@LUCS-03.NOVELL.LEEDS.AC.UK.

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Last Updated: 31 January 1997.