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.

A listing of books reviewed in `UK Nonlinear News` is available
at:
http://www.amsta.leeds.ac.uk/Applied/news.dir/uknonl-books.html

<< Move to

Last Updated: 31 January 1997.