Nonlinear Physics with Mathematica for Scientists and Engineers Richard H. Enns and George C. McGuire, Birkhauser Verlag AG, Basel etc. (2001) , price ca. € 70.-
	Nonlinear Physics with Maple for Scientists and Engineers, 2nd ed. Richard H. Enns and George C. McGuire, Birkhauser Verlag AG (2001), Basel etc., price ca. € 70.-
	Dynamical Systems with Applications using Maple Stephen Lynch, Birkhauser Verlag AG, Basel etc. (2001), price ca. € 65.-
Reviewer: Alois Steindl, Institute for Mechanics, Vienna University of Technology.

Level: medium.
During the last decades symbolic algebra packages have become a powerful mathematical tool. Programs like Mathematica or Maple provide sophisticated symbolic, numeric and graphical capabilities, which are very useful for the investigation of mathematical problems. It is especially convenient to formulate a set of equations within one program and let this program perform simulations, visualize the results, apply transformations or series expansions, just to name some important tasks for applications.

The books ``Nonlinear Physics with Mathematica for Scientists and Engineers'' and ``Nonlinear Physics with Maple for Scientists and Engineers'' by Richard H. Enns and George C. McGuire and ``Dynamical Systems with Applications using Maple'' by Stephen Lynch demonstrate how to treat simple and difficult systems arising in different branches of Applied Mathematics using the dominant computer algebra systems Mathematica and Maple.

Dynamical Systems with Applications using Maple
The paperback by Stephen Lynch concentrates mainly on continuous and discrete dynamical systems. After a short introduction into Maple simple ordinary differential equations are considered. The student learns to study the behaviour of (essentially) planar differential equations by drawing the vector fields and phase portraits. Also the linearization of the vector field at critical points and the Poincaré-Bendixson theorems are explained very well.
Further chapters deal with Hamiltonian systems, Lyapunov functions and bifurcation theory. It is shown how the dynamics of the system changes at a bifurcation point and that hysteresis effects are possible for certain parameter combinations.
In this chapter I would expect to find a short explanation of Center Manifold theory and Normal Form simplification, because these tools are essential for higher dimensional systems and can benefit quite a lot from algebra systems.
The chapters on Hilbert's sixteenth problem and limit cycles of Liénard systems demonstrate nicely how the graphical, symbolic and numeric capabilities of Maple can be used to deal with mathematical problems.
The following chapters cover discrete dynamical systems, complex iterative maps and fractals. As main application optical resonators are introduced in some detail and some rather complicated bifurcation diagrams for these devices are shown.
Finally the concept of ``controlling chaos'' is introduced and applied to several chaotic systems.

A student or scientist, who works through some chapters of the book, learns a good deal about the presented mathematical concepts and possibilities of the symbolic algebra package to assist the researcher in understanding his mathematical model.

Nonlinear Physics with Mathematica for Scientists and Engineers and Nonlinear Physics with Maple for Scientists and Engineers
Also the authors of these voluminous hardcover books emphasize that their main subject are the basic concepts and applied mathematical methods of nonlinear science. Additionally the second part of the books describes a series of experimental activities, which the reader/student should carry out to watch interesting phenomena, that can be understood with the theoretical background from the first part.
The books also contain CD-ROM media with the Maple worksheets or Mathematica notebooks to run the examples from the books and play around with the system.
After a short introduction (with many carefully selected examples) to the packages the books start with rather simple dynamical systems, e.g. from mechanics, population dynamics, and chemical reactions.
Then specific topics for nonlinear systems, like pattern formation, solitons and chaos are introduced and presented very clearly.
After this exposition topological methods like phase plane analysis and linearization are applied to several dynamical systems. Then the authors present several exact and approximative analytical methods to deal with nonlinearities. For these calculations the symbolic abilities of the programs are indispensable.
Chapter 6 discusses several numerical methods to integrate `normal' and stiff ODEs.
Chapter 7 is devoted to (self-excited) limit cycles; while the Poincaré-Bendixson theory is explained with sufficient detail, the Hopf bifurcation theorem, which also applies to higher order systems, isn't mentioned in this part. Next various aspects of forced oscillations and nonlinear maps are treated.
The following chapters are devoted to the analytic and numeric treatment of nonlinear PDEs, especially methods to calculate solitary waves.

Also these two books cover a large range of topics from simple differential equations to nonlinear PDEs. In addition to the presented theory many problems are stated as exercises to help the student to consolidate his new knowledge. Furthermore the provided programs for Mathematica and Maple demonstrate how to delegate the nasty calculations to computer programs and how to obtain reliable numerical results and impressive graphic output.

All three books demonstrate that it is quite straightforward to calculate stable solutions, which can also be observed in experiments. Unfortunately it isn't that easy to calculate unstable solution branches, which might also play an essential role for nonlinear systems, because they sometimes delimit domains of attraction or lead to `crises', that govern the periodic windows within chaotic regimes. I would have appreciated some discussion about the role of these unstable solutions and maybe some pointers to numerical or symbolic packages for their determination.
Nevertheless I recommend these three books as excellent textbooks for dynamical systems and good explanations how to apply (the most common) computer algebra packages for the analytical and numerical treatment and graphical representation of the mathematical models.