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65P30 Numerical analysis -- Numerical problems in dynamical systems -- Bifurcation problems

13P10 Commutative algebra -- Computational aspects and applications -- Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)

37M20 Dynamical systems and ergodic theory -- Approximation methods and numerical treatment of dynamical systems -- Computational methods for bifurcation problems En-ligne: Springerlink | Zentralblatt | MathSciNet

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This book elucidates the efficiency of computer algebra for the study of dynamical systems.

The first chapter recalls some particular aspects of the theory of Gröbner bases and introduces and compares variants of the basic algorithms (using Hilbert series, dynamic Buchberger algorithm, etc.). The second chapter is devoted to the art of computing invariants or equivariants via computer algebra methods. Here are discussed (from the points of view of both theory and performance) various examples of explicit computations, as well as general purpose algorithms. The third chapter applies to symmetric bifurcation theory. Here the author investigates problems of a local nature. She shows how useful computer algebra can be for local bifurcation analysis. Some source codes (in Maple) are available.

The last chapter is devoted to orbit space reduction (investigation of a system of differential equations modulo a group action). Several examples of symmetric systems are treated. Noether normalization is introduced, and its interest is explained via the example of the Taylor-Couette experiment.

This book is very valuable. It gives a new, broad and practical overview of computer algebra methods that can be useful in dynamical systems. Many references are given. (MathSciNet)

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