# Perturbation Methods, Bifurcation Theory and Computer Algebra With 10 Illustrations pdf

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## Perturbation Methods, Bifurcation Theory and Computer Algebra With 10 Illustrations by Richard H. Rand and Dieter Armbruster pdf

Perturbation Methods, Bifurcation Theory and Computer Algebra With 10 Illustrations by Richard H. Rand and Dieter Armbruster pdf free download. Our purpose in writing this book is to provide computer algebra programs which implement a number of popular perturbation methods. For each perturbation method, we present an introduction to the method, a couple of example problems, sample runs of the computer algebra programs and complete program listings. In addition, we include examples of various elementary bifurcations, such as folds, pitchforks and Hopf bifurcations.

### Perturbation Methods, Bifurcation Theory and Computer Algebra With 10 Illustrations by Richard H. Rand and Dieter Armbruster pdf

These arise in the example problems. Specifically, we treat Hopf bifurcations in autonomous nonlinear systems via Lindstedt’s method, the construction of center manifolds for simple, degenerate and nilpotent bifurcations in ordinary differential equations, the determination of normal forms for Hopf bifurcations and Takens-Bogdanov bifurcations, and averaging for autonomous and nonautonomous systems. Further, we use Lie transforms to determine normal forms in Hamiltonian systems. Bifurcation in partial differential equations, such as reaction diffusion equations or the Bernard convection problem, are treated via Liapunov-Schmidt reduction.

#### Perturbation Methods, Bifurcation Theory and Computer Algebra With 10 Illustrations by Richard H. Rand and Dieter Armbruster pdf

Moreover, we offer comparisons of the various methods. We compare averaging with normal forms, Liapunov-Schmidt reduction with center manifold reduction, Lindstedt’s method with normal form calculations, and so on. To help in making the comparisons we frequently treat the same problem by two or more methods. E.g., we derive the Hopf bifurcation formula both by Lindstedt’s method as well as via normal forms.

##### Perturbation Methods, Bifurcation Theory and Computer Algebra With 10 Illustrations by Richard H. Rand and Dieter Armbruster pdf

Our motivation for applying computer algebra to perturbation problems comes from the nature of the computations involved ~n these kinds of problems. viii The massive algebra usually required to obtain detailed results is more quickly and more accurately ~ccomplished by computer than by hand. Since our emphasis is on computation, we have dropped mathematical rigor in favor of intelligibility of the computational methods. However, we have provided the reader with references to standard mathematical textbooks or research papers.

## Perturbation Methods, Bifurcation Theory and Computer Algebra With 10 Illustrations by Richard H. Rand and Dieter Armbruster pdf

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