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Non-linear Dynamics, Symmetry and Perturbation Theory in

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Mathematics of Complexity and Dynamical Systems

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Article Outline

Glossary

Definition of the Subject

Introduction

Symmetry of Dynamical Systems

Perturbation Theory: Normal Forms

Perturbative Determination of Symmetries

Symmetry Characterization of Normal Forms

Symmetries and Transformation to Normal Form

Generalizations

Symmetry for Systems in Normal Form

Linearization of a Dynamical System

Further Normalization and Symmetry

Symmetry Reduction of Symmetric Normal Forms

Conclusions

Future Developments

Additional Notes

Bibliography

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Abbreviations

Perturbation theory :

A theory aiming at studying solutions of a differential equation (or system thereof), possibly depending on external parameters, near a known solution and/or for values of external parameters near to those for which solutions are known.

Dynamical system :

A system of first order differential equations \({\text{d} x^i / \text{d} t = f^i (x,t)}\), where \({x \in M}\), \({t \in {\mathbf R}}\). The space M is the phase space for the dynamical system, and \({\widetilde{M} = M \times {\mathbf R}}\) is the extended phase space. When f is smooth we say the dynamical system is smooth, and for f independent of t, we speak of an autonomous dynamical system.

Symmetry :

An invertible transformation of \({\widetilde{M}}\) mapping solutions into solutions. If the dynamical system is smooth, smoothness will also be required on symmetry transformations; if it is autonomous, it will be natural to consider transformations of M rather than of \({\widetilde{M}}\).

Symmetry reduction :

A method to reduce the equations under study to simpler ones (e. g. with less dependent variables, or of lower degree) by exploiting their symmetry properties.

Normal form :

A convenient form to which the system of differential equations under study can be brought by means of a sequence of change of coordinates. The latter are in general well defined only in a subset of M, possibly near a known solution for the differential equations.

Further normalization :

A procedure to further simplify the normal form for a dynamical system, in general making use of certain degeneracies in the equations to be solved in the course of the normalization procedure.

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    Giuseppe Gaeta

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Gaeta, G. (2012). Non-linear Dynamics, Symmetry and Perturbation Theory in. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_63

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