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Non-linear Dynamics, Symmetry and Perturbation Theory in

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

Given a differential equation or system of differential equations ? with independent variables \( { \xi^a \in \Xi \subseteq {\mathbf R}^q } \) and dependentvariables \( { x^i \in M \subseteq {\mathbf R}^p }\), a symmetry of ? is an invertible transformation of the extended phase space \( { \widetilde{M} = \Xi \times M } \) into itself which mapssolutions of ? into (generally, different) solutions of ?.

The presence of symmetries is a non-generic feature; correspondingly, equations with symmetry have some special features. These can beused to obtain information about the equation and its solutions, and sometimes allow one to obtain explicit solutions.

The same applies when we consider a perturbative approach to the equations: taking into account the presence of symmetries guarantees theperturbative expansion has certain specific features (e.?g. some terms are not allowed) and hence allows one to deal with simplified expansions andequations; thus this approach...

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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. (2009). Non-linear Dynamics, Symmetry and Perturbation Theory in. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_361

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