The complex dynamics of oscillators and maps with catastrophes

Catastrophe theory is a mathematical discipline aimed to analysis and classification of singularities of differentiable mappings. Following one of the popular books, you can say that this is a far-reaching generalization of a study of functions for maximum and minimum.

When people talk about the catastrophe theory, usually it is assumed that the space of images is one-dimensional, and the space of pre-images may be of any dimension. (Sometimes, the term is used in the broader sense, having in mind an arbitrary dimension for the space of images too.) It turns out that the number of types of singularities, which can occur generically with the number of control parameters up to five, is finite. That circumstance provides the basis for their classification, analysis, and applications.

A catastrophe of a certain type distinguished in the frame of this approach, is characterized by a special universal arrangement of the parameter space in a neighborhood of the catastrophe. The number of relevant parameters needed to reveal a complete picture of the parameter space is also determined by the type of the singularity. Each special catastrophe is equipped with the term ( fold, cusp, swallowtail, elliptic umbilic, hyperbolic umbilic, etc.), the formula, an algebraic relation in a form of a function, containing the relevant variables and parameters, and with a graphic image. For the high-order catastrophes these images are not trivial and aesthetically attractive (as an example, a picture of the parameter space for the swallowtail catastrophe is shown).

There is a direct correspondence between the catastrophe theory and a chapter of the nonlinear dynamics dealing with the gradient system, these is, in essence, a theory of bifurcations of equilibrium states in such systems. (A differential equation for a gradient system appears from equalizing of time derivatives of the generalized coordinates to the partial derivatives of the functions of the catastrophes in respect to these variables.)

In our studies, the concepts and results of the catastrophe theory are applied to other classes of systems, as an organizing principle for choosing concrete models to be considered. One of the directions consists in construction of nonlinear oscillators with potential functions associated with various catastrophes. In presence of damping and periodic external driving, they may demonstrate the complex dynamics, and the nature of the regimes varies depending on the parameters present in the catastrophe function [1,2]. If you have feedback that provides compensation for loss of energy from an external source of non-oscillatory nature, the object becomes a self-oscillator. For example, for the construction associated with the fold catastrophe you get the van der Pol oscillator , and in the situation of the cusp catastrophe it is the Fitz Hugh - Nagumo oscillator; they appear as the simplest representatives of some broad classification scheme [3, 4]. Another direction is design of discrete-time models (iterated maps) based on the functions of the catastrophe theory, which could demonstrate different dynamic regimes and bifurcations depending on the parameters [5].

[1] A.Yu.Kuznetsova, A.P.Kuznetsov, C.Knudsen, E.Mosekilde. Catastrophe theoretic classification of nonlinear oscillators. International Journal of Bifurcation and Chaos, 14, 2004, No. 4, 1241-1266
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[2] A.P.Kuznetsov and A.Yu.Potapova. Featires of complex dynamics of nonlinear oscillators with catastrophes of R.Thom. Applied Nonlinear Dynamics (Saratov), vol.8, 2000, No 6, pp.94-120 (in Russian).
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[3] A.P.Kuznetsov, Ju.V.Sedova, I.R.Sataev. Bifurcations and syncronization in van der Pol oscillators with potential functions determined by Thom's catostrophes. Basic Problems of Nonlinear Physics. Conference of Young Scientists. Abstracts. March 1 - 7, Nizhny Novgorod, 2008, pp. 139-140 (in Russian).

[4] D.V.Vizgalin. Self-oscillatory systems associated with the cusp catastrophe. In book: Nonlinear Days in Saratov for Youth - 2006. Saratov, 2007, pp.97-100 (in Russian).

[5] Yu.S.Ivanov, A.Yu.Kuznetsova. Mappings of Catastrophes. In book: Nonlinear Days in Saratov for Youth - 2003. Saratov, 2003, pp.46-50 (in Russian).
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Saratov group
of theoretical nonlinear dynamics