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Products of flows

Published online by Cambridge University Press:  24 October 2008

D. K. Arrowsmith
Affiliation:
Department of Mathematics, University of Leicester

Abstract

The aim of the paper is to exhibit the structure of products of flows that are suspensions of diffeomorphisms and show their relationship with suspensions of product diffeomorphisms.

The problem is important for applications of the following type. Suppose we have two dynamical systems each with a stable attractor. If the systems are brought into contact, as, for example, in the way a forced oscillation is imposed upon a natural oscillation in synchronization theory, then this is represented by taking the product flow to find the resulting behaviour of the joint system. However, the product of two stable attractors is no longer stable in general, and therefore it is important to analyse why, in order to understand what new stable attractors can arise from perturbations.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1973

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References

(1)Auslander, L. and Mackenzie, R. R.Introduction to differentiable manifolds (McGraw Hill, 1963).Google Scholar
(2)Arrowsmith, D. K. Ph.D. Thesis, Leicester University (unpublished).Google Scholar
(3)Husemoller, D.Fibre bundles (McGraw Hill, 1966).CrossRefGoogle Scholar
(4)Peixoto, M. M. Qualitative theory of differential equations and structural stability. Symposium on Differential Equations and Dynamical Systems (Puerto Rico) (Academic Press, New York, 1967).Google Scholar
(5)Smale, S.Stable manifolds for differential equations and diffeomorphisms. Ann. Scuola Norm. Sup. Pisa (3), 17 (1963), 97116.Google Scholar
(6)Smale, S.Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747817.CrossRefGoogle Scholar
(7)Tondeur, P.Introduction to Lie groups and transformation groups. Lect. Notes Math. (Springer-Verlag), 7.Google Scholar
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