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Bulletin of the London Mathematical Society Advance Access originally published online on May 29, 2008
Bulletin of the London Mathematical Society 2008 40(5):725-748; doi:10.1112/blms/bdn046
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© 2008 London Mathematical Society

Some third-order ordinary differential equations

Peter Swinnerton-Dyer

Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
Wilberforce Road
Cambridge
CB3 0WB
United Kingdom

Thomas Wagenknecht

Alan Turing Building
School of Mathematics
University of Manchester
Oxford Road
Manchester
M13 9PL
United Kingdom
thomas.wagenknecht@manchester.ac.uk

Received 29 August 2007.

The paper deals with periodic orbits in three systems of ordinary differential equations. Two of the systems, the Falkner–Skan equations and the Nosé equations, do not possess fixed points, and yet interesting dynamics can be found. Here, periodic orbits emerge in bifurcations from heteroclinic cycles, connecting fixed points at infinity. We present existence results for such periodic orbits and discuss their properties using careful asymptotic arguments. In the final part results about the Nosé equations are used to explain the dynamics in a dissipative perturbation, related to a system of dynamo equations.

2000 Mathematics Subject Classification 37G15, 34C25, 34C37.


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