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The invariant algebraic surfaces of the Lorenz system

Published online by Cambridge University Press:  17 June 2002

Isaac Newton Institute for Mathematical Sciences, 20 Clarkson Road, Cambridge CB3 0EH.


The object of this paper is to find all the irreducible algebraic surfaces which (for special values of the parameters b, r, s) are invariant under the Lorenz system

x˙ = X(x, y, z) = s(yx), y˙ = Y(x, y, z) = rxyxz, ż = Z(x, y, z) =−bz+xy. (1)

It is customary in considering the Lorenz system to require the parameters b, r, s to be all strictly positive; however for this particular problem we shall follow previous practice in only imposing the condition s ≠ 0. (If s = 0 the equations are trivially integrable and x is constant on any trajectory; thus x should be regarded as a parameter and the question discussed in this paper ceases to be a natural one.)

Research Article
2002 Cambridge Philosophical Society

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