On the Nature of the Torus in the Complex Lorenz Equations

Creators: Fowler, A. C.; McGuinness, M. J.

Abstract

The complex Lorenz equations are a nonlinear fifth-order set of physically derived differential equations which exhibit an exact analytic limit cycle which subsequently bifurcates to a torus. In this paper we build upon previously derived results to examine a connection between this torus at high and low r1 bifurcation parameter) and between zero and nonzero r2(complexity parameter); in so doing, we are able to gain insight on the effect of the rotational invariance of the system, and on how extra weak dispersion (r2 ≠ 0) affects the chaotic behavior of the real Lorenz system (which describes a weakly dissipative, dispersive instability).

Additional Information

© 1984 Society for Industrial and Applied Mathematics. Received by the editors February 11, 1983. We would like to acknowledge discussions with John Gibbon, and particularly to thank Colin Sparrow for providing a pre-publication view of his book [21] on the Lorenz equations. As will be seen, the development of §3 is largely based on Chapter VII of his book, and the style of piecing together a global analytic picture from bits of numerical and analytical evidence is very much his.

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