Prog. Theor. Phys. Supplement No.69 (1980) pp. 185-199

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Towards the Interpretation of the Global Bifurcation Structure of the Lorenz System

— A Simple One-Dimensional Model —

Kazuhisa Tomita and Ichiro Tsuda

Department of Physics, Kyoto University, Kyoto 606

(Received September 16, 1980)

Abstract:

A simple one-dimensional model is proposed which is capable of reproducing the global bifurcation structure of the Lorenz system as a function of the control parameter r, other parameters being kept constant.
The symmetry of the system leads to the equivalence between the first and third quadrants in x-y plane, and the map φ between these quadrants is much simpler to analyse than the local Lorenz map Φ = φ². The map φ is then considered as derived from a universal prototype of transfer function by restricting to an interval corresponding to the maximum possible span of the asymptotic invariant manifold. It is in addition assumed that the relative position of the universal transfer function and the restricting interval is the only quantity which is sensitive to the change of the control parameter r.
This assumption together with a parabolic prototype leads to a bifurcation structure fairly similar to that obtained for the Lorenz system by simulation.

URL : http://ptp.ipap.jp/link?PTPS/69/185/
DOI : 10.1143/PTPS.69.185

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Citing Article(s) :

Progress of Theoretical Physics Vol. 68 No. 1 (1982) pp. 64-84 :
Global Aspects of the Dissipative Dynamical Systems. I

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Progress of Theoretical Physics Vol. 70 No. 5 (1983) pp. 1249-1263 :
Symbolic Dynamics Approach to Intermittent Chaos

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Progress of Theoretical Physics Vol. 76 No. 2 (1986) pp. 335-355 :
Phase Description Method to Time Averages in the Lorenz System

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