Simplest Dissipative Chaotic Flow

J. C. Sprott
Department of Physics, University of Wisconsin, Madison, Wisconsin 53706
(Received 18 December 1996; accepted for publication 14 January 1997)

ABSTRACT

Numerical examination of third-order, autonomous ODEs with one dependent variable and quadratic nonlinearities has uncovered what appears to be the algebraically simplest example of a dissipative chaotic flow, d3x/dt3 + Ad2x/dt2 - (dx/dt)2 + x = 0. This system exhibits a period-doubling route to chaos for 2.017 < A < 2.082 and is approximately described by a one-dimensional quadratic map.

Ref: J. C. Sprott, Phys. Lett. A 228, 271-274 (1997)

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Fig. 1. Bifurcation diagram. Note that the scales have been reversed to emphasize the similarity to the familiar Feigenbaum diagram.
[Figure 1]

Fig. 2. Return map showing each value of xmax versus the previous value of xmax. The insert shows fractal structure at a magnification of 104.
[Figure 2]

Fig. 3. Chaotic time variation of x, v, a, and j for A = 2.107.
[Figure 3]

Fig. 4. Stereoscopic view of the attractor, which is approximately a Mobius strip. Also shown is the fixed point at the origin with its stable and unstable manifold.
[Figure 4]

GIF animated views of this attractor and its basin of attraction are available.