#
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)
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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, d^{3}*x*/d*t*^{3}
+
*A*d^{2}*x*/d*t*^{2} - (d*x*/d*t*)^{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.

Fig. 2. Return map showing each value of *x*_{max} versus
the previous value of *x*_{max}. The insert shows fractal
structure at a magnification of 10^{4}.

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

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.

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