# Elementary Chaotic Flow

Stefan J.
Linz and J. C. Sprott

*Theoretische Physik I, Instit fur Physik, Universitat Augsburg,
D-86135 Augsburg, Germany*

*Department of Physics, University
of Wisconsin, Madison, Wisconsin 53706, USA*

(Received 21 April 1999; accepted 20 June 1999)
### ABSTRACT

Using an extensive numerical search for the simplest chaotic
non-polynomial
autonomous three-dimensional dynamical systems, we identify an
elementary
third-order differential equation that contains only one control
parameter
and only one nonlinearity in the form of the modulus of the dynamical
variable.
We discuss general properties of this equation and the possibility of
chaotic
behavior in functionally closely related equations. Finally, we
present
its analytical solution in an algorithmic way.
Ref: S. J. Linz and J. C. Sprott, Phys.
Lett.
A **259**, 240-245 (1999)

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Fig. 1. Successive maxima of the long-term evolution of *x*(*t*)
generated
by Eq. (2) as function of the control parameter *A* in the
range 0.5 __<__ *A* __<__ 0.8. Initial
conditions
are given by d^{2}*x*/d*t*^{2} = d*x*/d*t*
= *x* = 0.

Fig. 2. Largest Lyapunov exponent (LE) as function of the control
parameter
*A*
in the same range as in Fig. 1.

Fig. 3. Stereoscopic view of the strange attractor of Eq. (3) for
the
control parameter value *A* = 0.6 in order to obtain a
three-dimensional
impression. The corresponding Lyapunov exponents are given by (*L*_{1},
*L*_{2},
*L*_{3})
= (0.035, 0, -0.635)

A GIF animated view of this
attractor
is available.