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 d2x/dt2 = dx/dt = x = 0.
[Figure 1]

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

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 (L1, L2, L3) = (0.035, 0, -0.635)
[Figure 3]

A GIF animated view of this attractor is available.