Simple Chaotic Flows with a Line Equilibrium
Department, Amikabir University of Technology, Tehran
J. C. Sprott
Department of Physics, University
Wisconsin, Madison, WI 53706, USA
Received 12 June 2013, Accepted 20 August 2013
Using a systematic computer search, nine simple chaotic flows with
quadratic nonlinearities were found that have the unusual feature of
having a line equilibrium. Such systems belong to a newly introduced
category of chaotic systems with hidden attractors that are
important and potentially problematic in engineering applications.
Ref: S. Jafari and J. C. Sprott, Chaos
Solitons & Fractals 57,
The complete paper is available in
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Fig. 1. State space plots of the cases in Table
1 projected on the xy-plane.
Fig. 2. The largest Lyapunov exponent and bifurcation diagram of
showing a period-doubling route to chaos.
Fig. 3. Cross section of the basins of attraction of the two
attractors in the xz-plane at y = 0 for case LE1
Initial conditions in the white region lead to unbounded orbits,
those in the red region lead to the strange attractor, and those
in the light blue region lead to the line equilibrium.
Fig. 4. Regions of different dynamic behavior in parameter space
for case LE1
. Light blue represents a static
equilibrium, and the black dots correspond to regions of chaos.
Each pixel uses a different random initial condition thereby
indicating the coexistence of static and chaotic attractors.