A dynamical system with a strange attractor and invariant tori

J.C.Sprott

Department of Physics, University of Wisconsin-Madison, Madison, WI 53706, USA
Received 15 January 2014
Received in revised form 13 March 2014
Accepted 17 March 2014
Available online 21 March 2014
Communicated by C.R. Doering

This paper describes a simple three-dimensional time-reversible system of ODEs with quadratic nonlinearities and the unusual property that it exhibits conservative behavior for some initial conditions and dissipative behavior for others. The conservative regime has quasi-periodic orbits whose amplitude depend on the initial conditions, while the dissipative regime is chaotic. Thus a strange attractor coexists with an infinite set of nested invariant tori in the state space.

Ref: J. C. Sprott, Phys. Lett. A 378, 1361-1363 (2014)

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Figure 1
Fig. 1. A strange attractor (red) interlinked with a coexisting invariant torus (green) for System (1).

Figure 2
Fig. 2. A different view of the strange attractor (red) intertwined with a symmetric repellor (black) and both interlinked with a coexisting invariant torus (green) for System (1).

Figure 3
Fig. 3. Cross section in the z = 0 plane showing nested tori surrounded by a multifractal strange attractor for System (1). Initial conditions in the conservative region are shown in blue, and initial conditions in the basin of attraction of the strange attractor are shown in yellow.