Christophe LETELLIER
04/08/2014

Roy B. Leipnik & Tyre A. Newton (University of California at Santa Barbara) proposed the system

with three quadratic interactions arising from a modified Euler’s rigid body equations by the addition of linear feedback [1]. In its original form, there is a factor 2 before the last term of the first equation : we removed this term because it breaks the rotation symmetry around the *z*-axis. Compared to the Lorenz system,
there is one additional nonlinear term, namely *byz* in the first equation. This system has five singular points : the origin of the state space and the four symmetry related
points whose coordinates are

where

For parameter values (*a* = 0.73 and *b* = 5) and the original factor 2 in the first equation, four attractors co-exist in the phase space as shown in Fig. 1a. The two attractors observed mainly with positive *z* values are topologically equivalent to the Burke and Shaw attractor observed before the first attractor merging crisis [2] [3]. Two limit cycles also co-exist in the phase space for negative *z*-values. They are observed by choosing different
initial conditions. These two cycles remain roughly unchanged when the *c*
parameter is increased to 0.152. Contrary to this, the two disconnected
attractors for positive *z*-values merge into a single attractor (Fig.1b).

**Fig. 1. Disconnected attractors solution to the Leipnik and Newton system.**

The two attractors with negative *z*-values, different from the first two
attractors with positive *z*-values, are obtained because a second symmetry
is broken by the factor 2 in the term *2byz* in the first equation. The symmetry can be restored by removing the coefficient 2 from this term. Thus, for slightly different parameter values (*a* = 0.60, *b* = 5, and *c* = 0.1428), four symmetry-related attractors are obtained as shown in Fig. 2. For values of *c* greater than 0.1428, the attractors merge and two disconnected attractors remain in the phase space. They are symmetry-related. The modified Leipnik-Newton system has a *S*_{4}-symmetry (which here combines a rotation about the *z*-axis through π/2 radians followed by a reflection in the *z* = 0 plane) as detailed in [4].

**Fig. 2. Four disconnected chaotic attractors solution to the modified Leipnik and Newton system.**

[1] **R. B. Leipnik & T. A. Newton**, Double strange attractors in rigid body motion with linear feedback control, *Physics Letters A*, **86**, 63-87, 1981.

[2] **R. Shaw**,
Strange attractor, chaotic behavior and information flow, *Zeitschrift für Naturforschung A*, **36**, 80-112, 1981

[3] **C. Letellier, P. Dutertre, J. Reizner & G. Gouesbet**,
Evolution of multimodal map induced by an equivariant vector field, *Journal of Physics A*, **29**, 5359-5373, 1996.

[4] **C. Letellier & R. Gilmore**, Symmetry groups for 3D dynamical systems, *Journal of Physics A*, **40** 5597–5620, 2007.