1981 Rigid body-motion with linear feedback

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 S₄-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.