1981 Rigid body-motion with linear feedback

Christophe LETELLIER

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

      \dot{x}=-ax+y+b \, yz \\[0.3cm]
      \dot{y}=-x-ay+b\, xz \\[0.3cm]

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

  S_{\pm \pm} = 
      x_{\pm \pm} = -\frac{\lambda c}{b(2\lambda^2-ac)} \\[0.1cm]   
      y_{\pm \pm} = + \frac{\lambda}{b} \\[0.1cm]
      z_{\pm \pm} = -\frac{\lambda^2}{b(2\lambda^2-ac)} 


  \lambda =\pm \sqrt{ {c \over 8a}} \sqrt{3+4a^2 \pm \sqrt{9+8a^2 }} \, .

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).

JPG - 18.3 ko
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].

JPG - 18.8 ko
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.

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