telech
ACCUEIL  ·>  Some dynamical systems ·>  Minimal systems

Minimal systems

No chaotic system was found with either only three or four terms on their righthand side. Although about 108 systems were randomly selected for examination [1], no chaotic system was found with either only three or four terms on their righthand side. Although about 108 systems were randomly selected for examination, Sprott’s method cannot guarantee that all the chaotic systems with six or fewer terms including a single nonlinearity or with five or fewer terms including two nonlinearities have been discovered. However, these partial results raise the natural question : what is the behaviour of three-dimensional quadratic flows with less than five terms on their right-hand side ?

The answer to this question is not fully known at this time. Zhang Fu and Jack Heidel [2] [3] have proved that quadratic dissipative systems with a nonzero but constant divergence and with fewer than five terms cannot exhibit chaos. Although dissipative systems with constant divergence appear quite often in the literature, this is a very restrictive assumption. For example, the Rössler system has not a constant divergence.

Regarding the minimal structure of chaotic flows with nonconstant divergence, Malasoma [4] proved the following Theorem

- Jerk equations \stackrel{...}{x}= J(x, \dot{x},\ddot{x}), where the jerk function J is a quadratic polynomial with only one or two terms and for which \frac{\partial J(x, y, z)}{\partial z} is not a constant function of time, do not exhibit chaos.

 

ATOMOSYD © 2007-2017 |  Suivre la vie du site  |  SPIP  |  scoty  |  MàJ . 19/07/2017