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Minimal systems

# A first class of minimal chaotic flows

Jean-Marc MALASOMA
31/10/2012

Sprott [1] [2] found two jerk functions, namely

$\stackrel{...}{x} + \alpha \ddot{x} + x - \dot{x}^2 = 0$

and

$\stackrel{...}{x} + \alpha \ddot{x} + x - x\dot{x} = 0$

constituted by three monomials including a single quadratic nonlinearity. These two set of equations are the simplest quadratic jerk equations that exhibit chaotic dynamics. Their representations as sets of ordinary differential equations are constituted by five monomials on their right-hand sides :

$\left\{ \begin{array}{l} \dot{x} = y \\ \dot{y} = z\\ \dot{z} = - \alpha z - x + y^2 \end{array} \right.$

and

$\left\{ \begin{array}{l} \dot{x} = y \\ \dot{y} = z\\ \dot{z} = - \alpha z - x + xy \end{array} \right.$

These two systems are algebraically simpler than any previously reported three-dimensional quadratic flows and it is not possible to find a simpler one with constant divergence. These two examples are equivalent by a linear change of variables. They form a first class of minimal systems. Some of them are listed below [3].

A first example

The system

$\left\{ \begin{array}{l} \dot{x} = y \\ \dot{y} = - \alpha y + z \\ \dot{z} = - x + xy \end{array} \right.$

produces a chaotic attractor in the tiny range $2.0168... < \alpha < 2.0577...$. A chaotic attractor solution to this first system and for , that is, just before the boundary crisis that destroys the attractor, ejecting the trajectory to infinity.

Fig. 1 : Chaotic attractor solution to the first example belonging to class I of minimal systems.

[1] J. C. Sprott, Simplest dissipative chaotic flow, Physics Letters A, 228 (4-5), 271-274, 1997.

[2] J. C. Sprott, Some simple chaotic jerk functions, American Journal of Physics, 65 (6), 537-543, 1997.

[3] J.-M. Malasoma, A new class of minimal chaotic flows, Physics Letters A, 305 (1-2), 52-58, 2002.

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