📂Dynamical Systems

Sprott Systems in Dynamics

Systems¹

Sprott systems refer to the following 19 three-dimensional dynamical systems.

A

$$ \begin{align*} \dot{x} =& y \\ \dot{y} =& - x + yz \\ \dot{z} =& 1 - y^{2} \end{align*} $$

B

$$ \begin{align*} \dot{x} =& y z \\ \dot{y} =& x - y \\ \dot{z} =& 1 - x y \end{align*} $$

C

$$ \begin{align*} \dot{x} =& y z \\ \dot{y} =& x - y \\ \dot{z} =& 1 - x^{2} \end{align*} $$

D

$$ \begin{align*} \dot{x} =& - y \\ \dot{y} =& x + z \\ \dot{z} =& x z + 3 y^{2} \end{align*} $$

E

$$ \begin{align*} \dot{x} =& y z \\ \dot{y} =& x^{2} - y \\ \dot{z} =& 1 - 4 x \end{align*} $$

F

$$ \begin{align*} \dot{x} =& y + z \\ \dot{y} =& - x + 0.5 y \\ \dot{z} =& x^{2} - z \end{align*} $$

G

$$ \begin{align*} \dot{x} =& 0.4 x + z \\ \dot{y} =& x z - y \\ \dot{z} =& - x + y \end{align*} $$

H

$$ \begin{align*} \dot{x} =& - y + z^{2} \\ \dot{y} =& x + 0.5 y \\ \dot{z} =& x - z \end{align*} $$

I

$$ \begin{align*} \dot{x} =& - 0.2 y \\ \dot{y} =& x + z \\ \dot{z} =& x + y^{2} - z \end{align*} $$

J

$$ \begin{align*} \dot{x} =& 2 z \\ \dot{y} =& - 2 y + z \\ \dot{z} =& - x + y + y^{2} \end{align*} $$

K

$$ \begin{align*} \dot{x} =& x y - z \\ \dot{y} =& x - y \\ \dot{z} =& x + 0.3 z \end{align*} $$

L

$$ \begin{align*} \dot{x} =& y + 3.9 z \\ \dot{y} =& 0.9 x^{2} - y \\ \dot{z} =& 1 - x \end{align*} $$

M

$$ \begin{align*} \dot{x} =& - z \\ \dot{y} =& - x^{2} - y \\ \dot{z} =& 1.7 + 1.7 x + y \end{align*} $$

N

$$ \begin{align*} \dot{x} =& - 2 y \\ \dot{y} =& x + z^{2} \\ \dot{z} =& 1 + y - 2 z \end{align*} $$

O

$$ \begin{align*} \dot{x} =& y \\ \dot{y} =& x - z \\ \dot{z} =& x + x z + 2.7 y \end{align*} $$

P

$$ \begin{align*} \dot{x} =& 2.7 y + z \\ \dot{y} =& - x + y^{2} \\ \dot{z} =& x + y \end{align*} $$

Q

$$ \begin{align*} \dot{x} =& - z \\ \dot{y} =& x - y \\ \dot{z} =& 3.1 x + y^{2} + 0.5 z \end{align*} $$

R

$$ \begin{align*} \dot{x} =& 0.9 - y \\ \dot{y} =& 0.4 + z \\ \dot{z} =& x y - z \end{align*} $$

S

$$ \begin{align*} \dot{x} =& - x - 4 y \\ \dot{y} =& x + z^{2} \\ \dot{z} =& 1 + x \end{align*} $$

Explanation

The Sprott systems are all systems that exhibit chaos, yet they have the distinctive feature of being algebraically remarkably simple. In extreme cases they possess only a single nonlinear term, and even that nonlinear term is at most a quadratic term. For example, Sprott-F exhibits chaos despite having nothing but $x^{2}$.

For reference, Sprott-A has the distinctive feature of not possessing any fixed points. Otherwise, no unique properties stand out for each individual system, but the 19 systems taken together as a set are also used in benchmarks and the like².