## Simplest Dissipative Chaotic Flow

The image above is the solution of the algebraically simplest
example of a dissipative chaotic flow: *x*''' + *Ax*'' - *x*'^{2}
+ *x* = 0, where *x*' = d*x*/d*t* and *A*
is a
parameter taken as 2.017 in the case above. It turned up in a search
of third-order, one-dimensional systems of ordinary differential
equations
with three terms and one quadratic nonlinearity. It has been rigorously
proved that there can be no simpler chaotic system with a quadratic
nonlinearity. Such an equation is called
a "jerk" equation because it involves
the
third derivative of *x* (or the time derivative of the
acceleration).
The strange attractor of this dissipative chaotic flow is approximately
a Mobius strip with a dimension of 2.0265. The Lyapunov
exponents (base-*e*) are 0.055, 0, and -2.072. In the case
above,
the *x*' axis is toward the top of the screen and the object
rotates
in the *x-x*'' plane about the *x*' axis. You can also see rotations
about the *x* axis and rotations
about the *x*'' axis as well as a rotational
animation of the basin of attraction, which resembles a hairy
tadpole.
To view these animations, you need a browser that supports animated GIF
files. The image above is also available as an AVI
movie file.

Refs: J. C. Sprott, Phys. Lett. A
228, 271-274 (1997)

J.
C. Sprott, Am. J. Phys. 65, 537-543 (1997)

Other Simple Chaotic Flows with Quadratic
Nonlinearities

Simple Chaotic Flow with ABS
Nonlinearity

Other animated chaotic systems

Back to Sprott's Fractal Gallery