The image above is the solution of the algebraically simplest example of a dissipative chaotic flow: x''' + Ax'' - x'² + x = 0, where x' = dx/dt 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