Hyperchaos

A hyperchaotic system is a dynamical system with a bounded attractor set, on which there are at least two positive Lyapunov exponents.^[1]

Folded-towel map attractor.

Since on an attractor, the sum of Lyapunov exponents is non-positive, there must be at least one negative Lyapunov exponent. If the system has continuous time, then along the trajectory, the Lyapunov exponent is zero, and so the minimal number of dimensions in which continuous-time hyperchaos can occur is 4.

Similarly, a discrete-time hyperchaos requires at least 3 dimensions.

Mathematical examples

The first two hyperchaotic systems were proposed in 1979.^[2] One is a discrete-time system ("folded-towel map"):

Folded-towel map attractor, animated.

$${\displaystyle {\begin{aligned}&x_{t+1}=3.8x_{t}\left(1-x_{t}\right)-0.05\left(y_{t}+0.35\right)\left(1-2z_{t}\right),\\&y_{t+1}=0.1\left[\left(y_{t}+0.35\right)\left(1-2z_{t}\right)-1\right]\left(1-1.9x_{t}\right),\\&z_{t+1}=3.78z_{t}\left(1-z_{t}\right)+0.2y_{t}.\end{aligned}}}$$Another is a continuous-time system:$${\displaystyle {\begin{array}{ll}{\dot {x}}=-y-z,&{\dot {y}}=x+0.25y+w,\\{\dot {z}}=3+xz,&{\dot {w}}=-0.5z+0.05w.\end{array}}}$$More examples are found in.^[3]

Experimental examples

Only a few experimental hyperchaotic behaviors have been identified.

Examples include in an electronic circuit,^[4] in a NMR laser,^[5] in a semiconductor system,^[6] and in a chemical system.^[7]