A hyperchaotic system is a

dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...

with a bounded

attractor In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain c ...

set, on which there are at least two positive Lyapunov exponents. 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. One is a discrete-time system ("folded-towel map"):

left(1-1.9 x_t\right), \\ & z_=3.78 z_t\left(1-z_t\right)+0.2 y_t . \end

Another is a continuous-time system:

\begin
\dot=-y-z, & \dot=x+0.25 y+w, \\
\dot=3+x z, & \dot=-0.5 z+0.05 w .
\end

More examples are found in.

Experimental examples

Only a few experimental hyperchaotic behaviors have been identified. Examples include in an electronic circuit, in a NMR laser, in a semiconductor system, and in a chemical system.

References

{{Reflist Chaotic maps Nonlinear systems Articles containing video clips