LYAPUNOV EXPONENTS

To be called chaotic a system should also show sensitive dependence on initial conditions in the sense that neighboring orbits separate exponentially fast, on average. This sensitive dependence could be quantified by defining the Lyapunov exponent for a chaotic differential equation.

Let us consider two neighboring trajectories. Assume that nearby to the initial point lies a point where the initial separation is extremely small. Let be the separation after iterations. If

then is called the Lyapunov exponent. A positive Lyapunov exponent is a signature of chaos.

A computationally useful formula for is

Note that depends on the initial state However, it is the same for all in the basin of attraction of a given attractor.

For stable fixed points and cycle, is negative; for chaotic attractors is positive.