Thus the sum of all Lyapunov exponents must be zero.

Chaos is defined as the existence of at least one positive Lyapunov exponent.

See the extensive discussion of the Lyapunov exponent, its inverse.

At least one Lyapunov exponent of a deterministically chaotic system is positive.

The Lyapunov exponent is the error growth-rate of a given sequence.

The Lyapunov exponent characterises the extent of the sensitivity to initial conditions.

For the calculation of Lyapunov exponents from limited experimental data, various methods have been proposed.

The divergence can have the trend of the positive maximal Lyapunov exponent, but not more.

The link between exponential contour growth and positive Lyapunov exponents is easy to see.

The Lyapunov exponent can be computed by different methods:

Contour growth rates will approximate the average of the large Lyapunov exponents:

If the system is dissipative, the sum of Lyapunov exponents is negative.

These eigenvalues are also called local Lyapunov exponents.

It is obvious that therefore the divergence cannot reflect the maximal Lyapunov exponent.

The theorem states conditions for the existence of the defining limits and describes the Lyapunov exponents.

A measure of the divergence of time series with nearly identical initial conditions is known as the Lyapunov exponent.

The maximal Lyapunov exponent can be defined as follows:

These are the Lyapunov exponents.

Furthermore, in a certain neighborhood of this zero solution almost all solutions of original system have positive Lyapunov exponents.

The definition does not require that all points from a neighborhood separate from the base point x, but it requires one positive Lyapunov exponent.

A measure of this sensitivity to initial conditions, the maximal Lyapunov exponent, is discussed in Chapter 13.

The real parts of the Floquet exponents are called Lyapunov exponents.

The values of the Lyapunov exponents are invariant with respect to a wide range of coordinate transformations.

Much greater fractal detail is revealed by plotting the Lyapunov exponent, as shown by the example below.

From the preceding discussion, it should be apparent that the decay rate, the fractal dimension and the Lyapunov exponents are all related.