Lyapunov exponent

In mathematics the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with initial separation vector formula_1 diverge (provided that the divergence can be treated within the linearized approximation) at a rate given by
where formula_3 is the Lyapunov exponent.
The rate of separation can be different for different orientations of initial separation vector. Thus, there is a spectrum of Lyapunov exponents—equal in number to the dimensionality of the phase space. It is common to refer to the largest one as the maximal Lyapunov exponent (MLE), because it determines a notion of predictability for a dynamical system. A positive MLE is usually taken as an indication that the system is chaotic (provided some other conditions are met, e.g., phase space compactness). Note that an arbitrary initial separation vector will typically contain some component in the direction associated with the MLE, and because of the exponential growth rate, the effect of the other exponents will be obliterated over time.
The exponent is named after Aleksandr Lyapunov.
Definition of the maximal Lyapunov exponent.
The maximal Lyapunov exponent can be defined as follows:
where formula_5 is the maximum integer such that the sum of the formula_5 largest exponents is still non-negative. formula_7 represents an upper bound for the information dimension of the system. Moreover, the sum of all the positive Lyapunov exponents gives an estimate of the Kolmogorov–Sinai entropy accordingly to Pesin's theorem.
Along with widely used numerical methods for estimating and computing the Lyapunov dimension there is an effective analytical approach, which is based on the direct Lyapunov method with special Lyapunov-like functions.
The Lyapunov exponents of bounded trajectory
and the Lyapunov dimension of attractor are invariant under diffeomorphism of the phase space.
The multiplicative inverse of the largest Lyapunov exponent is sometimes referred in literature as Lyapunov time, and defines the characteristic "e"-folding time. For chaotic orbits, the Lyapunov time will be finite, whereas for regular orbits it will be infinite.
Numerical calculation.
Generally the calculation of Lyapunov exponents, as defined above, cannot be carried out analytically, and in most cases one must resort to numerical techniques. An early example, which also constituted the first demonstration of the exponential divergence of chaotic trajectories, was carried out by R. H. Miller in 1964. Currently, the most commonly used numerical procedure estimates the formula_8 matrix based on averaging several finite time approximations of the limit defining formula_8.
One of the most used and effective numerical techniques to calculate the Lyapunov spectrum for a smooth dynamical system relies on periodic
Gram–Schmidt orthonormalization of the Lyapunov vectors to avoid a misalignment of all the vectors along the direction of maximal expansion.
For the calculation of Lyapunov exponents from limited experimental data, various methods have been proposed. However, there are many difficulties with applying these methods and such problems should be approached with care. The main difficulty is that the data does not fully explore the phase space, rather it is confined to the attractor which has very limited (if any) extension along certain directions. These thinner or more singular directions within the data set are the ones associated with the more negative exponents. The use of nonlinear mappings to model the evolution of small displacements from the attractor has been shown to dramatically improve the ability to recover the Lyapunov spectrum, provided the data has a very low level of noise. The singular nature of the data and its connection to the more negative exponents has also been explored.
Local Lyapunov exponent.
Whereas the (global) Lyapunov exponent gives a measure for the total predictability of a system, it is sometimes of interest to estimate the local predictability around a point "x"0 in phase space. This may be done through the eigenvalues of the Jacobian matrix "J" 0("x"0). These eigenvalues are also called local Lyapunov exponents. (A word of caution: unlike the global exponents, these local exponents are not invariant under a nonlinear change of coordinates).
Conditional Lyapunov exponent.
This term is normally used regarding synchronization of chaos, in which there are two systems that are coupled, usually in a unidirectional manner so that there is a drive (or master) system and a response (or slave) system. The conditional exponents are those of the response system with the drive system treated as simply the source of a (chaotic) drive signal. Synchronization occurs when all of the conditional exponents are negative.