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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Lyapunov exponent or Lyapunov characteristic exponent of 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 ...
is a quantity that characterizes the rate of separation of infinitesimally close
trajectories A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete traje ...
. Quantitatively, two trajectories in
phase space The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...
with initial separation vector \boldsymbol_0 diverge (provided that the divergence can be treated within the linearized approximation) at a rate given by , \boldsymbol(t) , \approx e^ , \boldsymbol_0 , where \lambda 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 Predictability is the degree to which a correct prediction or forecast of a system's state can be made, either qualitatively or quantitatively. Predictability and causality Causal determinism has a strong relationship with predictability. Perfec ...
for a dynamical system. A positive MLE is usually taken as an indication that the system is
chaotic Chaotic was originally a Danish trading card game. It expanded to an online game in America which then became a television program based on the game. The program aired on 4Kids TV (Fox affiliates, nationwide), Jetix, The CW4Kids, Cartoon Netwo ...
(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 diminish over time. The exponent is named after
Aleksandr Lyapunov Aleksandr Mikhailovich Lyapunov (Алекса́ндр Миха́йлович Ляпуно́в, – 3 November 1918) was a Russian mathematician, mechanician and physicist. He was the son of the astronomer Mikhail Lyapunov and the brother of t ...
.


Definition of the maximal Lyapunov exponent

The maximal Lyapunov exponent can be defined as follows: \lambda = \lim_ \lim_ \frac \ln\frac The limit , \boldsymbol_0, \to 0 ensures the validity of the linear approximation at any time. For discrete time system (maps or fixed point iterations) x_ = f(x_n) , for an orbit starting with x_0 this translates into: \lambda (x_0) = \lim_ \frac \sum_^ \ln , f'(x_i),


Definition of the Lyapunov spectrum

For a dynamical system with evolution equation \dot_i = f_i(x) in an ''n''–dimensional phase space, the spectrum of Lyapunov exponents \ \,, in general, depends on the starting point x_0. However, we will usually be interested in the
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 ...
(or attractors) of a dynamical system, and there will normally be one set of exponents associated with each attractor. The choice of starting point may determine which attractor the system ends up on, if there is more than one. (For Hamiltonian systems, which do not have attractors, this is not a concern.) The Lyapunov exponents describe the behavior of vectors in the tangent space of the phase space and are defined from the
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. If this matrix is square, that is, if the number of variables equals the number of component ...
J_(t) = \left. \frac \_ this Jacobian defines the evolution of the tangent vectors, given by the matrix Y, via the equation \dot = J Y with the initial condition Y_(0) = \delta_. The matrix Y describes how a small change at the point x(0) propagates to the final point x(t). The limit \Lambda = \lim_ \frac \log (Y(t) Y^T(t)) defines a matrix \Lambda (the conditions for the existence of the limit are given by the
Oseledets theorem In mathematics, the multiplicative ergodic theorem, or Oseledets theorem provides the theoretical background for computation of Lyapunov exponents of a nonlinear dynamical system. It was proved by Valery Oseledets (also spelled "Oseledec") in ...
). The Lyapunov exponents \lambda_i are defined by the eigenvalues of \Lambda. The set of Lyapunov exponents will be the same for almost all starting points of an
ergodic In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies th ...
component of the dynamical system.


Lyapunov exponent for time-varying linearization

To introduce Lyapunov exponent consider a fundamental matrix X(t) (e.g., for linearization along a stationary solution x_0 in a continuous system), the fundamental matrix is \exp\left( \left. \frac \_ t\right) consisting of the linearly-independent solutions of the first-order approximation of the system. The singular values \_^ of the matrix X(t) are the square roots of the eigenvalues of the matrix X(t)^*X(t). The largest Lyapunov exponent \lambda_ is as follows \lambda_= \max\limits_\limsup _\frac\ln\alpha_j\big(X(t)\big). Lyapunov proved that if the system of the first approximation is regular (e.g., all systems with constant and periodic coefficients are regular) and its largest Lyapunov exponent is negative, then the solution of the original system is asymptotically Lyapunov stable. Later, it was stated by O. Perron that the requirement of regularity of the first approximation is substantial.


Perron effects of largest Lyapunov exponent sign inversion

In 1930 O. Perron constructed an example of a second-order system, where the first approximation has negative Lyapunov exponents along a zero solution of the original system but, at the same time, this zero solution of the original nonlinear system is Lyapunov unstable. Furthermore, in a certain neighborhood of this zero solution almost all solutions of original system have positive Lyapunov exponents. Also, it is possible to construct a reverse example in which the first approximation has positive Lyapunov exponents along a zero solution of the original system but, at the same time, this zero solution of original nonlinear system is Lyapunov stable. The effect of sign inversion of Lyapunov exponents of solutions of the original system and the system of first approximation with the same initial data was subsequently called the Perron effect. Perron's counterexample shows that a negative largest Lyapunov exponent does not, in general, indicate stability, and that a positive largest Lyapunov exponent does not, in general, indicate chaos. Therefore, time-varying linearization requires additional justification.


Basic properties

If the system is conservative (i.e., there is no
dissipation In thermodynamics, dissipation is the result of an irreversible process that affects a thermodynamic system. In a dissipative process, energy ( internal, bulk flow kinetic, or system potential) transforms from an initial form to a final form, wh ...
), a volume element of the phase space will stay the same along a trajectory. Thus the sum of all Lyapunov exponents must be zero. If the system is dissipative, the sum of Lyapunov exponents is negative. If the system is a flow and the trajectory does not converge to a single point, one exponent is always zero—the Lyapunov exponent corresponding to the eigenvalue of L with an eigenvector in the direction of the flow.


Significance of the Lyapunov spectrum

The Lyapunov spectrum can be used to give an estimate of the rate of entropy production, of the
fractal dimension In mathematics, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the Scaling (geometry), scale at which it is measured. It ...
, and of the
Hausdorff dimension In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line ...
of the considered
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 ...
. In particular from the knowledge of the Lyapunov spectrum it is possible to obtain the so-called
Lyapunov dimension In the mathematics of dynamical systems, the concept of Lyapunov dimension was suggested by Kaplan and Yorke for estimating the Hausdorff dimension of attractors. Further the concept has been developed and rigorously justified in a number of paper ...
(or Kaplan–Yorke dimension) D_ , which is defined as follows: D_= k + \sum_^k \frac where k is the maximum integer such that the sum of the k largest exponents is still non-negative. D_ 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 In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special cas ...
accordingly to Pesin's theorem. Along with widely used numerical methods for estimating and computing the
Lyapunov dimension In the mathematics of dynamical systems, the concept of Lyapunov dimension was suggested by Kaplan and Yorke for estimating the Hausdorff dimension of attractors. Further the concept has been developed and rigorously justified in a number of paper ...
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 In the mathematics of dynamical systems, the concept of Lyapunov dimension was suggested by Kaplan and Yorke for estimating the Hausdorff dimension of attractors. Further the concept has been developed and rigorously justified in a number of paper ...
of attractor are invariant under
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Definit ...
of the phase space. The
multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a ra ...
of the largest Lyapunov exponent is sometimes referred in literature as
Lyapunov time In mathematics, the Lyapunov time is the characteristic timescale on which a dynamical system is chaotic. It is named after the Russian mathematician Aleksandr Lyapunov. It is defined as the inverse of a system's largest Lyapunov exponent. Use T ...
, 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 L matrix based on averaging several finite time approximations of the limit defining L. 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 vector Lyapunov (, in old-Russian often written Лепунов) is a Russian surname that is sometimes also romanized In linguistics, romanization is the conversion of text from a different writing system to the Roman (Latin) script, or a system for ...
s to avoid a misalignment of all the vectors along the direction of maximal expansion. The Lyapunov spectrum of various models are described. Source codes for nonlinear systems such as the Hénon map, the Lorenz equations, a delay differential equation and so on are introduced. 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 in phase space. This may be done through the
eigenvalues In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
of the
Jacobian In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to: *Jacobian matrix and determinant (and in particular, the robot Jacobian) *Jacobian elliptic functions *Jacobian variety * Jacobian ideal *Intermediate Jacobian In mat ...
matrix . These eigenvalues are also called local Lyapunov exponents. Local exponents are not invariant under a nonlinear change of coordinates.


Conditional Lyapunov exponent

This term is normally used regarding
synchronization of chaos Synchronization of chaos is a phenomenon that may occur when two or more dissipative chaotic systems are coupled. Because of the exponential divergence of the nearby trajectories of chaotic systems, having two chaotic systems evolving in synchrony ...
, 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.See, e.g.,


See also

*
Chaos Theory Chaos theory is an interdisciplinary area of Scientific method, scientific study and branch of mathematics. It focuses on underlying patterns and Deterministic system, deterministic Scientific law, laws of dynamical systems that are highly sens ...
*
Chaotic mixing In chaos theory and fluid dynamics, chaotic mixing is a process by which flow tracers develop into complex fractals under the action of a fluid flow. The flow is characterized by an exponential growth of fluid filaments. Even very simple flows, s ...
for an alternative derivation * Eden's conjecture on the Lyapunov dimension *
Floquet theory Floquet theory is a branch of the theory of ordinary differential equations relating to the class of solutions to periodic linear differential equations of the form :\dot = A(t) x, with x\in and \displaystyle A(t) \in being a piecewise continu ...
*
Liouville's theorem (Hamiltonian) In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical mechanics, statistical and Hamiltonian mechanics. It asserts that ''the phase space, phase-space distribution functi ...
*
Lyapunov dimension In the mathematics of dynamical systems, the concept of Lyapunov dimension was suggested by Kaplan and Yorke for estimating the Hausdorff dimension of attractors. Further the concept has been developed and rigorously justified in a number of paper ...
*
Lyapunov time In mathematics, the Lyapunov time is the characteristic timescale on which a dynamical system is chaotic. It is named after the Russian mathematician Aleksandr Lyapunov. It is defined as the inverse of a system's largest Lyapunov exponent. Use T ...
*
Recurrence quantification analysis Recurrence quantification analysis (RQA) is a method of nonlinear data analysis (cf. chaos theory) for the investigation of dynamical systems. It quantifies the number and duration of recurrences of a dynamical system presented by its phase space tr ...
*
Oseledets theorem In mathematics, the multiplicative ergodic theorem, or Oseledets theorem provides the theoretical background for computation of Lyapunov exponents of a nonlinear dynamical system. It was proved by Valery Oseledets (also spelled "Oseledec") in ...
*
Butterfly effect In chaos theory, the butterfly effect is the sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a later state. The term is closely associated w ...


References


Further reading

* * * Cvitanović P., Artuso R., Mainieri R., Tanner G. and Vattay
Chaos: Classical and Quantum
Niels Bohr Institute, Copenhagen 2005 – ''textbook about chaos available under Free Documentation License'' * * * * * *


Software



R. Hegger, H. Kantz, and T. Schreiber, Nonlinear Time Series Analysis, Tisean, TISEAN 3.0.1 (March 2007).

Scientio's ChaosKit product calculates Lyapunov exponents amongst other Chaotic measures. Access is provided online via a web service and Silverlight demo.

Dr. Ronald Joe Record's mathematical recreations software laboratory includes an X11 graphical client, lyap, for graphically exploring the Lyapunov exponents of a forced logistic map and other maps of the unit interval. Th
contents and manual pages
of the mathrec software laboratory are also available.

Software on this page was developed specifically for the efficient and accurate calculation of the full spectrum of exponents. This includes LyapOde for cases where the equations of motion are known and also Lyap for cases involving experimental time series data. LyapOde, which includes source code written in "C", can also calculate the conditional Lyapunov exponents for coupled identical systems. It is intended to allow the user to provide their own set of model equations or to use one of the ones included. There are no inherent limitations on the number of variables, parameters etc. Lyap which includes source code written in Fortran, can also calculate the Lyapunov direction vectors and can characterize the singularity of the attractor, which is the main reason for difficulties in calculating the more negative exponents from time series data. In both cases there is extensive documentation and sample input files. The software can be compiled for running on Windows, Mac, or Linux/Unix systems. The software runs in a text window and has no graphics capabilities, but can generate output files that could easily be plotted with a program like excel.


External links


Perron effects of Lyapunov exponent sign inversions
{{Authority control Dynamical systems