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 diverge Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. ... The Lorenz attractor is an example of a non-linear dynamical system. ... Mathematically the term trajectory refers to the ordered set of states which are assumed by a dynamical system over time (see e. ... For other senses of this term, see phase space (disambiguation). ...

where represents the modulus of the considered vectors.

The rate of separation can be different for different orientations of initial separation vector. Thus, there is a whole spectrum of Lyapunov exponents—the number of them is equal to the number of dimensions of the phase space. It is common to just refer to the largest one, i.e. to the Maximal Lyapunov exponent (MLE), because it determines the predictability of a dynamical system. A positive MLE is usually taken as an indication that the system is chaotic. Prediction of future events is an ancient human wish. ...

Definition of the maximal Lyapunov exponent

The maximal Lyapunov exponent can be defined as follows:

Definition of the Lyapunov spectrum

For a dynamical system with evolution equation f^t in a n–dimensional phase space, the spectrum of Lyapunov exponents

in general, depends on the starting point x₀. The Lyapunov exponents describe the behavior of vectors in the tangent space of the phase space and are defined from the Jacobian matrix

The J^t matrix describes how a small change at the point x₀ propagates to the final point f^t(x₀). The limit

defines a matrix L(x₀) (the conditions for the existence of the limit are given by the Oseldec theorem). If Λ_i(x₀) are the eigenvalues of L(x₀), then the Lyapunov exponents λ_i are defined by In mathematics, the Oseledec theorem provides the theoretical background for computation of Lyapunov exponents of a nonlinear dynamical system. ...

The set of Lyapunov exponents will be the same for almost all starting points of an ergodic component of the dynamical system. The Lorenz attractor is an example of a non-linear dynamical system. ...

Basic properties

If the system is conservative (i.e. there is no dissipation), 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. A wave that loses amplitude is said to dissipate. ...

If the system is a flow, 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 and of the fractal dimension of the considered dynamical system. In particular from the knowledge of the Lyapunov spectrum it is possible to obtain the so-called Kaplan-Yorke dimension D_KY, that is defined as follows: In fractal geometry, the fractal dimension is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. ... The Lorenz attractor is an example of a non-linear dynamical system. ...

,

where k is the maximum integer such that the sum of the k largest exponents is still non-negative. D_KY represents an upper bound for the information dimension of the system ^[1]. Moreover, the sum of all the positive Lyapunov exponents gives an estimate of the Kolmogorov-Sinai entropy accordingly to Pesin's theorem ^[2] In fractal geometry, the fractal dimension is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. ... In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of ergodic theory. ...

The 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. Categories: Wikipedia cleanup | Stub | Dynamical systems ...

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. The commonly used numerical procedures estimates the L matrix based on averaging several finite time approximations of the limit defining L.

One of the most used and effective numerical technique to calculate the Lyapunov spectrum for a smooth dynamical system relies on periodic Gram-Schmidt orthonormalization of the Lyapunov vectors to avoid a misalignement of all the vectors along the direction of maximal expansion ^{[3] [4]}. In mathematics and numerical analysis, the Gram-Schmidt process of linear algebra is a method of orthogonalizing a set of vectors in an inner product space, most commonly the Euclidean space Rn. ...

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

Local Lyapunov exponent

Whereas the (global) Lyapunov exponent gives a measure for the total predictability of a system, it is sometimes interesting to estimate the local predictability around a point x₀ in phase space. This may be done through the eigenvalues of the Jacobian matrix J ⁰(x₀). These eigenvalues are also called local Lyapunov exponents. The eigenvectors of the Jacobian matrix point in the direction of the stable and unstable manifolds. In linear algebra, a scalar λ is called an eigenvalue (in some older texts, a characteristic value) of a linear mapping A if there exists a nonzero vector x such that Ax=λx. ... In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In mathematics, and in particular the study of dynamical systems, the idea of stable and unstable sets or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repellor. ... In mathematics, and in particular the study of dynamical systems, the idea of stable and unstable sets or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repellor. ...

See also

• Aleksandr Lyapunov
• Oseledec theorem
• Liouville's theorem (Hamiltonian)
• Floquet theory
• Recurrence quantification analysis
• Finite Time Lyapunov Exponent
• Finite Size Lyapunov Exponent

Aleksandr Mikhailovich Lyapunov (Александр Михайлович Ляпунов) (June 6, 1857 – November 3, 1918, all new style) was a Russian mathematician, mechanician and physicist. ... In mathematics, the Oseledec theorem provides the theoretical background for computation of Lyapunov exponents of a nonlinear dynamical system. ... In mathematical physics, Liouvilles theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. ... Floquet theory is a branch of the theory of ordinary differential equations relating to the class of solutions to linear differential equations of the form, , with a continuous periodic function with period . ... Recurrence quantification analysis (RQA) is a system for analyzing nonlinear systems that works by looking at the pattern and number of recurrences present in a data signal. ...

References

1. ^ J. Kaplan and J. Yorke Chaotic behavior of multidimensional difference equations In Peitgen, H. O. & Walther, H. O., editors, ``Functional Differential Equations and Approximation of Fixed Points Springer, New York (1987)
2. ^ Y. B. Pesin, Characteristic Lyapunov Exponents and Smooth Ergodic Theory, Russian Math. Surveys, 32 (1977), 4, 55-114
3. ^ G. Benettin, L. Galgani, A. Giorgilli and J.M. Strelcyn, Meccanica, 9-20 (1980); ibidem, Meccanica, 21-30 (1980).
4. ^ I. Shimada and T. Nagashima, Prog. Theor. Phys. 61, 1605 (1979).

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

• Freddy Christiansen and Hans Henrik Rugh (1997). "Computing Lyapunov spectra with continuous Gram-Schmidt orthonormalization". Nonlinearity 10: 1063–1072. 

• Salman Habib and Robert D. Ryne (1995). "Symplectic Calculation of Lyapunov Exponents". Physical Review Letters 74: 70–73. 

• Govindan Rangarajan, Salman Habib, and Robert D. Ryne (1998). "Lyapunov Exponents without Rescaling and Reorthogonalization". Physical Review Letters 80: 3747–3750. 

• X. Zeng, R. Eykholt, and R. A. Pielke (1991). "Estimating the Lyapunov-exponent spectrum from short time series of low precision". Physical Review Letters 66: 3229. 

• E Aurell, G Boffetta, A Crisanti, G Paladin and A Vulpiani (1997). "Predictability in the large: an extension of the concept of Lyapunov exponent" 30: 1-26. 

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Software

• [1] R. Hegger, H. Kantz, and T. Schreiber, Nonlinear Time Series Analysis, TISEAN 2.1 (December 2000).

• [2] Scientio's ChaosKit product calculates Lyapunov exponents amongst other Chaotic measures. Access is provided free online via a web service.