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Lyapunov Characteristic Exponent


The Lyapunov characteristic exponent [LCE] gives the rate of exponential divergence from perturbed initial conditions. To examine the behavior of an orbit around a point X^*(t), perturb the system and write

 X(t)=X^*(t)+U(t),
(1)

where U(t) is the average deviation from the unperturbed trajectory at time t. In a chaotic region, the LCE sigma is independent of X^*(0). It is given by the Oseledec theorem, which states that

 sigma_i=lim_(t->infty)1/tln|U(t)|.
(2)

For an n-dimensional mapping, the Lyapunov characteristic exponents are given by

 sigma_i=lim_(N->infty)ln|lambda_i(N)|
(3)

for i=1, ..., n, where lambda_i is the Lyapunov characteristic number.

One Lyapunov characteristic exponent is always 0, since there is never any divergence for a perturbed trajectory in the direction of the unperturbed trajectory. The larger the LCE, the greater the rate of exponential divergence and the wider the corresponding separatrix of the chaotic region. For the standard map, an analytic estimate of the width of the chaotic zone by Chirikov (1979) finds

 deltaI=Be^(-AK^(-1/2)).
(4)

Since the Lyapunov characteristic exponent increases with increasing K, some relationship likely exists connecting the two. Let a trajectory (expressed as a map) have initial conditions (x_0,y_0) and a nearby trajectory have initial conditions (x^',y^')=(x_0+dx,y_0+dy). The distance between trajectories at iteration k is then

 dk=|(x^'-x_0,y^'-y_0)|,
(5)

and the mean exponential rate of divergence of the trajectories is defined by

 sigma_1=lim_(k->infty)1/kln((d_k)/(d_0)).
(6)

For an n-dimensional phase space (map), there are n Lyapunov characteristic exponents sigma_1>=sigma_2>=...>sigma_n. However, because the largest exponent sigma_1 will dominate, this limit is practically useful only for finding the largest exponent. Numerically, since d_k increases exponentially with k, after a few steps the perturbed trajectory is no longer nearby. It is therefore necessary to renormalize frequently every t steps. Defining

 r_(ktau)=(d_(ktau))/(d_0),
(7)

one can then compute

 sigma_1=lim_(n->infty)1/(ntau)sum_(k=1)^nlnr_(ktau).
(8)

Numerical computation of the second (smaller) Lyapunov exponent may be carried by considering the evolution of a two-dimensional surface. It will behave as

 e^((sigma_1+sigma_2)t),
(9)

so sigma_2 can be extracted if sigma_1 is known. The process may be repeated to find smaller exponents.

For Hamiltonian systems, the LCEs exist in additive inverse pairs, so if sigma is an LCE, then so is -sigma. One LCE is always 0. For a one-dimensional oscillator (with a two-dimensional phase space), the two LCEs therefore must be sigma_1=sigma_2=0, so the motion is quasiperiodic and cannot be chaotic. For higher order Hamiltonian systems, there are always at least two 0 LCEs, but other LCEs may enter in plus-and-minus pairs l and -l. If they, too, are both zero, the motion is integrable and not chaotic. If they are nonzero, the positive LCE l results in an exponential separation of trajectories, which corresponds to a chaotic region. Notice that it is not possible to have all LCEs negative, which explains why convergence of orbits is never observed in Hamiltonian systems.

Now consider a dissipative system. For an arbitrary n-dimensional phase space, there must always be one LCE equal to 0, since a perturbation along the path results in no divergence. The LCEs satisfy sum_(i)sigma_i<0. Therefore, for a two-dimensional phase space of a dissipative system, sigma_1=0,sigma_2<0. For a three-dimensional phase space, there are three possibilities:

1. (Integrable): sigma_1=0,sigma_2=0,sigma_3<0,

2. (Integrable): sigma_1=0,sigma_2,sigma_3<0,

3. (Chaotic): sigma_1=0,sigma_2>0,sigma_3<-sigma_2<0.


See also

Chaos, Hamiltonian System, Lyapunov Characteristic Number, Oseledec Theorem

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References

Sandri, M. "Numerical Calculation of Lyapunov Exponents." Mathematica J. 6, 78-84, 1996. http://library.wolfram.com/infocenter/Articles/2902/.Chirikov, B. V. "A Universal Instability of Many-Dimensional Oscillator Systems." Phys. Rep. 52, 264-379, 1979.Ramasubramanian, K. and Sriram, M. S. "A Comparative Study of Computation of Lyapunov Spectra with Different Algorithms" 1999. http://arxiv.org/abs/chao-dyn/9909029.Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, p. 24, 2004. http://www.mathematicaguidebooks.org/.

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Lyapunov Characteristic Exponent

Cite this as:

Weisstein, Eric W. "Lyapunov Characteristic Exponent." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LyapunovCharacteristicExponent.html

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