Characterize the rate of separation of infinitesimally close trajectories
estimates the Lyapunov exponent of the uniformly sampled time-domain signal
lyapExp
= lyapunovExponent(X
,fs
)X
using sampling frequency fs
. Use
lyapunovExponent
to characterize the rate of separation
of infinitesimally close trajectories in phase space to distinguish different
attractors. Lyapunov exponent is useful in quantifying the level of chaos in a
system, which in turn can be used to detect potential faults.
___ = lyapunovExponent(___,
estimates the Lyapunov exponent with additional options specified by one or more
Name,Value
)Name,Value
pair arguments.
lyapunovExponent(___)
with no output
arguments creates an average logarithmic divergence versus expansion step
plot.
Use the generated interactive plot to find an appropriate
ExpansionRange
.
Lyapunov exponent is calculated in the following way:
The lyapunovExponent
function first generates a delayed
reconstruction Y1:N with embedding dimension m, and lag τ.
For a point i
, the software then finds the nearest
neighbor point i* that satisfies such that , where MinSeparation
, the mean period,
is the reciprocal of the mean frequency.
Lyapunov exponent for the entire expansion range is calculated as,
where, Kmin and Kmax represent ExpansionRange
,
dt
is the sampling time and
A single value for the Lyapunov exponent is then calculated from the
earlier step using the polyfit
command as,
[1] Michael T. Rosenstein , James J. Collins , Carlo J. De Luca. "A practical method for calculating largest Lyapunov exponents from small data sets ". Physica D 1993. Volume 65. Pages 117-134.
[2] Caesarendra, Wahyu & Kosasih, P & Tieu, Kiet & Moodie, Craig. "An application of nonlinear feature extraction-A case study for low speed slewing bearing condition monitoring and prognosis." IEEE/ASME International Conference on Advanced Intelligent Mechatronics: Mechatronics for Human Wellbeing, AIM 2013.1713-1718. 10.1109/AIM.2013.6584344.
[3] McCue, Leigh & W. Troesch, Armin. (2011). "Use of Lyapunov Exponents to Predict Chaotic Vessel Motions". Fluid Mechanics and its Applications. 97. 415-432. 10.1007/978-94-007-1482-3_23.