Characterize the rate of separation of infinitesimally close trajectories
estimates the Lyapunov exponent of the uniformly sampled time-domain signal
lyapExp = lyapunovExponent(
X using sampling frequency
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.
Visualize and Estimate Largest Lyapunov Exponent
In this example, consider a Lorenz attractor describing a unique set of chaotic solutions.
Load the data set and sampling frequency
fs to the workspace, and visualize the Lorenz attractor in 3-D.
For this example, use the x-direction data of the Lorenz attractor. Since
Lag is unknown, estimate the delay using
phaseSpaceReconstruction. Set dimension to 3 since the Lorenz attractor is a three-dimensional system. The
lag parameters are required to create the logarithmic divergence versus expansion step plot.
xdata = data(:,1); dim = 3; [~,lag] = phaseSpaceReconstruction(xdata,,dim)
lag = 10
Create the average logarithmic divergence versus expansion step plot for the Lorenz attractor, using the
lag value obtained in the previous step. Set a sufficiently large expansion range to capture all the expansion steps.
eRange = 200; lyapunovExponent(xdata,fs,lag,dim,'ExpansionRange',eRange)
The first dashed, vertical green line (on the left) indicates the minimum number of steps used to estimate the expansion range, while the second vertical green line (on the right), represents the maximum number of steps used. Together, the first and second vertical lines represent the expansion range. The dashed red line indicates the linear fit line for the data, within the expansion range.
To compute the largest Lyapunov exponent, you first need to determine the expansion range needed for accurate estimation.
In the plot, drag the two dashed, vertical green lines to best fit the linear fit line to the original data line to obtain the expansion range: and .
Note the new values of the expansion range after dragging the two vertical lines for an appropriate fit.
Since expansion range can only be specified using whole numbers, round-off and to the nearest integer. Find the largest Lyapunov exponent of the Lorenz attractor using the new expansion range value.
Kmin = 21; Kmax = 161; lyapExp = lyapunovExponent(xdata,fs,lag,dim,'ExpansionRange',[Kmin Kmax])
lyapExp = 1.6834
A negative Lyapunov exponent indicates convergence, while positive Lyapunov exponents demonstrate divergence and chaos. The magnitude of
lyapExp is an indicator of the rate of convergence or divergence of the infinitesimally close trajectories.
X — Uniformly sampled time-domain signal
vector | array | timetable
Uniformly sampled time-domain signal, specified as a vector, array, or
X has multiple columns,
lyapunovExponent computes the largest Lyapunov
exponent by treating
X as a multivariate signal.
X is specified as a row vector,
lyapunovExponent treats it as a univariate
fs — Sampling frequency
Sampling frequency, specified as a scalar. Sampling frequency or sampling rate is the average number of samples obtained in one second.
fs is not supplied, a normalized frequency of 2π
is used to compute the Lyapunov exponent. If
specified as a timetable, the sampling time is inferred from it.
dim — Embedding dimension
scalar | vector
Embedding dimension, specified as a scalar or vector.
equivalent to the '
Dimension' name-value pair.
lag — Time delay
scalar | vector
Time delay, specified as a scalar or vector.
lag is equivalent to
Lag' name-value pair.
comma-separated pairs of
the argument name and
Value is the corresponding value.
Name must appear inside quotes. You can specify several name and value
pair arguments in any order as
Dimension — Embedding dimension
2 (default) | scalar | vector
Embedding dimension, specified as the comma-separated pair consisting
Dimension' and either a scalar or vector. When
Dimension is scalar, every column in
X is reconstructed using
Dimension is a
vector having same length as the number of columns in
X, the reconstruction dimension for column
Dimension based on the dimension of your
system, that is, the number of states. For more information on embedding
Lag — Delay in phase space reconstruction
1 (default) | scalar | vector
Delay in phase space reconstruction, specified as the comma-separated
pair consisting of '
Lag' and either a scalar or
Lag is scalar, every column in
X is reconstructed using
Lag is a vector having
same length as the number of columns in
reconstruction delay for column
The default value of
Lag is 1.
If the delay is too small, random noise is introduced in the data. In
contrast, if the lag is too large, the reconstructed dynamics do not
represent the true dynamics of the time series. For more information on
estimating optimal delay, see
MinSeparation — Mean period
ceil(fs/max(meanfreq(X,fs))) (default) | positive scalar integer
Mean period, specified as the comma-separated pair consisting of
MinSeparation' and a positive scalar
MinSeparation is the threshold value used to find
the nearest neighbor i* for a point
i to estimate the
largest Lyapunov exponent.
The default value of
ExpansionRange — Range of expansion steps
[1, 5] (default) | 1x2 positive integer array | positive scalar integer
Range of expansion steps, specified as the comma-separated pair
consisting of '
ExpansionRange' and either a 1x2
positive integer array or a positive scalar integer.
The minimum and maximum value of
used to estimate the local expansion rate to calculate the Lyapunov
ExpansionRange is specified as a scalar
M, then the range is set to be
ExpansionRange can only be
specified using positive whole numbers and the default value is
lyapExp — Largest Lyapunov exponent
Largest Lyapunov exponent, returned as a scalar.
lyapExp quantifies the rate of divergence or
convergence of close trajectories in phase space.
A negative Lyapunov exponent indicates convergence, while positive
Lyapunov exponents demonstrate divergence and chaos. The magnitude of
lyapExp is an indicator of the rate of convergence
or divergence of the infinitesimally close trajectories.
The ability to discern levels of divergence within data sets is useful in the field of engineering to estimate component failure by studying their vibration and acoustic signals, or to predict when a ship would capsize based on its motion.
estep — Expansion step used for estimation
Expansion step used for estimation, returned as an array.
estep is the difference between the maximum and
minimum expansion range split into an equal number of points defined by the
maximum value of
ldiv — Logarithmic divergence
Logarithmic divergence, returned as an array with the same size as
estep. The magnitude of each value in
ldiv corresponds to the logarithmic convergence or
divergence of each point in
Lyapunov exponent is calculated in the following way:
lyapunovExponentfunction 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.
From , the Lyapunov exponent for the entire expansion range is calculated as,
where, Kmin and Kmax represent
dtis the sampling time and
A single value for the Lyapunov exponent is then calculated from the earlier step using the
 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.
 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.
 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.