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*Y*with embedding dimension_{1:N}*m*, and lag*τ*.For a point

`i`

, the software then finds the nearest neighbor point*i*that satisfies $$\underset{{i}^{*}}{\mathrm{min}}\Vert {Y}_{i}-{Y}_{{i}^{*}}\Vert $$ such that $$\left|i-{i}^{*}\right|>MinSeparation$$, where^{*}`MinSeparation`

, the mean period, is the reciprocal of the mean frequency.Lyapunov exponent for the entire expansion range is calculated as,

$$\lambda (i)=\frac{1}{{K}_{\mathrm{max}}+{K}_{\mathrm{min}}+1}{\displaystyle \sum _{K={K}_{\mathrm{min}}}^{{K}_{\mathrm{max}}}\frac{1}{K*dt}\mathrm{ln}\frac{\Vert {Y}_{i+K}-{Y}_{{i}^{*}+K}\Vert}{\Vert {Y}_{i}-{Y}_{{i}^{*}}\Vert}}$$

where,

*K*and_{min}*K*represent_{max}`ExpansionRange`

,`dt`

is the sampling time and $$ldiv=\mathrm{ln}\frac{\Vert {Y}_{i+K}-{Y}_{{i}^{*}+K}\Vert}{\Vert {Y}_{i}-{Y}_{{i}^{*}}\Vert}$$A single value for the Lyapunov exponent is then calculated from the earlier step using the

`polyfit`

command as,$$\text{lyapExp=}\mathrm{polyfit}\left(\left[{K}_{\mathrm{min}}\text{}{K}_{\mathrm{max}}\right],\lambda (i)\right)$$

[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.