lorenz#

class peccary.examples.lorenz(s=10, r=20, b=2.667, initialVals=(0.0, 1.0, 1.05))[source]#

Bases: object

The lorenz class can be used to generate the x-, y-, and z-coordinates of the Lorenz strange attractor, which is chaotic for certain values of the input parameters \(\sigma\), \(\rho\), and \(\beta\).

References

[1] Code modified and expanded from a Matplotlib tutorial

Examples

To initialize the class using the default input parameters (\(\sigma = 10\), \(\rho = 20\), \(\beta = 2.667\)) and initial values (\(x_0 = 0\), \(y_0 = 1\), \(z = 1.05\)), run the following code:

>>> import peccary.examples as ex
>>> lorenzSys = ex.lorenz()

To integrate the system, use the integrate method. The timestep resolution can be controlled with the dt parameter (default 0.01) and either the number of timesteps nsteps (default 10000) or the duration parameter tDur for the number of “seconds”. The tDur parameter is by default not used, but when specified, it supersedes nsteps. To integrate the initialized system for 250.0 seconds, run:

>>> lor = lorenzSys.integrate(tDur=250.0)

The 3D data is stored in a peccary.timeseries.Timeseries class with the attributes x, y, and z, with the timesteps stored in the t attribute. The data can be extracted and plotted, e.g.,

>>> import matplotlib.pyplot as plt
>>> ax = plt.figure().add_subplot(projection='3d')
>>> ax.plot(lor.x, lor.y, lor.z, lw=0.5)
>>> plt.show()

Methods Summary

`getPartials`(xyz)	Get partial derivatives for initialized Lorenz system
`integrate`([dt, nsteps, tDur])	Integrate x/y/z timeseries for Lorenz strange attractor

Methods Documentation

getPartials(xyz)[source]#

Get partial derivatives for initialized Lorenz system

Parameters:

xyzndarray: 3D array containing points of interest in three-dimensional space

Returns:

ndarray: 3D array containing partial derivatives at xyz

integrate(dt=0.01, nsteps=10000, tDur=None)[source]#

Integrate x/y/z timeseries for Lorenz strange attractor

Parameters:

dtfloat, optional: Timestep resolution, by default 0.01
nstepsint, optional: Number of timesteps to integrate, by default 10000
tDurfloat, optional: Duration of time to integrate over, supersedes nsteps, by default None

Returns:

peccary.timeseries.Timeseries: Timeseries for x-, y-, and z- coordinates of Lorenz strange attractor, stored in x, y, and z attributes of Timeseries class

Attributes:

dtfloat: Timestep resolution
nstepsint: Number of timesteps used in integration
tDurNone or float: Duration of timseries, None unless specified
tNone or ndarray: Array of timesteps corresponding to timeseries
t0float: Initial time for integration