Lorenz Equations

class otp.lorenz63.Lorenz63Problem

Bases: otp.Problem

A simple continuous chaotic system.

The three variable Lorenz ‘63 problem [Lor63] of the form,

\[\begin{split} x' &= σ(y - x),\\ y' &= ρx - y - xz,\\ z' &= xy - βz, \end{split}\]

exhibits chaotic behavior for certain values of the parameters. Here $x$ roughly corresponds to the rate of convection of a fluid, $y$ corresponds to temperature variation in one direction, and $z$ is temperature variation in the other direction. For a full problem description see [Str18].

Notes

Type	ODE
Number of Variables	3
Stiff	not typically, depending on $σ$, $ρ$, and $β$

Example

>>> problem = otp.lorenz63.presets.Canonical;
>>> sol = problem.solve('MaxStep', 1e-3);
>>> problem.plotPhaseSpace(sol);

Parameters

class otp.lorenz63.Lorenz63Parameters

Bases: otp.Parameters

Parameters for the Lorenz ‘63 problem.

Constructor Summary

Lorenz63Parameters(varargin)

Create a Lorenz ‘63 parameters object.

Parameters:: varargin – A variable number of name-value pairs. A name can be any property of this class, and the subsequent value initializes that property.

Property Summary

Sigma: A representation of the Prandtl number.

Rho: A representation of the Rayleigh number.

Beta: A geometric factor.

Presets

class otp.lorenz63.presets.Surprise

Bases: otp.lorenz63.Lorenz63Problem

Lorenz ‘63 preset ‘surprise’ from [Str18] which uses time span $t ∈ [0, 60]$, σ = 10$, $ρ = 100$, $β = 8/3$, and intial conditions $y_0 = [2, 1, 1]^T$.

Constructor Summary

Surprise(): Create the surprise Lorenz ‘63 problem object.

class otp.lorenz63.presets.LimitCycle

Bases: otp.lorenz63.Lorenz63Problem

Lorenz ‘63 preset limit cycle, a non-chaotic preset, from [Str18] which uses time span $t ∈ [0, 60]$, $σ = 10$, $ρ = 350$, $β = 8/3$, and intial conditions $y_0 = [0, 1, 0]^T$.

Constructor Summary

LimitCycle(): Create the limit cycle Lorenz ‘63 problem object.

class otp.lorenz63.presets.Canonical

Bases: otp.lorenz63.Lorenz63Problem

Original Lorenz ‘63 preset presented in [Lor63] which uses time span $t ∈ [0, 60]$, $σ = 10$, $ρ = 28$, $β = 8/3$, and intial conditions $y_0 = [0, 1, 0]^T$.

Constructor Summary

Canonical(varargin)

Create the canonical Lorenz ‘63 problem object.

Parameters:

varargin –

A variable number of name-value pairs. The accepted names are

Sigma – Value of $σ$.
Rho – Value of $ρ$.
Beta – Value of $β$.