Draw langford attractors with different initial conditions. More...

Detailed Description

Draw langford attractors with different initial conditions.

Author: Mitch Richling https://www.mitchr.me

Standards: C++20

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Details

This code generates a fractal-like image from the Langford attractor. The image represents a rectangle inthe \(x\)- \(z\) axis plane of \(\mathbb{R}^3\). We start the IVP problem with initial conditions in this rectangle. The color represents the time required for the solution point to enter a sphere of radius \(\frac{1}{2}\) centered at \((0, 0, \frac{3}{4})\). This sphere intersects the central "stalk" in the Langford attractor, and the time to hit this sphere is a proxy for the time required for the solution to be "captured by" or "on" the attractor.

For reference the Langford system is defined by:

\[ \begin{array}{lcl} \frac{dx}{dt} & = & (z - \beta) x - \omega y \\ \frac{dy}{dt} & = & \omega x + (z - \beta) y \\ \frac{dz}{dt} & = & \lambda + \alpha z - \frac{1}{3}z^3 - (x^2 + y^2) (1 + \rho z) + \epsilon z x^3 \end{array} \]

Traditional parameter values are:

\[ \begin{array}{lcc} \alpha & = & \frac{19}{20} \\ \beta & = & \frac{7}{10} \\ \lambda & = & \frac{3}{5} \\ \omega & = & \frac{7}{2} \\ \rho & = & \frac{1}{4} \\ \epsilon & = & \frac{1}{10} \end{array} \]

We use Euler's method to solve the equations with a \(\Delta{t}\) of 0.001 and a maximum of \(65536 = 2^{16}\) iterations.

Definition in file 3da_frac_langford.cpp.