File:Quasi-periodic attractor of Langford equation.png

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English: Quasi-periodic attractor of Langford equation

The used parameters are (α, β, λ, ω, ρ, ε) = (1, 0.7, 0.6, 3.5, 0.25, 0). Thee orbits in red, blue, and green starts from different initial values, and they are attracted to a 2-dimentional torus. References for making this figure are as follows:

  • 徳永 隆治 (1990) "カオスとフラクタル" in カオス ―カオス理論の基礎と応用, Information & Computing 49 (初版 ed.), サイエンス社, p. 70 ISBN: 4-7819-0592-7.
  • 池口 徹・山田 泰司・小室 元政 (2000) 合原 一幸 , ed. カオス時系列解析の基礎と応用 (初版 ed.), 産業図書, p. 50 ISBN: 978-4-7828-1010-1.

The following is the original source by Langford:

  • Langford W.F. (1984) Numerical Studies of Torus Bifurcations. In: Küpper T., Mittelmann H.D., Weber H. (eds) Numerical Methods for Bifurcation Problems. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 70. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6256-1_19
Date
Source Own work
Author Yapparina

Scilab source

Scilab source
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 This media was created with Scilab, a free open-source software.

Here is a listing of the Scilab source used to create this file. 

clear;

//Parameters
alpha = 1;
beta = 0.7;
lambda = 0.6;
omega = 3.5;
rho = 0.25;
epsilon = 0;

// x=x(1), y=x(2), z=x(3) 
function dx = LANGFORD(t,x)
  dx(1) = (x(3) - beta) * x(1) - omega * x(2) ;
  dx(2) = omega * x(1) + (x(3) - beta) * x(2) ; 
  dx(3) = lambda + alpha * x(3) - x(3)^3/3 - (x(1)^2 + x(2)^2) * (1 + rho * x(3)) + epsilon * x(3) * x(1)^3 ; 
endfunction

// The time range and step for calculation
Tin = 0;
Tout = 70;
Tstep = 0.01;
t = Tin:Tstep:Tout;

// The initial values and ode solvers for 3 cases
x10 = -0.5;
y10 = -0.5;
z10 = 2.5;
X10 = [x10; y10; z10];

X1 = ode(X10,0,t,LANGFORD);

x20 = 0.3;
y20 = 0.3;
z20 = 2.4;
X20 = [x20; y20; z20];

X2 = ode(X20,0,t,LANGFORD);

x30 = 0.6;
y30 = -0.6;
z30 = 2.3;
X30 = [x30; y30; z30];

X3 = ode(X30,0,t,LANGFORD);

clf;

param3d(X1(1,:), X1(2,:), X1(3,:), 'r')
e=gce()
e.foreground=color('red');

param3d(X2(1,:), X2(2,:), X2(3,:), 'r')
h=gce()
h.foreground=color('blue');

param3d(X3(1,:), X3(2,:), X3(3,:), 'r')
h=gce()
h.foreground=color('green');

xlabel('x')
ylabel('y')
zlabel('z')

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I, the copyright holder of this work, hereby publish it under the following license:
Creative Commons CC-Zero This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.
The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.

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