Strange Attractor [gnuplot]-Hiro's Soliloquy

Contents[hide]

1.Lorenz equation
- 1.1.PLT files
  - 1.1.1.lorenz.plt
  - 1.1.2.make.plt
- 1.2.lorenz.gif
2.Rössler equation
3.Langford equation
4.Other examples

Lorenz equation

$\begin{eqnarray*} \frac{\mathrm{d}x}{\mathrm{d}t}&=&-\sigma(x-y) \\ \frac{\mathrm{d}y}{\mathrm{d}t}&=&Rx-y-xz\\ \frac{\mathrm{d}z}{\mathrm{d}t}&=&xy-bz \end{eqnarray*}$

$(R=28,~b=\frac{8}{3},~\sigma=10)$

PLT files

lorenz.plt

	set terminal png enhanced font "Times" 20 size 986, 554
	set tics font 'Times,18'
	set nokey
	set grid
	set xr[-30:30]
	set yr[-30:30]
	set zr[0:60]
	set xl "x" font"Times:Italic, 22"
	set yl "y" font"Times:Italic, 22"
	set zl "z" font"Times:Italic, 22"
	set view 45,30,1,2
	DATA = "lorenz.data"
	set print DATA

	#=================== Parameters ====================
	dt　= 0.01 # Time step　[s]
	dh　= dt/6.

	sigma　= 10
	b　= 8./3.
	R　= 28

	n_max　= 100*150
	n　= 0
	cut　= 10 #　Number of frames: n_max/cut
	cnt　= 0

	#=================== Functions ====================
	f1(x, y, z) = -sigma*(x-y)
	f2(x, y, z) = Rx-y-xz
	f3(x, y, z) = xy-bz

	rk4x(x, y, zw) = (k1 = f1(x, y, z),\
	k2 = f1(x + dtk1/2., y + dtk1/2., z + dt*k1/2.),\
	k3 = f1(x + dtk1/2., y + dtk1/2., z + dt*k1/2.),\
	k4 = f1(x + dtk3, y + dtk3, z + dt*k3),\
	dh * (k1 + 2k2 + 2k3 + k4))
	rk4y(x, y, z) = (k1 = f2(x, y, z),\
	k2 = f2(x + dtk1/2., y + dtk1/2., z + dt*k1/2.),\
	k3 = f2(x + dtk1/2., y + dtk1/2., z + dt*k1/2.),\
	k4 = f2(x + dtk3, y + dtk3, z + dt*k3),\
	dh * (k1 + 2k2 + 2k3 + k4))
	rk4z(x, y, z) = (k1 = f3(x, y, z),\
	k2 = f3(x + dtk1/2., y + dtk1/2., z + dt*k1/2.),\
	k3 = f3(x + dtk1/2., y + dtk1/2., z + dt*k1/2.),\
	k4 = f3(x + dtk3, y + dtk3, z + dt*k3),\
	dh * (k1 + 2k2 + 2k3 + k4))

	# Initial value
	t　= 0.
	x　= 0.
	y　= 0.3
	z　= 0.

	call "make.plt"

view raw lorenz.plt hosted with ❤ by GitHub

make.plt

	# Ordinary differential equations
	x = x + rk4x(x, y, z)
	y = y + rk4y(x, y, z)
	z = z + rk4z(x, y, z)

	print x, y, z # Write initial values on DATA file

	if(n%cut==0){
	cnt = cnt + 1
	filename = sprintf("png/img_%04d.png",cnt)
	time = sprintf("{/Times:Italic t} = %4.2f",t)
	set title time font 'Times,18'
	set output filename
	splot DATA using 1:2:3 with line linecolor rgb "red"
	}
	t = t + dt
	n = n+1

	if(n <= n_max) reread
	n = 0

	set out

view raw make.plt hosted with ❤ by GitHub

lorenz.gif

Rössler equation

$\begin{eqnarray*} \frac{\mathrm{d}x}{\mathrm{d}t}&=&-y-z \\ \frac{\mathrm{d}y}{\mathrm{d}t}&=&z+ay\\ \frac{\mathrm{d}z}{\mathrm{d}t}&=&b+xz-cz \end{eqnarray*}$

$(a=0.3,~b=0.3,~c=5.7)$

Langford equation

$\begin{eqnarray*} \frac{\mathrm{d}x}{\mathrm{d}t}&=&(z-\beta)x-\omega y \\ \frac{\mathrm{d}y}{\mathrm{d}t}&=&\omega x+(z-\beta)y\\ \frac{\mathrm{d}z}{\mathrm{d}t}&=&\lambda+\alpha z-\frac{z^3}{3}-(x^2+y^2)(1+\rho z)+\varepsilon zx^3 \end{eqnarray*}$

$(\alpha=1,~\omega=3.5,~\beta=0.7,~\rho=0.25,~\lambda=0.6,~\varepsilon=0)$