RKF 1.0.6

Implementation of Runge-Kutta Fehlberg method to numerically solve system of ODEs.

RKF

Implementation of Runge-Kutta Fehlberg method to numerically solve system of ODEs.

Developed by Amirhossein Rezaei.

The Lorenz system

The Lorenz system is a system of ordinary differential equations first studied by Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the Lorenz attractor is a set of chaotic solutions of the Lorenz system. In popular media the 'butterfly effect' stems from the real-world implications of the Lorenz attractor, i.e. that in any physical system, in the absence of perfect knowledge of the initial conditions (even the minuscule disturbance of the air due to a butterfly flapping its wings), our ability to predict its future course will always fail. This underscores that physical systems can be completely deterministic and yet still be inherently unpredictable even in the absence of quantum effects. The shape of the Lorenz attractor itself, when plotted graphically, may also be seen to resemble a butterfly.

Overview

In 1963, Edward Lorenz, with the help of Ellen Fetter, developed a simplified mathematical model for atmospheric convection. The model is a system of three ordinary differential equations now known as the Lorenz equations:

The equations relate the properties of a two-dimensional fluid layer uniformly warmed from below and cooled from above. In particular, the equations describe the rate of change of three quantities with respect to time: is proportional to the rate of convection, to the horizontal temperature variation, and to the vertical temperature variation. The constants , , and are system parameters proportional to the Prandtl number, Rayleigh number, and certain physical dimensions of the layer itself.

The Lorenz equations also arise in simplified models for lasers, dynamos, thermosyphons, brushless DC motors, electric circuits, chemical reactions and forward osmosis. The Lorenz equations are also the governing equations in Fourier space for the Malkus waterwheel. The Malkus waterwheel exhibits chaotic motion where instead of spinning in one direction at a constant speed, its rotation will speed up, slow down, stop, change directions, and oscillate back and forth between combinations of such behaviors in an unpredictable manner.

From a technical standpoint, the Lorenz system is nonlinear, non-periodic, three-dimensional and deterministic. The Lorenz equations have been the subject of hundreds of research articles, and at least one book-length study.

Analysis and code

One normally assumes that the parameters , , and are positive. Lorenz used the values , and . The system exhibits chaotic behavior for these (and nearby) values. Further explanation on wiki.

import numpy as np
import matplotlib.pyplot as plt 
import time
from RKF import rkf
import matplotlib.animation as animation

def lorenz(t,u):
    s=10
    r=24
    b=8/3
    x,y,z=u
    vx=s*y-s*x
    vy=r*x-x*z-y
    vz=x*y-b*z
    return np.array([vx,vy,vz])

x0=[2,2,2]

t,u  = rkf( f=lorenz, a=0, b=1e+1, x0=x0, atol=1e-8, rtol=1e-6 , hmax=1e-1, hmin=1e-40,plot_stepsize=True).solve()

x,y,z= u.T

plt.style.use('dark_background')
fig = plt.figure()
ax = fig.gca(projection='3d')
ax.set_axis_off()
ax.plot(x,y,z,lw=0.5,c='whitesmoke')
# plt.show()

def rotate(angle):
    ax.view_init(elev=7.,azim=angle)

print("Making animation")
rot_animation = animation.FuncAnimation(fig, rotate, frames=np.arange(0, 600, 2), interval=36)
rot_animation.save('lorenz.gif', dpi=400, writer='imagemagick')