numerical methods - Solving the Lorentz model using Runge Kutta 4th Order in Python without a package

Question

I wish to solve the Lorentz model in Python without the help of a package and my codes seems not to work to my expectation. I do not know why I am not getting the expected results and Lorentz attractor. The main problem I guess is related to how to store the various values for the solution of x,y and z respectively.Below are my codes for the Runge-Kutta 45 for the Lorentz model with 3D plot of solutions:

import numpy as np
import matplotlib.pyplot as plt
#from scipy.integrate import odeint

#a) Defining the Runge-Kutta45 method

def fx(x,y,z,t):
    dxdt=sigma*(y-z)
    return dxdt

def fy(x,y,z,t):
    dydt=x*(rho-z)-y
    return dydt

def fz(x,y,z,t):
    dzdt=x*y-beta*z
    return dzdt


def RungeKutta45(x,y,z,fx,fy,fz,t,h):
    k1x,k1y,k1z=h*fx(x,y,z,t),h*fy(x,y,z,t),h*fz(x,y,z,t)
    k2x,k2y,k2z=h*fx(x+k1x/2,y+k1y/2,z+k1z/2,t+h/2),h*fy(x+k1x/2,y+k1y/2,z+k1z/2,t+h/2),h*fz(x+k1x/2,y+k1y/2,z+k1z/2,t+h/2)
    k3x,k3y,k3z=h*fx(x+k2x/2,y+k2y/2,z+k2z/2,t+h/2),h*fy(x+k2x/2,y+k2y/2,z+k2z/2,t+h/2),h*fz(x+k2x/2,y+k2y/2,z+k2z/2,t+h/2)
    k4x,k4y,k4z=h*fx(x+k3x,y+k3y,z+k3z,t+h),h*fy(x+k3x,y+k3y,z+k3z,t+h),h*fz(x+k3x,y+k3y,z+k3z,t+h)
    return x+(k1x+2*k2x+2*k3x+k4x)/6,y+(k1y+2*k2y+2*k3y+k4y)/6,z+(k1z+2*k2z+2*k3z+k4z)/6



sigma=10.
beta=8./3.
rho=28.
tIn=0.
tFin=10.
h=0.05
totalSteps=int(np.floor((tFin-tIn)/h))

t=np.zeros(totalSteps)
x=np.zeros(totalSteps) 
y=np.zeros(totalSteps) 
z=np.zeros(totalSteps)

for i in range(1, totalSteps):
    x[i-1]=1. #Initial condition
    y[i-1]=1. #Initial condition
    z[i-1]=1. #Initial condition
    t[0]=0. #Starting value of t
    t[i]=t[i-1]+h
    x,y,z=RungeKutta45(x,y,z,fx,fy,fz,t[i-1],h)


#Plotting solution

from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm

fig=plt.figure()
ax=fig.gca(projection='3d')
ax.plot(x,y,z,'r',label='Lorentz 3D Solution')
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('z')
ax.legend()

Answer 1

I changed the integration step (btw., classical 4th order Runge-Kutta, not the adaptive RK54) to use the python core concept of lists and list operations extensively to reduce the number of places where the computation is defined. There were no errors there to correct, but I think the algorithm itself is more concentrated.

You had an error in the system that changed it into a system that rapidly diverges. You had fx = sigma*(y-z) while the Lorentz system has fx = sigma*(y-x).

Next your main loop has some strange assignments. In every loop you first set the previous coordinates to the initial conditions and then replace the full arrays with the RK step applied to the full arrays. I replaced that completely, there are no small steps to a correct solution.

import numpy as np
import matplotlib.pyplot as plt
#from scipy.integrate import odeint


def fx(x,y,z,t): return sigma*(y-x)
def fy(x,y,z,t): return x*(rho-z)-y
def fz(x,y,z,t): return x*y-beta*z

#a) Defining the classical Runge-Kutta 4th order method

def RungeKutta45(x,y,z,fx,fy,fz,t,h):
    k1x,k1y,k1z = ( h*f(x,y,z,t) for f in (fx,fy,fz) )
    xs, ys,zs,ts = ( r+0.5*kr for r,kr in zip((x,y,z,t),(k1x,k1y,k1z,h)) )
    k2x,k2y,k2z = ( h*f(xs,ys,zs,ts) for f in (fx,fy,fz) )
    xs, ys,zs,ts = ( r+0.5*kr for r,kr in zip((x,y,z,t),(k2x,k2y,k2z,h)) )
    k3x,k3y,k3z = ( h*f(xs,ys,zs,ts) for f in (fx,fy,fz) )
    xs, ys,zs,ts = ( r+kr for r,kr in zip((x,y,z,t),(k3x,k3y,k3z,h)) )
    k4x,k4y,k4z  =( h*f(xs,ys,zs,ts) for f in (fx,fy,fz) )
    return (r+(k1r+2*k2r+2*k3r+k4r)/6 for r,k1r,k2r,k3r,k4r in 
            zip((x,y,z),(k1x,k1y,k1z),(k2x,k2y,k2z),(k3x,k3y,k3z),(k4x,k4y,k4z)))



sigma=10.
beta=8./3.
rho=28.
tIn=0.
tFin=10.
h=0.01
totalSteps=int(np.floor((tFin-tIn)/h))

t = totalSteps * [0.0]
x = totalSteps * [0.0]
y = totalSteps * [0.0]
z = totalSteps * [0.0]

x[0],y[0],z[0],t[0] = 1., 1., 1., 0.  #Initial condition
for i in range(1, totalSteps):
    x[i],y[i],z[i] = RungeKutta45(x[i-1],y[i-1],z[i-1], fx,fy,fz, t[i-1], h)

Using tFin = 40 and h=0.01 I get the image

looking like the typical image of the Lorentz attractor.