Computational Science Asked by Anirban Majumdar on March 1, 2021
I want to study the dynamics of a 2d linear flow, whose dynamical equation is- $begin{pmatrix} dot{x_1} dot{x_2} end{pmatrix}=begin{pmatrix} 1 & 1 4 & -2 end{pmatrix}begin{pmatrix} x_1 x_2 end{pmatrix}$. Now I have tried to solve and plot y vs. x of this coupled differential equation using RK4 in python for the initial condition $(y_0=2, x_0=-1)$. My code is following, but the graph is not correct [In the graph origin should be a saddle point, $begin{pmatrix} -0.25 1 end{pmatrix}$ axis should be stable manifold and $begin{pmatrix} 1 1 end{pmatrix}$ axis should be unstable manifold]-
import numpy as np
from math import sqrt
import matplotlib.pyplot as plt
# Equations:
def V(u,t):
x1, x2, v1, v2 = u
return np.array([ v1, v2, (x1+x2), -(4*x1-2*x2)])
def rk4(f, u0, t0, tf , n):
t = np.linspace(t0, tf, n+1)
u = np.array((n+1)*[u0])
h = t[1]-t[0]
for i in range(n):
k1 = h * f(u[i], t[i])
k2 = h * f(u[i] + 0.5 * k1, t[i] + 0.5*h)
k3 = h * f(u[i] + 0.5 * k2, t[i] + 0.5*h)
k4 = h * f(u[i] + k3, t[i] + h)
u[i+1] = u[i] + (k1 + 2*(k2 + k3) + k4) / 6
return u, t
u, t = rk4(V, np.array([2.0, 0., -1, 1.]) , 0. , 1. , 1000)
x1, x2, v1, v2 = u.T
plt.plot(x1,x2)
plt.show()
Can anyone helps me to write this code.
You are implementing the system $$ pmatrix{ddot x_1ddot x_2} = pmatrix{1&1-4&2} pmatrix{x_1x_2} $$ in its first-order version with $v_k=dot x_k$. It is not surprising that this different system results in different solutions.
For your stated system you would have to use
def V(u,t):
x1, x2 = u
return np.array([ (x1+x2), (4*x1-2*x2)])
The difference in state space dimension should be readily noticeable.
Correct answer by Lutz Lehmann on March 1, 2021
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