How to graph the second derivatives of coupled non-linear second order ODEs in Python?

How to graph the second derivatives of coupled non-linear second order ODEs in Python?

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How to graph the second derivatives of coupled non-linear second order ODEs in Python?
Tag : python , By : Jimmy
Date : November 28 2020, 04:01 AM

To fix the issue you can do The return value of odeint is the solution to z(t) which you have defined to be z = [x,y,x',y']. Therefore the second derivative is not a part of the solution returned by odeint. You can approximate the second derivative of x and y by taking finite differences of the returned values of the first derivatives.
For example:
import numpy as np
from scipy.integrate import odeint
from numpy import sin, cos, pi, array
import matplotlib.pyplot as plt

init = array([0,pi/18,0,0]) 

def deriv(z, t):
    x, y, dxdt, dydt = z

    return np.array([dxdt, dydt, dx2dt2, dy2dt2])

time = np.linspace(0.0,10.0,1000)
sol = odeint(deriv,init,time)

x, y, xp, yp = sol.T

# compute the approximate second order derivative by computing the finite
# difference between values of the first derivatives
xpp = np.diff(xp)/np.diff(time)
ypp = np.diff(yp)/np.diff(time)

# the second order derivatives are now calculated at the midpoints of the
# initial time array, so we need to compute the midpoints to plot it
xpp_time = (time[1:] + time[:-1])/2

plt.plot(time, x, label='x')
plt.plot(time, y, label='y')
plt.plot(time, xp, label="x'")
plt.plot(time, yp, label="y'")
plt.plot(xpp_time, xpp, label="x''")
plt.plot(xpp_time, ypp, label="y''")
plt.plot(time, deriv(sol.T,time)[2], label="x''")
plt.plot(time, deriv(sol.T,time)[3], label="y''")

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Python complex coupled ODEs error

Tag : python , By : xie renhui
Date : March 29 2020, 07:55 AM
I hope this helps you . At the moment, I am trying to solve a system of coupled ODEs with complex terms. I am using scipy.integrate.ODE, I have successfully solved a previous problem involving a coupled ODE system with only real terms. In that case I used odeint, which is not suitable for the problem I am facing now. The system has 9 coupled ODEs, and this is my code: , This should solve this issue:
def resol(t, y, par):
    p1111, p1212, p2121, p1112, p1121, p1122, p1221, p1222, p2122 = y
    gamma, h, H = par

Solving two coupled ODEs by matrix form in Python

Tag : python , By : Mihai Mocanu
Date : March 29 2020, 07:55 AM
like below fixes the issue The initial value to odeint must be an array, not a matrix. Try use y0=np.hstack((y_init, m_init)) and put that as the initial value (y0 is the second argument to odeint).

python two coupled second order ODEs Runge Kutta 4th order

Tag : python , By : Marcos de Carvalho
Date : March 29 2020, 07:55 AM
fixed the issue. Will look into that further Cutting away unused and contradicting variables and statements a running reduction and correction of your code is
import numpy as np
from math import sqrt

import matplotlib.pyplot as plt

# Parameters
mr = 2
m = 1
c = 5
k = 36
F = 3

# Equations:

def V(u,t):
    x1, x2, v1, v2 = u
    dx = x2-x1;
    dv = v2-v1;

    return np.array([ v1, v2, (c*dv+k*dx)/mr, -(F+c*dv+k*dx)/m ])

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] + 0.5*h)
        u[i+1] = u[i] + (k1 + 2*(k2 + k3) + k4) / 6
    return u, t

u, t  = rk4(V, np.array([0., 0., 1., 1.]) , 0. , 1. , 10)
x1, x2, v1, v2 = u.T
plt.plot(t, x1, t, x2)

How do I convert the x and y values in polar form from these coupled ODEs to to cartesian form and graph them?

Tag : python , By : Lee KW
Date : March 29 2020, 07:55 AM
this one helps. The first solution should give you the expected result, but there is a mistake in the implementation of the ode.
The function you pass to odeint should return an array containing the solutions of a 1st-order differential equations system.
import numpy as np
from scipy.integrate import odeint
from numpy import sin, cos, pi, array
import matplotlib.pyplot as plt

def deriv(z, t):

    x, y, dxdt, dydt = z

    dx2dt2 = (0.415 + x) * (dydt)**2 - 50 / 1.006 * x + 9.81 * cos(y)
    dy2dt2 = (-9.81 * 1.006 * sin(y) - 2 * (dxdt) * (dydt)) / (0.415 + x)

    return np.array([dxdt, dydt, dx2dt2, dy2dt2])

init = array([0, pi / 18, 0, 0])
time = np.linspace(0.0, 10.0, 1000)
sol = odeint(deriv, init, time)

plt.plot(sol[:, 0], sol[:, 1], label='hi')
x = sol[:,0]
y = sol[:,1]

def plot(h):
    x, y = h
    n = (0.4 + x) * sin(y)
    u = (0.4 + x) * cos(y)
    return np.array([n, u])

n,u = plot( (x,y))

Using python built-in functions for coupled ODEs

Tag : python , By : ArdentRogue
Date : March 29 2020, 07:55 AM
To fix this issue You can wrap your existing functions into a function accepted by odeint (option tfirst=True) and solve_ivp as
def odesys(t,u):
    theta,omega = u[:n],u[n:]; # or = u.reshape(2,-1);
    return [ *f(omega), *g(theta,omega) ]; # or np.concatenate([f(omega), g(theta,omega)])

u0 = [*theta0, *omega0]
t = linspan(t0, tf, timesteps+1);
u = odeint(odesys, u0, t, tfirst=True);


res = solve_ivp(odesys, [t0,tf], u0, t_eval=t)
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