import numpy as np import matplotlib. pyplot as plt from scipy. integrate import odeint # Define the system of differential equations def system(y, t): [z,p] = y dzdt= 2 * z - z* 2- zp dpdt = p**2 p * (z2) return [dzdt, dpdt] # Function to calculate the Jacobian matrix at an equilibrium point def jacobian_matrix (point): z, p = point return np.array ( [ [2 - 2+ z-p, -z], [-2 pz, 2 p - z2]]) # Equilibrium points equilibrium_points = [(0, 0), (1, 1), (2, 0), (-2, 4)] # Set up the plot fig, axs plt.subplots (2, 2, figsize=(10, 8)) fig.suptitle('Phase Portraits for Linearized Solutions') # Plot linearized solutions at each equilibrium point for i, point in enumerate(equilibrium_points): # Calculate the Jacobian matrix at the equilibrium point A = jacobian_matrix (point) # Eigenvalues and eigenvectors eigenvalues, eigenvectors = np. linalg.eig (A) # Initial conditions for the linearized system ye = np.concatenate ([point, eigenvectors[:, 0]]) # Time points t = np. linspace(0, 5, 100) # Solve the linearized system linearized_solution = odeint (system, ye, t) #Plot the phase portrait axs[i // 2, 1 % 2].plot(linearized_solution[:, 0], linearized_solution[:, 1]) axs [i // 2, 1 % 2].set_title(f'Equilibrium Point: {point}') #Set common labels for ax in axs.flat: ax.set (xlabel='z', ylabel='p') # Adjust layout plt.tight_layout(rect= [0, 0.03, 1, 0.95]) # Show the plot plt.show()

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 70EQ

See similar textbooks

Related questions

Question

what is wrong with the python code? Trying to plot phase portraits for the linearized systems for each equilibrium point of a non-linear system of differential equations, but command line 7 defining the system -- [z,p]=y --- gives me the error: too many values to unpack (expected 2)