Engineering Math - Chaos Theory

  

 

 

 

Duffing Oscillator

 

Duffing oscillator is a classic example used to demonstrate chaotic behavior and dynamics. It is represented in mathematical form as follows (the details of the form and symbols may vary depending on the documents, but overall form would be similar to this).

 

 

The meaning of each terms and coefficients are as follows :

    x = displacement

    x' = velocity

    x'' = acceleration

    δ = damping coefficient

    ω0 = natural frequency of oscillation

    β = nonlinearity coefficient

    γ = forcing amplitude

    ω = forcing frequency

    φ = phase shift of forcing function

 

The characteristics of this equation can be summarized as follows :

  • The x3 term introduces nonlinearity which enables chaotic dynamics.
  • The system exhibits chaotic, bounded oscillations for certain parameter ranges.
  • Varying the forcing frequency ω relative to the natural frequency ω0 impacts the behavior.
  • The oscillations switch between periodic and chaotic as parameters change.
  • Slight changes in initial conditions yield dramatically different trajectories over time.
  • The phase shift φ of the external forcing alters when chaotic regimes emerge.
  • Tools like phase portraits and Poincaré sections reveal the underlying attractor.

 

You can plot out the solution of this equation using a simple script below. Play with parameters and initial conditions and see how the plot changes.

 

 

Matlab

    %Save the following contents in a .m file and run the .m file

    % this is tested only in Matlab, not in Octave

    delta = 0.06;

    beta = 1.0;

    w0 = 1.0;

    w = 1.0;

    gamma = 6.0;

    phi = 0;

     

    dy_dt = @(t,y) [y(2);...

                    -delta*y(2)-(beta*y(1)^3 + w0^2*y(1))+gamma*cos(w*t+phi)];

     

    odeopt = odeset ('RelTol', 0.00001, 'AbsTol', 0.00001,'InitialStep',0.5,'MaxStep',0.5);

    [t,y] = ode45(dy_dt,[0 100], [3.0 4.1],odeopt);

    subplot(1,3,[1 2]);plot(t,y(:,1),'r-',t,y(:,2),'g-'); xlabel('time'); legend('y(1)','y(2)');

    subplot(1,3,3);plot(y(:,1),y(:,2)); xlabel('y(1)'); ylabel('y(2)');

 

 

Python

    # make it sure that you installed all these packages

    import numpy as np

    from scipy.integrate import odeint

    import matplotlib.pyplot as plt

     

    # Parameters

    delta = 0.06

    beta = 1.0

    w0 = 1.0

    w = 1.0

    gamma = 6.0

    phi = 0

     

    # Derivatives function  

    def duffing(y, t):

     

        y1, y2 = y

        dydt = [y2,

               -delta*y2 - beta*y1**3 + w0**2*y1 + gamma*np.cos(w*t + phi)]

        return dydt

     

    # Initial conditions

    y0 = [3.0, 4.1]

     

    # Integrate  

    t = np.linspace(0, 100, 3000)

    sol = odeint(duffing, y0, t)

     

    # Plot

     

    fig = plt.figure(figsize=(12, 4))

     

    ax1 = plt.subplot2grid((1, 3), (0, 0), colspan=2)

    ax1.plot(t, sol[:,0], 'r-', t, sol[:,1], 'g-')

    ax1.set_xlabel('Time')

    ax1.legend(['y(1)','y(2)'])

     

    ax2 = plt.subplot2grid((1, 3), (0, 2))

    ax2.plot(sol[:,0], sol[:,1])

    ax2.set_xlabel('y(1)')

    ax2.set_ylabel('y(2)')

     

    plt.tight_layout()

    plt.show()