Duffing Oscillator¶

Overview¶

The Duffing oscillator is a nonlinear second-order differential equation used to model systems with a restoring force that is not purely linear. It has applications in physics, engineering, and even biology.

The equation is:

Where: - \(x\) is the displacement. - \(\delta\) is the damping coefficient. - \(\alpha\) is the linear stiffness. - \(\beta\) is the nonlinear stiffness. - \(\gamma\) is the amplitude of the driving force. - \(\omega\) is the angular frequency of the driving force.

Python Code Example: Simulating and Visualizing the Duffing Oscillator¶

import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import solve_ivp

# Define the Duffing oscillator
def duffing(t, state, delta, alpha, beta, gamma, omega):
    x, dx_dt = state
    ddx_dt = -delta * dx_dt - alpha * x - beta * x**3 + gamma * np.cos(omega * t)
    return [dx_dt, ddx_dt]

# Parameters
delta = 0.2  # Damping coefficient
alpha = -1.0  # Linear stiffness
beta = 1.0  # Nonlinear stiffness
gamma = 0.3  # Driving force amplitude
omega = 1.2  # Driving force frequency

# Initial conditions
initial_state = [0.1, 0.0]  # Initial displacement and velocity

# Time span for the simulation
time_span = (0, 100)
time_eval = np.linspace(time_span[0], time_span[1], 10000)

# Solve the Duffing oscillator
solution = solve_ivp(duffing, time_span, initial_state, args=(delta, alpha, beta, gamma, omega), t_eval=time_eval, method='RK45')

# Extract the results
x, dx_dt = solution.y

# Plot the displacement over time
plt.figure(figsize=(12, 6))
plt.plot(solution.t, x, label="Displacement", color="blue")
plt.title("Duffing Oscillator: Displacement Over Time")
plt.xlabel("Time")
plt.ylabel("Displacement")
plt.grid()
plt.legend()
plt.savefig("docs/Mechanics/pic/duffing_time.png")
plt.show()

# Plot the phase plane
plt.figure(figsize=(8, 8))
plt.plot(x, dx_dt, color="purple")
plt.title("Phase Plane of the Duffing Oscillator")
plt.xlabel("Displacement")
plt.ylabel("Velocity")
plt.grid()
plt.savefig("docs/Mechanics/pic/duffing_phase.png")
plt.show()

Key Insights¶

1. Nonlinear Dynamics: The cubic term introduces nonlinearity, leading to rich dynamics, including bifurcations and chaos.

2. Phase Space Behavior: The phase plane can reveal limit cycles, chaotic attractors, or other patterns depending on parameters.

3. Driven and Damped System: The interplay of damping, driving force, and nonlinearity determines the behavior of the oscillator.

Suggested Projects¶

1. Bifurcation Analysis: Vary parameters like \(\gamma\) or \(\omega\) to explore transitions between different dynamical regimes.

2. Lyapunov Exponent: Compute the Lyapunov exponent to identify chaotic regions.

3. Comparison with Real Systems: Model physical systems like beam vibrations or electronic circuits.

4. Energy Analysis: Study how energy evolves in the system over time, especially in chaotic regimes.

This example demonstrates the nonlinear dynamics of the Duffing oscillator. Experiment with parameters and initial conditions to uncover its fascinating behavior!