Duffing Oscillator

Duffing Oscillator

The Duffing oscillator is a driven nonlinear oscillator modelling a mass in a double-well potential with damping and periodic forcing:

$\dot{x} = y, \quad \dot{y} = -\delta y - \alpha x - \beta x^3 + \gamma \cos(\omega t)$

With parameters δ = 0.3, α = −1, β = 1, γ = 0.5, ω = 1.2, the system exhibits a rich variety of behaviour: periodic orbits, period-doubling cascades, and fully chaotic attractors depending on the forcing amplitude γ.

The term −αx − βx³ creates a double-well potential (two stable equilibria). The periodic driving can kick the system from one well to the other chaotically — a model for physical phenomena from buckled beams to certain electronic circuits.

Implement the following functions:

• duffing_deriv(x, y, t, delta, alpha, beta, gamma, omega) — compute the derivatives (ẋ, ẏ) at time t
• duffing_euler(x0, y0, delta, alpha, beta, gamma, omega, dt, steps) — integrate using Euler's method, advancing t by dt each step, returning final (x, y)