Duffing oscillator is a nonlinear oscillator.

$\ddot{x} + \beta \dot{x} + x + \delta x^3 = \Gamma \cos{\omega t}$

Physical meaning

The oscillator corresponds to a nonlinear spring with either hardening for $\delta > 0$ or softening for $\delta < 0$ (for amplitude not too large, as then it's motion becomes unbounded).

Free (unforced) Duffing oscillator

Free undamped Duffing oscillator

The system can be integrated to obtain an energy, and the system is then a Hamiltonian system:

$E(t) \equiv \frac{1}{2} \dot{x}^2 + \frac{1}{2} x^2 + \frac{1}{4} \delta x^4 = \text{const}$

Free damped Duffing oscillator

When $\beta > 0$ , $E(t)$ satisfies:

$\frac{d E(t)}{dt} = - \beta \dot{x}^2 \leq 0$

One can easily show that this is indeed a Lyapunov function and the origin is globally asymptotically stable

Forced Duffing oscillator

More interesting. Nonlinear resonances. Shows chaotic behaviour, intermittency (jump phenomena), etc. See Lakes of Wada..

Nonlinear resonances

Treat with multiple scales method

Primary resonance

Secondary resonances

Subharmonic

Superharmonic

Onset of chaos

Period-doubling cascade

Reverse period doubling and reverse cascade (bubbles)

Intermittency

Lakes of Wada

Other?