# Duffing Oscillator

Duffing Oscillator

The Duffing oscillator moves in a double well potential, sometimes characterized as nonlinear elasticity, with sinusoidal external forcing. It is described by the equation x+γ-x+ϵ=Γcos(Ωt). We consider the parameters , , , , , and =0. Solutions to the oscillator equation can exhibit extreme nonlinear dynamics, including limit cycles, strange attractors, and chaotic behavior. The system is, as expected, highly sensitive to the initial conditions.

2

d

d

2

t

dx

dt

2

ω

3

x

γ=0.1

ϵ=0.25

ω=1

Ω=2

x(0)=1

dx

dt

t=0

When the periodic force () that drives the system is large, the motion can become chaotic and the phase space diagram can develop a strange attractor. A Poincaré section can be plotted by taking one phase space point in each period of the driving force. In the simplest cases, when the system enters a limit cycle, the Poincaré section reduces to a single point. A strange attractor is usually associated with a complicated fractal curve.

Γ