Duffing Resonator — Phase Portrait & Poincaré Map (2D)

Drive & Damping

Drive frequency ω/ω₀ 0.800

Drive amplitude F₀ 1.00

Quality factor Q 80

Nonlinearity α +1.5

Frequency Sweep

Live Readouts

Amplitude |x|

0.00

ω/ω₀ now

0.800

Poincaré spread

–

Strobe count

Last jump detected

none yet — sweep ω slowly

How it works (2D model)

This is the same nanomechanical Duffing resonator as the 3D beam-bending version, but viewed the way a nonlinear-dynamics lab actually reads out a real device: not the physical shape of the beam, but its trajectory in phase space — position x against velocity v — sampled once per drive cycle. The governing equation is unchanged:

x'' + (ω₀/Q)·x' + ω₀²x + α·x³ = F₀·cos(ωt)

The left panel plots the fading trail of (x, v) as it's integrated (RK4, independent of the 3D engine). Once transients die out, a period-1 response traces a single closed loop every drive cycle; sampling (x, v) at t = 0, T, 2T, … (one drive period T = 2π/ω apart) — the Poincaré map — collapses that loop to a single point if the motion is periodic, or to several/many points if it isn't. The right strip is the scrolling waveform x(t) against the drive signal.

Poincaré spread reports the distance between the two most recent strobe points — it shrinks toward zero as the trajectory settles onto a period-1 orbit, and stays large while still transient or riding an unstable/quasi-periodic state.
α > 0 (stress-stiffening) bends the resonance toward higher frequency; α < 0 (softening) bends it toward lower frequency — past a critical drive amplitude the response becomes bistable, and sweeping ω up vs down makes the phase portrait's steady loop jump abruptly between the two amplitude branches at different frequencies (hysteresis).
Use Auto sweep to ramp ω slowly through resonance and watch the trail and Poincaré point jump size between orbits when the branch flips.

Real-world relevance: a Poincaré/stroboscopic map is exactly how a real NEMS lab distinguishes a clean single-tone response from period-doubling or noise-driven jitter in a resonant mass sensor — a single tight cluster of strobe points means trustworthy readout; a smeared cloud means the sensor has left its well-behaved operating regime.

Trajectory trail Current state Poincaré points x(t) waveform