Duffingoscillator

The Duffing oscillator is a prototypical nonlinear driven damped oscillator used to model nonlinear oscillatory systems. It is commonly described by the second-order differential equation x'' + δ x' + α x + β x^3 = γ cos(ω t), where x(t) is the displacement, δ > 0 is the linear damping coefficient, α and β determine the linear and cubic stiffness, and γ and ω are the amplitude and angular frequency of the external drive.

Depending on the signs and magnitudes of α and β, the conservative part of the system can have a single-well or double-well effective potential. The cubic term introduces either hardening or softening of the resonance as the drive amplitude grows; β > 0 typically yields hardening behavior, while β < 0 yields softening behavior.

Under periodic forcing and damping, the Duffing oscillator exhibits a wide range of dynamical phenomena, including nonlinear resonance, jump phenomena and hysteresis, period-doubling bifurcations, quasi-periodicity, and, for suitable parameters, chaotic motion with broadband power spectra and positive largest Lyapunov exponent.

Analysis commonly uses numerical integration, phase-space plots, Poincaré sections, and bifurcation diagrams to map parameter regimes and transitions between behaviors.

Applications span mechanical systems with nonlinear stiffness (beams, cantilevers), electrical circuits with nonlinear inductance or capacitance, and micro- and nanoelectromechanical systems, as well as signal processing and energy harvesting research.

Historical note: the model is named after Georg Duffing, who studied nonlinear oscillations in the early 20th century; it remains a canonical model in nonlinear dynamics and chaos theory.