Duffing Oscillator - Interactive Visualization

Explore nonlinear dynamics, chaos theory, and the rich behavior of the Duffing oscillator with real-time phase portraits, Poincaré sections, and potential energy visualization

Duffing Equation:

ÿ + δẏ + αy + βy³ = γ cos(ωt)

Time Domain

Time (t) Position (x)

Phase Portrait

Position (x) Velocity (ẋ)

Poincaré Section

Position (x) Velocity (ẋ)

Potential Energy

Position (x) V(x)
Kinetic Energy: 0.000
Potential Energy: 0.000
Total Energy: 0.000
Max Position: 0.000
Max Velocity: 0.000
Simulation Time: 0.00

System Parameters

Initial Conditions

Simulation Settings

Presets

Visualization Options

Theory and Background

Overview

The Duffing oscillator is a classic example of a nonlinear dynamical system that exhibits a rich variety of behaviors, including periodic motion, period doubling, and chaos. It models a damped, driven oscillator with a nonlinear restoring force.

The Equation

The equation is: ÿ + δẏ + αy + βy³ = γ cos(ωt), where δ is damping, α and β are linear and nonlinear stiffness coefficients, γ is drive amplitude, and ω is drive frequency.

Double-Well Potential

When α < 0 and β > 0, the system has a double-well potential with two stable equilibrium points. The particle can oscillate in one well or jump between wells, leading to complex dynamics.

Chaos and Sensitivity

For certain parameter values, the system exhibits chaotic behavior characterized by sensitivity to initial conditions, strange attractors in phase space, and a broad power spectrum.

Parameter Guide

Damping (δ)

Controls energy dissipation. Higher values lead to faster decay of oscillations. δ = 0 gives conservative motion.

Linear Coefficient (α)

Determines potential shape. α > 0: single well (hard spring). α < 0: double well with two stable equilibria.

Nonlinear Coefficient (β)

Controls strength of cubic nonlinearity. β > 0 gives hardening spring effect, β < 0 gives softening.

Drive Amplitude (γ)

Strength of periodic driving force. Increasing γ can lead to period doubling and transition to chaos.

Drive Frequency (ω)

Frequency of periodic driving. Resonance occurs near natural frequency, leading to large amplitude oscillations.

Visualization Guide

Time Domain Plot

Shows position x(t) over time. Periodic motion shows repeating patterns, while chaos appears irregular and unpredictable.

Phase Portrait

Plots velocity vs position. Closed loops indicate periodic motion. Strange attractors with fractal structure indicate chaos.

Poincaré Section

Samples the state once per drive period. Periodic motion shows discrete points. Chaos shows fractal-like point distributions.

Potential Energy

Shows the potential energy surface V(x) = -½αx² + ¼βx⁴. Double wells have two minima. Single wells have one minimum.

Applications
  • Mechanical vibrations and structural engineering
  • Nonlinear electrical circuits
  • Biological oscillators and neural systems
  • Climate dynamics and population models
  • Quantum mechanics analogies