Nonlinear Dynamics & Chaos

Poincaré Section

Simulate a driven damped pendulum $\ddot{\theta} + b\dot{\theta} + \omega_0^2 \sin\theta = F_0 \cos(\omega_d t)$. Sampling phase space $(\theta, \omega)$ once per drive period reveals intricate fractal Poincaré strange attractors.

Drive Amplitude F₀ 1.20

Sampled Points 1450

Angular Speed ω 0.42 rad/s

Drive & Damping Parameters

Drive Force F₀ 1.20

Damping b 0.50

Drive Frequency ω_d 0.67

Visual Overlay

Show Live Pendulum Simulation

Show Continuous Phase Trajectory

Show Mass & Phase Space Badges

Controls

Driven Damped Pendulum Equation

Differential Equation: d²θ/dt² + b·dθ/dt + sin θ = F₀·cos(ω_d t)
Poincaré Sampling Condition: Sample (θ_k, ω_k) at times t_k = k · (2π / ω_d)

Developer Reference

Core Algorithm & Standalone Script

// Driven damped pendulum Poincaré section solver
const canvas = document.getElementById('canvas');
const ctx = canvas.getContext('2d');
// d2theta/dt2 + b * dtheta/dt + sin(theta) = F0 * cos(wd * t)