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Explore deterministic chaos, Lyapunov exponent divergence, strange attractors, and 50-pendulum optical wave harmonics in real-time 60 FPS.
Numerical integration, butterfly divergence, and phase space analysis
Calculates coupled non-linear Euler-Lagrange equations of motion at 60 FPS with micro-substep numerical precision and exact Hamiltonian conservation.
Launch two pendulums with a microscopic 0.0001 radian difference to witness sensitive dependence on initial conditions and exponential trajectory divergence.
Explore standing wave interference, traveling serpentine modes, and chromatic phase alignment with up to 50 mathematically tuned pendulums.
The motion of coupled pendulums is governed by L = T - V, yielding two coupled second-order non-linear differential equations.
A positive Lyapunov exponent indicates that nearby trajectories separate at an exponential rate, the definitive hallmark of chaos.
Tuning length to integer oscillation counts creates standing wave nodes, traveling serpentine beats, and complete periodic reconvergence.
Everything you need to know about Chaos Pendulum Lab
The double pendulum is a classic example of deterministic chaos. While governed by exact Lagrangian equations of motion, its non-linear coupled dynamics exhibit positive Lyapunov exponents, making long-term trajectory prediction impossible without infinite precision.
In Twin Perturbation mode, two identical double pendulums are released with an initial angle difference of just 0.0001 radians (0.0057°). You can watch them track identically for a few seconds before suddenly diverging into completely distinct, uncorrelated trajectories.
A pendulum wave consists of 10 to 50 independent pendulums whose lengths are tuned according to L_i = g * (T / (2*pi*(N + i)))^2. In a period T (e.g. 60 seconds), the i-th pendulum completes N + i oscillations, creating mesmerizing traveling wave patterns.
Deepen your understanding of classical mechanics, fluid dynamics, and orbital physics.
Map real-time angle versus angular velocity ($ heta, omega$) state-space portraits to discover strange attractors and quasi-periodic limit cycles.
The simulation uses the 4th-Order Runge-Kutta (RK4) numerical integration algorithm with multiple sub-steps per frame, preventing numerical energy drift and maintaining Hamiltonian total energy conservation.
Phase space plots the state of a dynamical system by graphing generalized coordinates (such as angle theta) against generalized momenta (angular velocity omega). Chaotic systems trace intricate, fractal-like 'strange attractors' in phase space.
Lower gravity (like on the Moon, g = 1.62 m/s^2) slows oscillation frequencies and delays the onset of high-speed chaotic flips. Higher gravity (Jupiter, g = 24.79 m/s^2) accelerates energy exchange, triggering rapid chaotic loops.
Total mechanical energy E is the sum of rotational kinetic energy T and gravitational potential energy V. In an undamped conservative system (damping = 0), E remains strictly constant throughout the entire simulation.
Yes! You can download a time-series CSV log containing timestamps, theta1, theta2, omega1, omega2, energy, and Lyapunov divergence distances, as well as high-resolution PNG trajectory snapshots.
At high initial energy levels (angles > 60°), the double pendulum enters a non-periodic chaotic regime where trajectories wander ergodically through phase space without ever exactly retracing the same path.
No installations are required. The laboratory executes entirely in your web browser at 60 FPS using HTML5 Canvas 2D and modern JavaScript numerical math.
Adding damping (gamma > 0) removes kinetic energy over time. The system gradually decays from chaos into regular damped harmonic oscillations before settling at the stable downward equilibrium point (0, 0).
Yes! It is designed as an interactive laboratory for classical mechanics, Lagrangian dynamics, non-linear differential equations, and introductory chaos theory.