Quantitatively and Visually Analyzing Chaos in the Double Pendulum

Overview

Chaos, discovered in the 1960s by Edward N. Lorenz, has a broad variety of applications from describing many systems in our daily life to analyzing systems such as electrical pulses in the heart and concepts in quantum mechanics. Chaos describes a minuscule change from initial conditions of a system that results in a largely different outcome. A well-known example of chaos theory is the butterfly effect, where a butterfly that flaps its wings leads to the formation of a tornado. Chaotic systems, however, can be difficult to perceive. This project aims to quantify the chaos of a double pendulum and use visual models to make it accessible for non-experts. Three different numerical integrators were tested to simulate a double pendulum: first-order Euler method, second-order Runge-Kutta (RK2) and fourth-order Runge-Kutta (RK4). Based on our data, RK4 was the most accurate and was chosen to simulate the double pendulum. The divergence plot, Poincaré section, chaos map, and Lyapunov exponent clearly show the chaotic nature of the system. Specifically, the Lyapunov exponent quantifies a system’s chaos, and represents it as a single value, 1.286 for this system. The positive Lyapunov exponent defines the system as chaotic. Additionally, the chaos map provides an intuitive visualization of the double pendulum's unpredictability. This project was successful in meeting our objective by clearly displaying the chaos of a double pendulum and showing that chaos can be deterministic, quantifiable and most importantly, understandable.

Competition history

• CSEF 2026 Physics & Astronomy (Senior Division) · Entry S-17-16

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