The Onset of Chaos - Double Pendulum

Equations of Motion

$$\begin{aligned} \ddot{\theta}_1 &= \frac{-g(2M_1+M_2)\sin\theta_1 - M_2 g \sin(\theta_1-2\theta_2) - 2\sin(\theta_1-\theta_2)M_2(\dot{\theta}_2^2 L_2 + \dot{\theta}_1^2 L_1 \cos(\theta_1-\theta_2))}{L_1(2M_1+M_2-M_2\cos(2\theta_1-2\theta_2))} \\[10pt] \ddot{\theta}_2 &= \frac{2\sin(\theta_1-\theta_2)(\dot{\theta}_1^2 L_1(M_1+M_2) + g(M_1+M_2)\cos\theta_1 + \dot{\theta}_2^2 L_2 M_2 \cos(\theta_1-\theta_2))}{L_2(2M_1+M_2-M_2\cos(2\theta_1-2\theta_2))} \end{aligned}$$

Onset of Chaos

Base Angle $\theta_1$ Initial Position 45.0°

Base Angle $\theta_2$ Initial Position 90.0°

Separation $\Delta\theta$ Chaos Sensitivity 0.100°

Mass $M_1$ 1.5

Mass $M_2$ 1.5

Gravitational $g$ 9.81

Trace Length $N$ 4000

Time Scale 1.0