Butterfly Effect

Inspired by 2swap: https://www.youtube.com/watch?v=dtjb2OhEQcU

A project never hurts from some design; Dieter Rams and his thoughts on ‘good design’ served as the artistic guidance!

It’s pretty boggling (and invaluable in a way) to think that the tiniest changes in the past could have meant such a varying future. If we hadn’t slept in one day, or had a particular saturday-impulse to go out for some coffee, who knows if we would’ve met some of our closest people.

Likewise, double pendulums are pretty unique since they showcase the butterfly effect (how tiny changes in initial settings produce wildly disparate motions).

In a different lens, as in economics, they suggest how difficult it really is in forecasting markets for longer durations.

Kinematic equations:

\begin{aligned} x_1 &= L_1\sin(\theta_1) &\qquad \dot{x}_1 &= \dot{\theta}_1L_1\cos(\theta_1) \\[0.6em] y_1 &= -L_1\cos(\theta_1) & \dot{y}_1 &= \dot{\theta}_1L_1\sin(\theta_1) \\[0.6em] x_2 &= L_1\sin(\theta_1)+L_2\sin(\theta_2) & \dot{y}_2 &= \dot{\theta}_1L_1\sin(\theta_1) + \dot{\theta}_2L_2\sin(\theta_2) \\[0.6em] y_2 &= -L_1\cos(\theta_1)-L_2\cos(\theta_2) & \dot{x}_2 &= \dot{\theta}_1L_1\cos(\theta_1) + \dot{\theta}_2L_2\cos(\theta_2) \end{aligned}

Lagrangian:

\mathcal{L}=T-V

\begin{aligned} \mathcal{L} ={}& \frac{1}{2} M_1\dot{\theta}_1^{\,2}L_1^{\,2} \\[0.4em] &+ \frac{1}{2}M_2 \left[ \dot{\theta}_1^{\,2}L_1^{\,2} + \dot{\theta}_2^{\,2}L_2^{\,2} + 2\dot{\theta}_1\dot{\theta}_2 L_1L_2\cos(\theta_1-\theta_2) \right] \\[0.4em] &+ M_1gL_1\cos\theta_1 + M_2g \left( L_1\cos\theta_1 + L_2\cos\theta_2 \right) \end{aligned}

Euler-Lagrangian & principle of least action:

\frac{\partial\mathcal{L}}{\partial\theta} - \frac{d}{dt} \left( \frac{\partial\mathcal{L}} {\partial\dot{\theta}} \right) = 0

S=\int L\,dt

Final equations:

\begin{aligned} &(M_1+M_2)g\sin\theta_1 + (M_1+M_2)L_1\ddot{\theta}_1 \\[0.35em] &\qquad + M_2L_2\ddot{\theta}_2 \cos(\theta_1-\theta_2) + M_2L_2\dot{\theta}_2^{\,2} \sin(\theta_1-\theta_2) = 0 \end{aligned}

\dot{\theta}_1^{\,2}L_1 \sin(\theta_1-\theta_2) - g\sin\theta_2 - \ddot{\theta}_2L_2 - \ddot{\theta}_1L_1 \cos(\theta_1-\theta_2) = 0

Now, simulating the motion on the computer:

\(\mathbf{s}=(\theta_1,\theta_2,\omega_1,\omega_2)\)

\begin{aligned} k_1 &= f(t_n,\mathbf{s}_n), \\[0.5em] k_2 &= f\left( t_n+\frac{h}{2}, \mathbf{s}_n+\frac{h}{2}k_1 \right), \\[0.5em] k_3 &= f\left( t_n+\frac{h}{2}, \mathbf{s}_n+\frac{h}{2}k_2 \right), \\[0.5em] k_4 &= f\left( t_n+h, \mathbf{s}_n+hk_3 \right). \end{aligned}

\boxed{ \mathbf{s}_{n+1} = \mathbf{s}_n + \frac{h}{6} \left( k_1+2k_2+2k_3+k_4 \right) }

The maps are pretty fun in that thousands of simulated pendulums make up the background of the screen, based on different presettings of thetas and omegas.

Colorings (from black to blue/green to white) are based on the Lyapunov exponent (how chaotic a pendulum is), so clicking on dark “islands of stability” will produce more familiar and orderly patterns.