# Dynamical Systems and Chaos

## Back to the Double Pendulum!

A double pendulum consists of one pendulum attached to another. Double pendula are an example of a simple physical system which can exhibit chaotic behavior. Consider a double bob pendulum with masses m1 and m2 attached by rigid massless wires of lengths l1 and l2. Further, let the angles the two wires make with the vertical be denoted theta1 and theta2, as illustrated above. Finally, let gravity be given by g. Then the positions of the bobs are given by:

 (1) (2) (3) (4)

The Potential Energy for the system is given by:

 (5) (6)

And Kinetic Energy for the system is given by:

 (7) (8)

Then the Lagrangian is given by:

 (9)
Therefore, for :

 (10) (11) (12)

so the Euler-Lagrange differential equation for becomes

 (18)

Dividing through by , this simplifies to

 (19)

Plots for and :

The coupled second-order ordinary differential equations (14) and (19) can be solved numerically for and , as illustrated above for one particular choice of parameters and initial conditions. Plotting the resulting solutions quickly reveals the complicated motion.

The equations of motion can also be written in the Hamiltonian formalism. Computing the generalized momenta gives:

 (20) (21)

The Hamiltonian is then given by:

 (22)

Solving (20) and (21) for and and plugging back in to (22) and simplifying gives:

 (23)

This leads to the Hamilton's equations:

 (24) (25) (26) (27)

where:

 (28)

and:

 (29)

So now we have completely explained the dynamics of double-hinged pendulum system in a rigorous mathematical fashion. From looking at our equations for Potential and Kinetic Energy we can clearly see that they are not linear and are in fact polinomial of degree 2! This means that solutions are nonlinear and our system is primed to exhibit chaotic behavior under certain initial conditions.

## Small Initial Displacement - Lissajous Behavior

We've seen the chaotic nature of the pendulums evolve, and derived their equations of motion. Now let's see what structure we can tease out of plots.

Let's give the pendulum a small initial displacement and plot the two pendulum angles as we watch it evolve:

It traces out the familiar form of a Lissajous curve. This small motion is highly regular and structured, why? Let's turn to the small angle approximation. The small angle approximation means that when theta1 and theta2 are small, we can approximate their sin() and cos() as θ and 1 respectively. Additionally, because their momenta will be small (less than 1), the product of their momenta will be very small and can be ignored. this converts the nonlinear systems of equations into a nice set of linear ones. A good exercise for the reader would be to apply the small angle approximation to the equations of motion. If you look at the small angle equations of motion long enough they begin to look familiar. Almost like the system of equations for two masses connected by a spring.

## Medium Initial Displacement - Bounded Nonlinear Behavior

Now what happens if we push the initial theta1 and theta2 far enough that the small angle approximation breaks down?

Both initial angles in the GIF above are .7 radians, which is about 40 degrees. This is outside the range of where the small angle approximation holds, but the small angle approximation isn't binary. It doesn't stop working after a certain angle. It slowly gets worse, and that's what we see here. Instead of the nice rectangular Lissajous curve, we now get a Lissajous curve that's been stretched and bent. It's easy to see our trajectory is still bounded, but we have a much more complex nonlinear shape.

## Large Initial Displacement - Unbounded Nonlinear Behavior

What if now we increases the initial angles to be much larger than the small angle approximation bounds?

We see the nonlinear dynamics in full effect with it's wild trajectory. Note: We have chosen not to regularize the angles to remain between 0 and 2Pi because it doesn't add additional structure and I liked the visual of the thetas walking around.

The unbounded chaotic behavior is plainly evident. Notice how it starts of with seemingly bounded nonlinear behavior but after a few iterations quickly becomes chaotic.