Dynamical Systems and Chaos


Well hello there! Here we are going to consider the dynamics of nonlinear systems that display chaotic tendencies under cetain conditions. Before we dig deeper into the topic, let us consider one of my favorite visual examples of one of the priciple traits of chaos: sensitivity to initial conditions. The three double pendulumns below all start form nearly identical starting parameters. For a time they follow the same differential, but them after just a few iterations linearity breaks down and the chaotic, nonlinear nature is easily seen!

Demonstrating Chaos.gif

So, What Is Chaos?

While Chaos Theorey lays out a very precise defenition for what chaos is mathematically, there is still no general consensus among the mathematically community as to what a rigorous definition actually is. That being said there are generally three properties which most mathematicians agree define a Chaotic Dynamical System:

Sensativity to Initial Conditions

Topological Transitivity

Dense Periodic Orbits

In some cases, the last two properties above have been shown to actually imply sensitivity to initial conditions. Chaotic dynamics, in a nutshell, means that a system is extremely sensitive to initial conditions. That means a small change in where the system begins, becomes a big difference in where it ends up. A lot of people say that chaos means that we cant predict what the system will do, and this is not exactly true. Chaotic systems, including the one we are looking at today, can be deterministic. This means that if we know it's initial conditions, and can integrate it forward in time with infinite precision, we could predict its motion out arbitrarily into the future. Unfortunately, in the real world, there's always some measurement error in determining the initial conditions and computers only have finite precision. This means that in reality we often cant predict chaotic systems out arbitrarily into the future.


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 Eric Weisstein's World of Math 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:

smallOffset.gif

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?

mediumOffset.gif

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?

LageOffset.gif

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.

I hope you enjoyed the lesson the chaos! -Matt