Chemical oscillators that behave like a double pendulum

Question

Oscillating chemical reactions such as BZ-reaction or chemical clock typically result in sequential systematic changes, most notable of which is arguably coloring. Though underlying mechanism can be extremely complicated, we always expect the same sequenced cycling set of properties. For iodine clock it's always will be

... dark-blue > colorless > dark-blue > colorless ...

colors that we observe. Just like ordinary pendulum. But are there oscillating chemical reactions with the non-linear outcome and "chaotic" sequences for each cycle, e.g. behaving like a double pendulum?

And if such systems do exist, can we also just use similar mathematical apparatus like a system of differential equations used for a double pendulum to describe such behavior?

Answer 1

Before considering a double pendulum, let us consider a simple pendulum, which is modeled by

$$\ddot \theta + \frac{g}{\ell} \sin (\theta) = 0$$

Choosing units that make $\frac{g}{\ell} =1$, we have

$$\ddot \theta + \sin (\theta) = 0$$

Let $x_1 := \theta$ and $x_2 := \dot \theta$. Hence, the 2nd order ODE above can be rewritten as follows

$$\begin{array}{rl} \dot x_1 &= x_2\\ \dot x_2 &= - \sin(x_1)\end{array}$$

The 2nd ODE is not merely nonlinear, it is non-polynomial (due to the sine). However, the ODEs that model elementary chemical reactions should be polynomial due to the law of mass action. Hence, it should be impossible to simulate the simple pendulum using a chemical analog computer. Simulating the double pendulum should also be impossible.

What may be possible is to simulate an approximation. For example, if $x_1$ is sufficiently "close" to zero, then we could use the approximation $\sin(x_1) \approx x_1$, which produces a system of coupled linear (and, thus, polynomial) ODEs

$$\begin{array}{rl} \dot x_1 &= x_2\\ \dot x_2 &= - x_1\end{array}$$

which approximate the original pendulum provided that $x_1$ remains "small" (in absolute value).

Lastly, since concentrations cannot be negative, we have non-negativity constraints $x_1, x_2 \geq 0$.

Answer 2

I would like to supplement this excellent answer by a different argument. Chemical reactions are typically done in bulk, involving numbers of particles on the order of $10^{17}$ - typically greater. In statistical thermodynamics, this is considered an ensemble, and while a single pair of molecules (that can react which each other) cannot be predicted in their behavior (when will they react? what will be the product?), the behavior of the ensemble can usually be predicted. I think that the same may be true for an ensemble of double pendulums. Thus, the analogy would hold, but not in the way the question intended; rather, it would be on a molecular level.