I’m a bit confused with the correct way of calculating the Lyapunov exponent for a particular system that I study.
Imagine we can place $k$ foo-balls and $k$ goo-balls on a surface. Each foo-ball is attracted by any other goo-ball, and when they get close enough, they annihilate each other. There is no closed-form solution of the system and it needs to be solved numerically. The system starts from an initial positional arrangement of these particles in a 2D plane, $S=[[x_1,y_1], [x_2,y_2],...]$ and they all annihilate in different locations one by one, $F=[[x'_1,y'_1],...]$.
In order to calculate the Lyapunov exponent for this system, I dislocate the $S$ a little ($\delta_0$) and this dislocates the $F$. What I cannot figure out is how to calculate the exponent when the initial conditions are more than one. Like simulating the solar system for example. A displacement in the initial condition could be just moving the earth for 1 meter or moving all the planets for 1 meter. Which would be right and how the exponent would be calculated from the final fixed point displacement?
The reason I am trying to find the exponent is that I have seen some simulation results which have made me think this system has a chaotic behavior and I am trying to check it.
After all balls have annihilated in your system, there is no dynamics anymore or the dynamics has converged to a fixed point – depending on your perspective. Anyway, there is no long-term chaotic behaviour.
However, the short-term behaviour may be chaotic – a so-called chaotic transient. You can indeed use Lyapunov exponents to characterise this behaviour. However, for this the annihilation locations are bad observables for several reasons:
Each annihilation point is associated with a different time point.
Annihilation is part of ending the chaotic transient. Thus its details do not yield good information about the transient itself. You are looking too late.
Whether your balls annihilate does not inform the early dynamics. You could probably conceive a system without annihilation whose early dynamics is identical to the one you are observing.
Thus, instead of looking at the annihilation points, I recommend that you look at the position of your balls after a short time and in particular before any annihilations happen. More precisely, consider:
$$ \frac{ | \hat{S}(t)-S(t) | }{ | \hat{S}(0)-S(0) | },$$
where $S'$ is the displaced system and your initial displacement $|\hat{S}(0)-S(0)| = δ_0$ is as small as reasonably possible – I recommend two orders of magnitude above the numerical accuracy threshold. If you average the above for many initial conditions and you really have chaos, it should exponentially grow with $t$, as long as $|\hat{S}(t)-S(t)|$ is small compared to other length scales in the system, such as the usual minimum distance between balls. The exponential growth rate then is the Lyapunov exponent.
That should not matter. The important thing is that the displacement is small.
Strictly speaking, there are some pathologic displacement directions (orthogonal to the largest Lyapunov vector), which will yield a misleading result. However, even if you want to hit this direction, you usually cannot because of numerical accuracies. If you want to be super sure, just randomise your displacement in direction, so the probability that you hit such a pathologic direction is zero.