# Question about a Attractors in Non-linear Systems

I've recently been reading up on non-linear dynamics and came across the concept of attractors. I'd like to ask if the concept of attractors can be used for pedestrian egress from a room? Since pedestrians converge at the exit (for simplicity, let's assume that there is just one exit), is the exit position (or vicinity of the exit) considered an attractor? I am not a physics major and am very new to this topic, so my apologies if this question is too general and lacking in details.

• I don't think that it is an attractor, because people get past it. If you go to the other side of the exit you will not be "attrected" to go back. You can also not consider it to be an repeller. You maybe could consider it as an attractor, if your coordinate system ends at the exit (but that would not represent the physical situation). – MrYouMath May 2 '16 at 9:19
• That's very interesting, and thank you- I hadn't thought of that. However, if the system is run only until everyone has evacuated, and if the coordinate system ends at the exit, it is an attractor then? To (over)simplify it further, if pedestrians who have evacuated are constrained to remain at the exit for the time the simulation is run, would it satisfy the requirements for the exit being an attractor? – ogevery May 2 '16 at 9:26

is the exit position (or vicinity of the exit) considered an attractor?

Sort of.

Let’s first consider a single pedestrian who wants to exit the room and whose position in the room is $(x,y)$. Let’s further assume that the exit is located at $(0,0)$. Then we may describe the pedestrian’s position with the following differential equations:

$$\dot{x} = \begin{cases}\frac{-vx}{\sqrt{x^2+y^2}} &\text{if } x≠0\\ 0 &\text{if } x=0\end{cases}, \qquad \dot{y} = \begin{cases}\frac{-vy}{\sqrt{x^2+y^2}} &\text{if } y≠0\\ 0 &\text{if } y=0\end{cases}.$$

So, our pedestrian moves towards the exit with a constant speed $v$ and stays there when he reaches it. For this system, the exit is indeed an attractor. More precisely, the state [pedestrian at exit] ($x=0,y=0$) is the attractor. However, it is not anymore, if we let our pedestrian to actually exit the room and continue his movement outside the room.

If we consider more than one pedestrian, things get complicated, as two pedestrians cannot be in the same place (except for the exit). Thus the differential equations denoted above become more complex. Also, looking at this from the perspective of dynamical systems, one would conflate all the pedestrians’ positions into one state. So if we have three pedestrians, the state of the system would look like $(x_1, y_1, x_2, y_2, x_3, y_3)$, where the indices denote the different pedestrians. If again, we let the pedestrians stay at the exit when they reach it (and allow them to overlap there), the state [all pedestrians at the exit] ($(0,0,0,0,0,0)$ for three pedestrians) is an attractor of the entire system.

Note that in most dynamical systems based on movement, one would also require the speed or momentum of an object to fully describe the system’s state. This is not necessary in the above example, as the pedestrian moves with a constant speed and does not accelerate or similar. If the speed were part of the state, then the attractor of our one-pedestrian system would be something like [pedestrian standing still at the exit].

• Thank you for the detailed explanation. I'll read up more accordingly. – ogevery May 2 '16 at 14:00

An attractor is defined in phase space. Phase space is the space of all degrees of freedom of your system. So in your example it cannot be a spatial location such as a room exit.

Instead you have to imagine how many parameters describe the motion of one person (a lot), then how many persons there are, multiply the two and you will get the size of phase space.

In that space, a single point represents everybody in the room ! The motion of that point, again in phase state, represents the motion of everybody in the room as a whole.

Now an attractor would be a more or less complex trajectory limited to a subspace of phase space. When the single point describing all the people follows the attractor, then you will notice regularities in the way the people behave. For example, if the attractor is a closed curve then after a while you will find everybody at the same place, doing the same thing, exactly as they did some time before.

• Thank you for your answer. I realize the phase space could be very large. I do have a question- if the trajectories of the pedestrians are provided, does that include the interactions the pedestrians have with their environment? Like the boundary repulsions, pedestrian interactions etc. With that information, is it possible to evaluate system stability- like calculation of Lyapunov exponents? – ogevery May 2 '16 at 9:36
• I'm sorry if that is wrong- I'm just trying to understand this better. – ogevery May 2 '16 at 9:37
• The main point I think is that phase state allows the complete description of the system. So everything is included, all interactions, the environment if it acts on the people, whatever you need to fully describe the situation. Note that only the dimension (the number of degrees of freedom) is then fixed: how you distribute the degrees of freedom into a set of parameters is up to you. You could correlate different persons together if you wish. It is an abstract space; its basis depends on the specific way you want to model the system. – Stéphane Rollandin May 2 '16 at 9:50