# What causes the chaotic trajectory of a balloon going flat? [duplicate]

I think this is not a duplicate of the question John mentions. That question asks specifically why the balloon makes a spiraling movement, which in general is not the case (the answers pay only attention to two variables: the nozzle and air resistance, for which there isn't a fixed centre of drag) while I ask the broader question which variables make the movement chaotic (which it is for sure while its velocity responds in a non-linear way to these variables). The non-spiraling movement can be seen as a sequence of (infinite) little pieces of circular trajectories because there are other variables influencing the motion of the balloon too (like the chaotic air movement around the moving balloon, which causes a non-fixed center of drag).

When a balloon goes flat it makes a chaotic movement through the air. Is it the interaction with the air, the behavior of the part through which the air escapes (in which case the balloon will also trace out a chaotic trajectory on the moon), or what?

• This is just a guess, but I think it's probably interaction between the air flow and the flexible tube through which the air escapes, through the same mechanism as the snaking of a fire hose, which I described here: physics.stackexchange.com/questions/47726/… Apr 6 '17 at 7:43
• How do you know its chaotic? Apr 7 '17 at 8:28
• Because the response of the balloon's velocity to the other variables is non-linear. Were that not the case then the balloon would follow a straight line. The question is what the other variables are. Apr 7 '17 at 10:31
• Being nonlinear doesn't mean that it is necessarily chaotic. Isn't it? Apr 7 '17 at 14:57
• Possible duplicate of Why does a balloon spiral in air instead of moving in a straight line? Apr 8 '17 at 11:19

Pointing out a single factor for the chaos is impossible in this case. It can be anything from vertical flow of surrounding air to surface deformities of the balloon membrane to even radiation pressure(this may be a small factor). The list may go on. Analyzing such a complex dynamical system without proper lab equipments is impossible. Even then I doubt if one can find the sole cause. Although, the main reasons may be:

1) non-uniformity near the nozzle of the balloon, where the flow supposedly becomes turbulent and therefore chaotic.

2)surrounding atmospheric phenomenon.

• I disagree. How do you know that the motion is chaotic in the first place? Apr 7 '17 at 8:51
• Chaos is said to occur in a system when given slightly different initial conditions, the evolution result is 2 very different system. If you fill up a balloon with a fixed quantity of air and let it go and trace its trajectory, can you gurantee if you repeat the experiment again when a slight breeze is blowing, the trajectory will be same? It won't be. In fact they may be widely different. That's why this is a chaotic dynamical system. There exit very few natural processes which are absolutely not chaotic. Apr 7 '17 at 11:51
• The velocity of the balloon is reacting in a non-linear way to the other variables, i.e. the motion is chaotic. The velocity of an accelerating rocket in empty space, on the other hand, is linearly proportional to the mass of the rocket and its acceleration and therefore the motion of the rocket is non-chaotic. Apr 7 '17 at 16:28

Chaos and irregular motion are not one and the same thing!

Chaotic motion is a precisely defined concept and every irregular motion that you see in a physical system need not be chaotic. However, chaotic motion always seems to be irregular. To be specific, chaotic motion refers to the deterministic motion that sensitively depends on the initial condition of the system. Also, the motion should satisfy things like "boundedness" and "topological mixing".

This means that if you have many obvious random variables in the system, you should not use the term "chaos" to describe the observed irregularities. For example, atomic transitions are irregular, but not chaotic. In the case of the balloon you are describing, we do not have (at least I do not know of) any model that is deterministic. If you have such a model, then one can try to find our whether it exhibits chaotic motion (at least for some ranges of its parameters) and only then you would be able to ask questions like which parameters cause chaos in the model and so on.

In fact, as I see it, the balloon's behavior, in this case, seems to be only a transient to me (because it stops after all the air has escaped) and hence cannot be chaotic. To understand more about the actual concept of chaos, you can try to read this.

• "Although no universally accepted mathematical definition of chaos exists" is said directly in the Wikipedia article and contradicts your initial statement. It seems we don't have a rigorous definition, and instead several proposed descriptions. I'm not sure exactly how it's often defined, but this system should be highly sensitive to initial conditions. It is a pretty complex coupling of systems that are already quite sensitive themselves. It seems like even knowing initial conditions to a fairly large degree of accuracy, I doubt you could estimate it's final position very well.
– JMac
Apr 7 '17 at 9:30
• @JMac : What is the exact problem with my answer though that somebody felt like down-voting it? Apr 7 '17 at 9:32
• @JMac : How do we know the actual cause in case of balloon? Yes we can't estimate the final position. But is that because of the chaotic motion or because of the random fluctuations of the air molecules or something like that? For example, in a random coin tossing model, you can't predict the outcome but it's not chaotic. Apr 7 '17 at 9:35
• I down voted it for the exact reasons I outlined in my comment. Your wikipedia article contradicts your opening statement, which seems to be the main point of your answer. It also doesn't answer the question at all.
– JMac
Apr 7 '17 at 9:35
• @JMac : No it doesn't! It is true that there is no universally accepted definition of chaos in physical systems. However, it is also true that if you take those three properties together, they define a precise mathematical concept. And still, my answer does not talk too much about that but rather about the fact that balloon' motion is a transient and by any possible definition of chaotic motion, can't be chaotic. Apr 7 '17 at 9:38