It´s clear that a moving flag is a chaotic system. But is it nontheless possible, under certain conditions and a uniform wind velocity, to make the flag look frozen in time, i.e., flat without movement?
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The simplest answer to your question would be: Yes, put the flag out on a day when there is no wind. If you want to be able to distinguish which country it represents, tilt it so that its pole is horizontal. But I guess that this is not the situation that you have in mind. I imagine that you are asking if it is possible that wind will be strong enough to hold the flag up and regular enough so that the flag appears motionless.
If that is your question, I would guess: 'maybe in theory, but no in practice'.
Keep in mind that it is not the dynamics flag that are chaotic, but the dynamics of the air. As long as it is light enough, the flag just follows the wind. Flowing fluids such as wind have a laminar (smooth) motion when the flow is slow, but as it speeds up, the smooth motion becomes unstable and even the slightest perturbation will produce turbulence. Usually there is a critical velocity above which the laminar flow is unstable.
If I were to try to realise the situation that you describe, I would use a flag that is very stiff (so that it holds up almost by its self) and very heavy (so that it is not affected by the wind). For example, a flag that is made out of a 1cm thick slab of metal should not move under most weather conditions. If you insist that the flag be made out of cloth then I would try a flag that is very light (so that a weak wind can lift it) and use a very thin pole (so that the laminar flow is stable at high velocities).
I have been thinking on the way to estimate what parameters would work, but this is an incredibly complex problem that I do not have the time to try to solve. Moreover, it depends on a lot of parameters that you have not specified. For example, what is the stiffness of the flag, it's mass, the radius of the pole, the height of the pole? When you say that the flag is perfectly still: Is it allowed to have small and slow movement that is imperceptible? How small? How slow?
I would say that the most efficient way to solve your problem is to go out with a flag on a windy day and do the experiment. If you flag moves, try again on a day that is less windy. If your flag falls, try another lighter one. If you can do it, then the answer to your question is 'yes'. If you can't, try a thinner pole and a different kind of flag. Keep trying until you succeed or believe the the answer is 'no'.
Steven Mathey
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This is not unlike the question "can you theoretically balance a perfect pencil on its tip". The answer is always "no".
No, because the initial conditions cannot be perfectly obtained; and the equations of motion are such that a small perturbation from the initial condition will grow. You cannot generate perfectly laminar flow on a molecular scale, indefinitely: the mechanisms used to generate the wind will themselves leave small perturbations in the air, and these will be sufficient to initiate motion of the flag.
