Chaotic motion on a surface

[nslay]

Assume gravity and resistance are the only forces that act on the object.  Suppose we create a vector function that denotes the position of the object along a surface after 1 unit of time given its current position along the surface.  Iterating, for example, this function n times, yields the position n units of time later.  Is it possible to find a tangible surface in which the object behaves chaotically?  By tangible I mean that the surface exists on a finite interval and has a finite range.  This implies a few things, the vector function never converges, the object succeeds in perpetual motion, the vector function is not periodic (that means there does not exist some n for which f_n(x)=x).  You would think that there are no such surfaces since there is a resisting force acting on the object.

[Darkness]

How about a universal definition of chaotic? I would think that if there exists a function that describes it's motion, it wouldn't be chaotic by definition. But that's just what I've been taught.

[Rule]

I think that many chaotic systems have equations that can describe their motion perfectly, given that all the initial conditions are precisely defined. 

[nslay]

I think that many chaotic systems have equations that can describe their motion perfectly, given that all the initial conditions are precisely defined. 

Now, that depends.  Consider a numerical example, there are numbers that we can't even write.  In fact, most of the real numbers are "untouchable."  How are you so sure that the set of chaotic equations is countable?  Suppose the chaotic equations are uncountable?  Then there would indeed be "untouchable" expressions.

[Adron]

Now, that depends.  Consider a numerical example, there are numbers that we can't even write.  In fact, most of the real numbers are "untouchable."  How are you so sure that the set of chaotic equations is countable?  Suppose the chaotic equations are uncountable?  Then there would indeed be "untouchable" expressions.

There are many numbers we can't write down numerically exactly. Pi is one. Would you say that means Pi doesn't perfectly describe the relationship between the diameter and circumference of a circle?

[nslay]

Now, that depends.  Consider a numerical example, there are numbers that we can't even write.  In fact, most of the real numbers are "untouchable."  How are you so sure that the set of chaotic equations is countable?  Suppose the chaotic equations are uncountable?  Then there would indeed be "untouchable" expressions.

There are many numbers we can't write down numerically exactly. Pi is one. Would you say that means Pi doesn't perfectly describe the relationship between the diameter and circumference of a circle?

Ugh, that's not the point.  Pi describes the diameter and circumference perfectly if treated symbolically.  If you write it numerically, it really isn't Pi now is it?
The point is, that there could be unexpressable chaotic equations.

[Adron]

Ugh, that's not the point.  Pi describes the diameter and circumference perfectly if treated symbolically.  If you write it numerically, it really isn't Pi now is it?
The point is, that there could be unexpressable chaotic equations.

What you said was that most of the real numbers can't be written down numerically, and you're right. But if there is some particular number we want to write down, we can make a symbol for it, and presto, it can be written down.