Chaos vs purely exponentially growing systems

[Sunny Singh]

I have just started reading chaos from the MIT OpenCourseWare and the following passage has confused me.

"The sensitivity to initial conditions is important to chaos but does not itself differentiate from simple exponential growth, so the aperiodic behavior is also important. In the definition of this somewhat undescriptive phrase we include that the system should undergo Topological Mixing. This means that any points starting in a region (open set) of the phase space will evolve to overlap any other region of the phase space, so chaotic systems tend to explore a larger variety of regions of the phase space"

IF the solution of a system described by linear equations is exponential, then it is aperiodic too right? it won't fall into a periodic orbit for sure. Then how is it not considered to be chaotic? I might be misunderstanding "Topological mixing" here. How does this topological mixing thing leads to such a system not getting called chaotic?

 

[mfb]

If it is exponential, it is still predictable: while the absolute error diverges, the relative uncertainty stays the same. Start with a 1% uncertainty and you still have a 1% uncertainty later. That is not considered chaotic. To make a more formal definition, this topological mixing is introduced: While the system cannot be exactly periodic, it should come close to its original conditions again (to avoid cases like eternal exponential growth), and then you get proper chaotic behavior.

 

[hilbert2]

If you make a 1-particle system that is like a harmonic oscillator, but has a "wrong" sign in the potential energy: $V(x)=-\frac{1}{2}kx^2$, it has an unstable equilibrium point at $x=0$ and an arbitrarily small deviation from that equilibrium will grow exponentially, but it's not something that would be called chaotic. Systems with actual chaos always have higher than 2nd powers in the potential energy function.

 

[Sunny Singh]

If it is exponential, it is still predictable: while the absolute error diverges, the relative uncertainty stays the same. Start with a 1% uncertainty and you still have a 1% uncertainty later. That is not considered chaotic. To make a more formal definition, this topological mixing is introduced: While the system cannot be exactly periodic, it should come close to its original conditions again (to avoid cases like eternal exponential growth), and then you get proper chaotic behavior.

So this means that given enough time, the system will necessarily return arbitrarily close to its initial conditions? Does it also mean that the system eventually explores the full volume of its phase space accessible to it with the constraints of energy? Can you please give me some link to topological mixing that isn't too hard on mathemaics? because most of the places i found about it were articles from mathematics departments and i find their notations a bit hard to follow. Thank you so much helping me out.