We've finally reached chaos in my last ever lecture course (we've been working down and to the right across the dynamical view of the world). Today we covered the Lorentz strange attractor and learnt that is was a fractal (that is, it's a set of points with infinite surface area but zero volume); a 'surface' defined by a line that never crosses itself. It never crosses because of the uniqueness theorem: There can only be one answer for any set of values (ie position in phase space) so for the system to remain chaotic and aperiodic it must never cross or merge with itself. These chaotic properties arise from nonlinearities in the equations themselves, not from measurement difficulties or noise.
Well, this is a problem I think, because we analyse fractals numerically. You can only float a variable to a finite number of places, so rounding errors when plotting the trajectories are effectively noise. The finite precision of variables means you are plotting the trajectories on a lattice of discrete points, and since the trajectory always stays near the strange attractor (although this has never been proved it appears to be the case) eventually the trajectory will overlap and we'll get periodic behaviour. Two consequences: Firstly we can't get chaos with a computer; secondly trajectories are diverging faster than they should due to the effect of noise.
Ignore the second point at the moment. Imagine plotting the Lorentz equations with a certain precision and seeing how long it takes (on average, we do this several times from several different starting points) for the system to become periodic and what period it has. Now increase the precision and repeat. Continue doing this and plot the period against the precision. We know that at infinite precision the period is infinite, so that's the extreme - but how do we get there? Could this graph tell us anything useful? That's another project for the Summer.
Here's another one: We know that for points around the attractor the trajectories are towards it. If we had an initial sphere of points around the Lorentz attractor, how would that sphere deform? What would it look like? Another project, but for that I'll need to program in Java or something because I'll need graphics.
We've finally reached chaos in my last ever lecture course (we've been working down and to the right across the dynamical view of the world). Today we covered the Lorentz strange attractor and learnt that is was a fractal (that is, it's a set of points with infinite surface area but zero volume); a 'surface' defined by a line that never crosses itself. It never crosses because of the uniqueness theorem: There can only be one answer for any set of values (ie position in phase space) so for the system to remain chaotic and aperiodic it must never cross or merge with itself. These chaotic properties arise from nonlinearities in the equations themselves, not from measurement difficulties or noise.
Well, this is a problem I think, because we analyse fractals numerically. You can only float a variable to a finite number of places, so rounding errors when plotting the trajectories are effectively noise. The finite precision of variables means you are plotting the trajectories on a lattice of discrete points, and since the trajectory always stays near the strange attractor (although this has never been proved it appears to be the case) eventually the trajectory will overlap and we'll get periodic behaviour. Two consequences: Firstly we can't get chaos with a computer; secondly trajectories are diverging faster than they should due to the effect of noise.
Ignore the second point at the moment. Imagine plotting the Lorentz equations with a certain precision and seeing how long it takes (on average, we do this several times from several different starting points) for the system to become periodic and what period it has. Now increase the precision and repeat. Continue doing this and plot the period against the precision. We know that at infinite precision the period is infinite, so that's the extreme - but how do we get there? Could this graph tell us anything useful? That's another project for the Summer.
Here's another one: We know that for points around the attractor the trajectories are towards it. If we had an initial sphere of points around the Lorentz attractor, how would that sphere deform? What would it look like? Another project, but for that I'll need to program in Java or something because I'll need graphics.