chaos theory vs catastrophe theory

marellasunny

I am taking a course in non-linear dynamics and I read that Lorenz systems exhibit 'chaotic behaviour' and the spruce-budworm non-linear D.E follows the criteria of 'catastrophe theory'.Is there a difference between these 2 theories?If yes,does this mean that small changes in the spruce-budworm model do not exhibit the 'butterfly effect'?Please explain(higher math is also understandable for me,you could use Thom's taylor series proof etc.)

Number Nine

Chaotic and catastrophic behaviours are different things, though a chaotic system may have catastrophes and vice-versa. Chaos usually looks like bounded but non-periodic behaviour; often what happens is that you have an extremely complex bifurcation structure, and minute changes in the system are knocking the state into subtly different trajectories/bifurcations, and so extremely similar initial conditions will tend to diverge very quickly. In a catastrophe, usually the appearance or disappearance of a fixed point is causing the system to abruptly change its state. You can imagine a situation where a system is sitting nicely at a fixed point, and then a bifurcation causes the fixed point to disappear and the system collapses to a fixed point some distance away.

AlephZero

I think the two theories are about different things.

Chaos theory is about the sensitivity of a system to its initial conditions.

Catastrophe theory is about the different ways the system response can change at a bifurcation point.

A bifurcation point doesn't necessarily imply that the solution is chaotic, and a chaotic system need not have any bifurcation points.

marellasunny

AlephZero and all,could you please confirm that again-
"A chaotic system does not have bifurcation points",but we all know about systems that undergo a process called period doubling diverging into 'chaos' as a rapid succession of bifurcations brings it towards the phase space basin of a strange attractor.(taken from a website)

So,can I classify the 2 theories like this?

Catastrophe theory applied to -'finite number of bifurcations'

Chaos theory applied to -'infinite number of bifurcations'

If yes,why isn't there a connect between the 2 theories in terms of bifurcations?

rollingstein

Are there any good real world examples of catastrophe theory?

marellasunny

AS I referred to in my question:the spruce-budworm model exhibits the principles of catastrophe theory(look it up).For some reason,nearly every book I read on non-linear dynamics has very little on catastrophe theory and much more on chaos theory.The author justifies this by saying catastrophe theory has fallen way-side while chaos theory grows,...wonder why.?

AlephZero

I think catastrophe theory has become a sub-topic of the more general study of bifurcations in nonlinear systems.

Not to mention the fact that Thom, its "inventor", sort of drifted away from math and spent the last 20 years of his life re-evaluatiing Aristotelian philosophy.

marellasunny

Sorry if I am dragging the post,but I would really like to know if catastrophe theory is a speculative theory or has anyone of its models been proved realistic?Take for example the boat floating on water with a certain dead weight and live weight(humans).If there is excess movement,the boat would flip to its other stable state and back again provided there exists some force(don't know if i can call it hysteresis) .Wouldn't this also come under catastrophe theory?(with weights and buoyancy as the parameters and boat displacement and wave velocity as variables).??

Or take for example an ecosystem flipping between states due to human interventions like forest fires or shooting down deer ...etcetc

If catastrophe theory is really used to descirbe such realistic systems,why has it fallen out-of-place?(besides Thom shifting interests :p).

Number Nine

I'm not sure what "speculative" is supposed to mean here; a catastrophe is something that happens in certain dynamical systems. It happens; there's no debate over whether or not it happens. If you're asking for applications, then a wiki search will provide several.