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Chaos theory vs catastrophe theoryby marellasunny
Tags: chaos theory 
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#1
Feb113, 08:59 PM

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I am taking a course in nonlinear dynamics and I read that Lorenz systems exhibit 'chaotic behaviour' and the sprucebudworm nonlinear 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 sprucebudworm 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.)



#2
Feb113, 09:31 PM

P: 772

Chaotic and catastrophic behaviours are different things, though a chaotic system may have catastrophes and viceversa. Chaos usually looks like bounded but nonperiodic 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.



#3
Feb113, 09:35 PM

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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. 


#4
Feb113, 09:45 PM

P: 236

Chaos theory vs catastrophe theory
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? 


#5
Feb113, 10:29 PM

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#6
Feb113, 11:54 PM

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#7
Feb213, 08:56 AM

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I think catastrophe theory has become a subtopic 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 reevaluatiing Aristotelian philosophy. 


#8
Feb213, 10:25 PM

P: 236

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 outofplace?(besides Thom shifting interests :p). 


#9
Feb313, 01:12 PM

P: 772




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