>>Glenn, a question for you (since I'm not an expert in nonlinear dynamics):
>>While chaos theory describes the way a system's time-evolution is quite
>>sensitive to initial conditions, aren't there also mathematical equations
>>of motion which lead to very similar time evolution no matter what the initial
>>conditions? (Is this a "strange attractor"?) The existence of such systems
>>could be appealed to by those who seek to blunt the force of an anthropic
>>principle which depends on "fine-tuning" of initial conditions.
I was unsubscribed for the weekend, didn't get around to resubscribing til
today, and haven't looked thoroughly at the archives, so this may be a bit
redundant.
Stan is right. Chaotic systems can exhibit strange attractors as well as
sensitive dependence on initial conditions. The way nonlinear dynamics was
taught 30 years ago when I was in school, there were only two types of
attractors: points and limit cycles. Once a trajectory settles down to a
stable point attractor it stays there. A limit cycle is a closed curve in
the phase space. Once a trajectory settles down to a stable limit cycle it
repeatedly traces out the same curve in the phase space. A strange
attractor is a trajectory which tends to stay in some (frequently
complicated) region of the phase space. The behavior of the trajectory on
a strange attractor may appear very organized, but the trajectory never
passes through the same point twice. Strange attractors appear to fill
regions of space, although if you magnify them you find there's space
between the successive traces.
In a real nonlinear dynamical system there are likely to be not one, but
many strange attractors, and which attractor the behavior settles down to
depends on initial conditions and later disturbances. The sensitivity to
initial conditons (and to disturbances) can in principle enable a miniscule
disturbance to cause the trajectory to switch from one strange attractor to
another. In other words you have a system that is ideal for a) stably
maintaining a given behavior most of the time, and b) changing to another
behavior when an omniscient designer/overseer imposes the correct,
undetectible perturbation :-).
Bill Hamilton | Chassis & Vehicle Systems
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