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Hi Len,
Len said:
That's not quite right. The Butterfly Effect is that given a
sensitive dependence on initial conditions, two similar systems
will evolve to very disimilar states in a short period of time.
It is the effect of amplification that occurs when conditions
are coupled by non-linear equations that determine unpredictable
or uncertain future states.
Didier replies:
I do not see what you found not right in what I said Len. I said that
"butterfly effect is associated to chaotic systems having one or several
attractor". The last statement implies that since this model is based on
state space (this is where you plot the attractors) and then uses vector
analysis it will be dependent on the initial conditions. However, the
nature of the attractors will not end with the same result for the same
initial conditions. Nonetheless, to create models of chaotic systems, we
need to map them into vector spaces/state spaces. The problem could be
to model them in state space when there are too much possible states or
if the states do not match the actually known attractor categories.
There is also another way to study networked systems based on stochastic
models. In these models the attractors could potentially be at each step
of a network path and look like a random walk (polynomial with
probabilities for each node). These systems are non continuous, cannot
be modeled with Riemannian integrals but more with Lebesgue integrals.
This is why I said that it is easier to model these phenomenons with
percolation models than with classic chaotic system dynamics involving
different mathematics. This said, a lot of people are actually studying
stochastic phenomenons through the perspective of attractors. Off
course, since understanding more these type of phenomenon will also help
to model the stock market :-) :-) So this is why I said modestly that I
do not know if the VHS/beta phenomenon maps well with chaotic system
models but I know that they map better or are better modeled with
percolation models involving stochastic calculus. Utilitarian? Maybe,
practical? Yes indeed.
Cheers
Didier PH Martin
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