Re: chaos

[P8475423@xxxxxxxxxxxxxxxxxxxxxxxx]

Allin asks:
Is that right?  I thought that for a system exhibiting chaos,
the equilibrium takes the form of neither a fixed point nor
a limit cycle, but a 'strange' (fractal) attractor.  If you
stochastically perturb a chaotic system, will it not typically
return to this equilibrium (i.e. move back onto the attractor)
quite quickly?
.. re my description of various classes of chaos.

No, a whole variety of behaviours are possible, of which "strange"
attractors are only one. The best known of these is Lorenz's
(incredibly simple) system of 3 equations for (I think!) the
x, y and z displacements of air:

dx/dt = a*(y-x)
dy/dt = (b-z)*x - y
dz/dt = x*y -c*z

This generates a system with two unstable equilibria, and hence two
"orbits" form around these equilibria, with those orbit regions
(though not the orbits themselves!) intersecting. Hence the system
will start around one, seem to start to curve in towards it, then
suddenly fly off to the other; and back again.

My system generates a single unstable equilibria, which is an
attactor up to a point and a repeller from that point in; so the
system moves towards the attractor (cyclically), then moves away,
then in, then out again, effectively "wrapping itself around" an
orbit which marks the border between the attractive and repulsive
regions.

Other systems which are potentially chaotic (such as for
example the iterated logistic model) can have single equilibria
to which the system converges for a range of parameters, multiple
equilibria, and the total apparent randomness which is called
chaos.

Probably that's the source of the confusion. There are a class of
models which can generate behaviour described as chaos; but
given some parameter values, they will demonstrate convergence to
an equilibrium (and they may even be analytically soluble, as
with the logistic equation). So to say that a model is chaotic
doesn't necesarily mean that for all parameter values it will
demonstrate the apparent randomness called chaos.
Cheers,
Steve K


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