so let's summarize Unit 8 the topic of
Unit 8 was strange
attractors and I introduced strange
attractors through several examples
first example was the hen on map this is
a discret two-dimensional mapping
they're two variables X and Y and they
jump forward in time from Time 1 to 2 to
three by just applying this rule this
says that the next X and Y value is a
function of the current X and Y values
and then there are these two parameters
A and B and we observe chaotic behavior
for some parameter values here's one of
those chaotic parameter values this was
the standard one that n on
discovered and um this so shows AP
periodic Behavior sensitive dependence
on initial conditions but what's new in
this chapter is plotting X against y
plotting it in an XY plane we see a
complicated and structured
attractor and we used some software to
zoom in on this and we saw that it had a
fractal structure that um every time you
think you see one line it's actually
made up of two lines and this there
bends within bends within bends here in
this shape so it's a complicated highly
ordered and structured attractor it
looks um very different than the
disordered irregular AP periodic
Behavior one sees here
so uh this is a strange
attractor so what is a strange attractor
well it's two things it's an
attractor um and remember that means
that nearby orbits get pulled into it
it's stable if you're on the attractor
and you get bumped off a bit you get
pulled back into the
attractor however for strange attractors
motion on the attractor is chaotic and
that means that orbits are AP periodic
and have sensitive dependence on initial
conditions so it is an attractor that is
itself chaotic an attracting chaotic
orbit so that was a hon map and then we
did another
example the Loren equations this is a
system of three differential equations
it's a dynamical system it's a rule for
telling us how X Y and Z depends on time
how it how they change over time and the
rule here is specified in terms of
derivatives rates of change and there
are three parameters in Greek now Sigma
row and Beta And if we fix those at a
particular value we can solve for x y
and z using a computer and here um is
one example of doing that we see a
periodic cycling in all three variables
there's some um sort of time regularity
but the amplitude of these Wiggles and
whether or not they're up or down veryy
um seemingly at
random so so um again we can plot now X
Y and Z against each other so plot this
trajectory in a three-dimensional phase
space and if we do that we get the Loren
attractor and this
orbit um lies on a strange attractor and
similar to the Hanan attractor it's a
strange attractor because it's
attracting and because motion on the
attractor itself um has sensitive
dependence on an conditions the
butterfly effect again a reminder that
since this is a deterministic system
lines in Phase phase space can't cross
it looks like they do but that's because
one is going over the
other so then in order to talk about
this idea of stretching and folding I
introduced one more um dynamical system
which is maybe a little less well-known
but um is one of my favorite ones and
that's the um rustler equations
so like the Loren equations these are a
system of three differential equations
and sorry I just realized now that this
DT got um cro just a little bit so um
it's three differential equations dxdt
dydt
dzdt three parameters a b and c and we
choose parameter values we can solve
these equations on a computer and we get
an X solution and a y solution and a Z
solution so again we see there's some
sort of time regularity to their Wiggles
but um the amplitude of The
Wiggles um varies seemingly at random
and I'm particularly fond of the Z
trajectory this is just an aesthetic
observation um this seems
particularly I mean these are all
chaotic they're all AP perodic but I
don't know somehow this just looks more
chaotic than everything else if you had
a function that was behaving this way it
at least I wouldn't at first guess oh it
must be following a simple rule anyway
as with the Loren equations we can plot
X versus Y versus T plot these in a
three-dimensional phase bace and if we
do that we see another uh strange
attractor known as the rustler attractor
so here this is a higher resolution
image um here's the the source for this
it's from uh from Wikipedia
and um orbits move around sometimes they
go up high but they're sort of always
cycling around in this direction but
sometimes they get pulled up and down
and then I showed with a number of
different diagrams that what this is
doing this uh Solutions Move Along here
the lines move along here they're
stretched up in this direction and then
they they're folded back down across so
this is is stretching and folding
stretching and folding stretching and
folding and I argued that this
stretching and folding is at the core of
chaos so stretching and folding I said
is the key geometric ingredient or
ingredients of chaos the stretching
pulls nearby orbits apart and that's
where sensitive dependence not initial
conditions come from if all the orbits
are being pulled apart then small
differences grow very rapidly but we
also need some sort of a fold
otherwise orbits would just get pulled
apart forever and go off to Infinity so
we need some sort of a stretch and then
a fold stretch and then a fold like one
does when kneading
dough so the stretching and folding that
one can see I hope in these
three-dimensional strange attractors
particularly the rustler attractor the
stretching and folding occurs in
one-dimensional maps as well so the
logistic equation anything that's shaped
like the logistic equation is performing
a stretch and a fold and this explains
maybe not fully explains but provides
some uh indication for how
onedimensional Maps can capture some
features of higher dimensional systems
they all involve stretching and
folding so lastly because it's an
important and I hope interesting and fun
idea just want to underscore strange
attractors again so first of all just to
point out that they're complex
structures these are interesting uh
shapes they're usually fractal and they
arise from very simple dynamical systems
the motion on the attractor itself is
chaotic but all orbits get pulled into
the attractor so they combine elements
of order and disorder if I start the
Loren equations anywhere I can't predict
what the exact orbit will be but I know
it's going to trace out that same
strange attractor shape so it's
predictable in that I know what the
overall shape will be but I don't know
exactly how it will Traverse that shape
so it combines order and disorder
predictability and unpredictability
another way to say that is that motion
on the attractor is locally unstable
locally unstable because nearby orbits
get pushed apart it's globally stable
because um no matter
what happens the the orbit is going to
trace out that strange attractor in the
same
way
I want to underscore this final point
because I think it's important it's one
of the most important realizations that
emerges from the study of dynamical
systems and that is that strange
attractors systems with strange
attractors combine order and disorder
predictability and unpredictability in
the following way as we' seen motion on
an attractor is chaotic that means it
has sensitive dependence on initial
conditions the butterfly effect it's so
impossible sensitive on those initial
conditions that it behaves as if it were
random or maybe depending on your point
of view it actually is random but the
point is it's not something that can be
predicted however I do know even if I
don't know the exact trajectory that a a
curve will take through phase space I
know that it's going to trace out that
strange attractor it has to lie in that
attractor um and it will sort of trace
it out in similar ways and what that
means is is that certain average or
statistical properties of that orbit are
entirely predictable I could state with
a high degree of certainty what fraction
of the time say the um trajectory will
have a positive x value or a negative x
value or how often it will reach a zv
value above a certain uh a certain
height so this global stability gives it
a different type of predictability a
statistical predictability we can make
very confident statements about average
properties but we can't make statements
about the particular trajectory followed
by a particular initial
condition so uh an analogy might help
shed some light on this and it may even
not be an exact um it may be stronger
than an analogy but the analogy is that
um the strange attractor the fact that
there's some stable statistical or
Global properties is sort of like the
climate things that don't change over
time or change very little over time the
um exact orbit which is unpredictable
could be like the weather so we say the
weather is unpredictable but the climate
is predictable there are well-defined
stable average properties of the climate
again they change very slowly compared
to you know they might change on the
under typical times under the scales of
what hundreds or thousands of years at
least
so um we can have unpredictable
short-term Behavior with statistically
predictable long-term Behavior right so
sometimes people in these silly climate
debates climate deniers will say how can
you predict the climate when you can't
even predict the weather well there the
the point of strange attractors a
phenomenon of strange attractor says
there's no logical inconsistency at all
between long-term uh structural or
statistical stability and predictability
and shortterm or uh local instability or
unpredictability so stranger tractors
Prov um
combine predictability and
unpredictability Order and disorder from
the same system the same set of
equations in really interesting and
important ways now of course the climate
is I think much more than a three an
attractor in threedimensional
space but this gives the right idea and
the main point um I'm not saying that
the climate is a strange attractor but
the main point is is that there's no
logical inconsistency at all in fact
it's a very normal State of Affairs to
have systems that are locally
unpredictable but globally and uh
predictable and uh in a statistical or
structural sense okay so this brings us
to the end of Unit 8 which was about
strange attractors they're fun
interesting mathematical objects I think
they make beautiful shapes and I think
they tell us how order and disorder are
related and can be combined in some
interesting and important ways
what the exact applications are in
different areas of science will probably
vary depending on those areas but I
think it's an important
result so the next unit unit 9 we'll
take a look at pattern formation this
unit will be a little bit different in
character it'll be less mathematical and
more just showing you results from a few
mathematical models and we'll see that
simple dynamical systems can produce
strange attractors they can produce
disordered or apparently disordered
Behavior the butterfly effect but they
also can produce uh strikingly complex
patterns as well so we'll see you next
week