In this video I want to draw out and
highlight some of the key themes and
realizations from the study of dynamical systems.
Some of the things that I think that are
most important and interesting.
And I should mention that I don't think
there's one single lesson or set of
lessons that come from dynamical systems.
And in particular I think the importance
of dynamical systems and the realizations
that come from it will be different in different fields.
Certainly what's important to a
mathematician might be different than
what's important to a biologist or an economist.
And, also I should mention that there's
certainly not universal agreement on what
those important things are.
So one of the goals of this course is to
give you a mathematical grounding in the
basic phenomena of chaos and strange
attractors and the like. So you can form
your own opinions about all this.
Nevertheless, I hope these thoughts
or comments are of interest, and at least
give you something to think about.
So, let's see. One of the things that I
think is most important about the study
of dynamical systems is this realization
that simple dynamical systems, like the
logistic equation or the Rossler equations
do not possess, or do not necessarily
possess, simple dynamical properties.
And another way I like to think about this
is that there can be a difference between
the nature of a process and the nature of
the thing it produces.
So for example, the logistic equation is a
deterministic equation. It's a simple
iterative rule. It's about as simple and
almost as boring as can be, but the simple
deterministic equation produces aperiodic
and random, or seemingly random output.
So there's this distinction between the
characteristics of the generating process;
in this process a simple deterministic
equation, and it's output which in this
case is an aperiodic, in a sense, random
sequence. So, I think this has all sorts
of important implications some of which
we talked about in Unit 3. So, for example
if you see a process that's behaving
apparently at random, wild fluctuations or
something of the sort, you might think,
"oh, there must be something, some
external process going on. It's not
following a simple rule. It's being moved
about at random." And that could be the
case. It also could be the case that
there's very simple intrinsic dynamics
that are making that apparently random
behavior. You can have a system that's
rule bound and describe in simple terms
but behaves in ways that are seemingly
opposite to that.
Another example where we can have behavior
that seems opposite to that which is
producing it is in the study of bifurcations.
So when we looked at the logistic equation
with harvest, we saw that there's a
stable population and as you increase
harvest the stable population decreases.
But there is this critical point, this
critical harvest rate, and if you
increase it just a little bit beyond that
harvest rate the stable population doesn't
go slowly down to zero but it just disappears.
So it's a sudden discontinuous change.
The differential equation itself is
perfectly continuous. But the behavior
of the equilibrium points, or the fixed
points, can be discontinuous.
So if you see something, a system that
has a sudden change. You had a steady
population of 100, and a little while
later the population is just gone, you
might very logically expect some external
influence, and some large external
influence must've been required to move
that population from 100 to 0 all of a sudden.
And indeed, that could be the case, but
dynamical systems also shows us that there
could be an intrinsic cause for that;
internal to the dynamics of the system and
in particular, it could be that a very
small change in the parameter leads to a
very big change in the equilibrium value
of the population. So a continuous system,
a differential equation can have
discontinuities built into it. Again, the
process generating a phenomena is in some
ways opposite in character from the
process that is generated.
Another example of this phenomena I'm
trying to describe is pattern formation,
which we talked about in the last
unit. You can have a simple dynamical
system, spatially extended in this sense,
but something that's local. That each
point is the future value of each point is
a function of just what is going on at
that point, and you can even add diffusion
into it. This force, so to speak, that
tends to smooth everything out. And that
even in the presence of all that a local
diffusive rule with some reactive terms
in there as well, one can get stable
spatial structures. So these patterns come
seemingly from nowhere, from an almost
perfectly homogenous initial condition
that diffusion doesn't smooth it out but
it can make these patterns emerge.
So if one sees a patterned outcome,
in physics or biology or chemistry or
economics, it could be that somebody made
that, drew it up like one would make
blueprints for a house. Or it could be
that is just what the system does without
any designer or direction from above, it
self-organizes. Certain types of patterns
can be formed spontaneously.
So again we see an example where the
process generating a phenomenon, in this
case a local deterministic rule, is in
some ways opposite to the outcome of this
global stable pattern. So I think one of
the key themes, key ideas, from dynamical
systems is this separation between a
process's output, what the output looks
like, it's characteristics, and the
process that generates it. I think that's
an important realization for all of
science and certainly for the study of complex systems.
Another important realization that comes
from the study of dynamical systems, is
this idea that order and disorder are not
necessarily opposites; that they
intermingle and coexist in a lot of
interesting ways. One way we've already
talked about is sensitive dependence on
initial conditions, or the butterfly effect.
Where we have a system that's
deterministic but nevertheless is
impossible to predict over the long term.
Because of sensitive dependence on initial
conditions. And in fact, it's almost as if
the function is so deterministic, it
depends so much on the initial conditions,
it obeys that rule so precisely over time
that precision that makes it unpredictable
because one would need to know the initial
condition, the initial value, to an
impossible precision in order to do long
term prediction. So in some ways, that's
a negative result. That says that there's
certain processes, predicting the weather
perhaps, that we'll never be able to do
long term prediction on. But I also think
it's a positive result for a number of
reasons. One, it's a positive result to
know limits of knowledge. It's certainly
better to know our limits than to not know them.
But also, there's certain types of
chaotic systems that may not be
predictable in some sense but may be
predictable in an average sense.
And we saw that in strange attractors.
Which is a really nice and vivid example
of how order and disorder can coexist in
the same system. So in a strange attractor
we have chaotic motion, sensitive
dependence on initial conditions, but the
structure or the shape in phase space
along which those, the butterfly effect
exists, along which trajectories go, it is
a stable structure, it's an attractor.
So if we have a lot of initial conditions,
far away, they'll get pulled into the
attractor and the ones on the attractor
will move around chaotically, aperiodic
with sensitive dependence on initial
conditions. So for motions that are
described by a strange attractor, we can't
do predictions of an individual
trajectory, but we can make statistical
predictions precisely because the strange
attractor, the overall shape that the
trajectory moves along in phase space,
is stable. That is predictable. So we can
make very clear statements about how
often the system will be in this region of
phase space as opposed to this region or
whatever the situation may be. So we can
have a certain statistical, or long term,
global stability with a local instability
that gives rise to sensitive dependence on
initial conditions. So it's an interesting
mixture, this combination, in one
phenomena of disorder and order.
More generally, I think there's often a
presumption in science and in modelling
maybe, that order and disorder are just
completely different things. If you see
something that behaves kind of ordered
and kind of disordered, you'll have to
understand the ordered system one way with
a nice set of equations and the disordered
part you'll need to do statistics or a
stochastic process on or something.
And that can be a fine approach but this
also says that order and disorder can
exist in the same system. In a sense can
be just different sides of the same coin.
So, in a lot of different ways, I think
dynamical systems challenge this idea that
order and disorder are opposite and it
also challenges the idea that simplicity
and complexity are opposites. We see lots
of examples of complex behavior coming
from simple equations.
Another issue that comes up in dynamical
systems, and elsewhere but we've talked
about it in this course that's a theme
that's worth highlighting, is the
different ways mathematical models are
used. So, very roughly speaking, one can
imagine two approaches to making a model
of a phenomenon. One is to try to get
every detail correct and try to have a
model that can be as predictive as
possible. That's a valuable approach but
it's also sometimes of interest to come up
with a model that's deliberately simple;
that leaves out certain features but
preserves maybe some features you don't
care about but preserves some features
that you're most interested in. So in this
sense this type of model is more like a
caricature. It's designed to just capture
some essential element of a system and not
try to reproduce every detail about that
system. Often those types of models can
lead to better understanding, or at least
a different type of understanding, than
those really large models that try to
capture everything. So these simple models
can just suggest maybe there's some
simpler common mechanisms for similar
features that are seen across a number of
different phenomena. Maybe the best
example of that, is the idea of universality.
So that period doubling route to chaos is
universal in the sense that certain
aspects of how that occurs, ratios of
lengths of successive periodic regions,
will be the same not just for mathematical
systems but for physical systems as well.
So that is a stunning example of how a
very simple mathematical model can give
rise to not just qualitative understanding
but numerical predictions about physical
phenomena. Personally, I tend to think
that universality will not be seen in the
same strong sense that we see it in
dynamical systems in complex systems like
ecosystems or complex networks or
economies or the like. So I think that
universality in the strict sense might be
a little bit of an exception; that we
won't see that all the time. However, I
do think it's the case that these simple
models provide really valuable input into
origins of complex behavior and random
behavior as well. It's not the only type
of input you'll want but I think it's a
really important realization.
So, lastly, there's a question of "What is
chaos?", "What is dynamical systems?" and
how to think about in relation to the rest
of science and the rest of math.
Is it a revolution or a paradigm shift
like quantum mechanics is? Well, we
discussed this some in the interview with
Stephen Kellert, the philosopher of
science from Hamline University. There I
think, to me at least, chaos and dynamical
systems is not a revolution in the way
that quantum mechanics was for instance.
Quantum mechanics completely changed our
picture of the universe and required not
just accepting some new things but
rejecting some other things. Chaos, I
don't think quite reaches that level.
It doesn't really say that Newtonian
mechanics is wrong, it just says that
Newtonian mechanics, this rule based idea
of the universe, this mechanistic
clockwork universe, is actually a lot more
interesting. It turns out clocks can do
all sorts of really cool things; not just
be clocks. They can be chaotic and random
and make patterns as well. So anyway I
don't think chaos and dynamical systems
requires us to necessarily reject physical
theories, or entire physical theories, but
it does require us to rethink certain
concepts and certain ideas.
A nice description of this is in a history
of science paper by David O'Bahn and Amy
Delmitico. They argue that rather than
thinking about chaos as a revolution,
sudden change, they take a longue durée
view of history. So a long arc, many
different streams of thought flowing
together and what emerges in this area of
chaos and dynamical systems, they call a
vast socio-disciplinary reconfiguration.
It changes the conceptual landscape.
So disciplines and trains of thought flow
together and come up with, perhaps a
common language or common framework
that reconfigures certain categories; like
order and disorder like we talked about
before. Realizing those don't have to be
opposites. Realizing you can get aperiodic
behavior from a repetitious iterated
function and the like. So it reconfigured
some of those ideas and disciplines have
in a sense flowed together so there is a
common language, at least in part, across
biology, economics, math, and physics.
So these notions of strange attractor,
phase space, Lyupanov exponents, butterfly
effect, sensitive dependence on initial
conditions are all part of this new
framework. It's a way of communicating
ideas and it's a way of doing certain
types of mathematical modelling and maybe
it's a way of looking at the world.
Hopefully, through this course, you're now
part of this convergence of ideas in chaos
and dynamical systems as well.