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.