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So let's summarize what
we've done in Unit 7.
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I introduced two and three
dimensional dynamical systems.
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Two and three dimensional
differential equations
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and two dimensional
iterated functions.
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The main new thing or object to think
about from this unit was the idea of
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phase space.
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A phase plane or a phase space
or in the discrete dynamical system case
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for the Hénon map, those xy-plots.
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I started by introducing the
Lotka-Volterra equation.
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This was the main example of a two
dimensional differential equation.
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So we have two equations...uh oh, that
should be an R for rabbits.
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Good thing I noticed that.
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So we've got two differential equations
keeping track of two variables, R and F.
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R, the rabbits, depend on the rabbits and
the foxes and the foxes depend on the
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rabbits and the foxes.
So we'd say they're coupled.
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The fates of the rabbits and
the foxes are intertwined.
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These are known as Lotka-Volterra
equations and they are in ecology and
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perhaps in economics as well.
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Sort of the basic model, the most simple
model of a predator-prey interaction.
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We can solve these differential equations
using Euler's method or the like and we
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have two solution curves.
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This is rabbits as a function of time and
these are the foxes as a function of time.
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We notice that they oscillate.
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So we have these two solution curves.
To try to get a qualitative picture of
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what's going on and how R and F, rabbits
and foxes, are related to each other
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we can plot R against F
and get rid of time.
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That's just a two dimensional version of
the phase lines we worked with before.
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Doing so gives us a picture like this.
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We plot R and F against each other and it
shows how R and F are related.
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In this case we move
this way on the diagram.
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The rabbit population increases then the
fox population increases while the rabbits
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decrease and the foxes decrease and
then the rabbits come back again.
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So we see how these
two cycles are related.
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So that's the phase plane.
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It's often useful to show several
different solutions on the same phase
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plane. If there are multiple equilibria,
maybe show those. And that's sometimes
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called a phase portrait. It gives an
overall view of the qualitative features
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of the differential equations. By
qualitative I mean the long-term fate
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of orbits. So this would be the phase
portrait for the Lotka-Volterra system.
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We see that for different starting values
of rabbits and foxes we get different
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cycles but in all cases we have cycles.
These cycles turn out to be neutral.
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Meaning that they are neither attracting
nor repelling.
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If you're on this cycle and move off a
little bit you're just on another cycle.
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You don't get repelled away or pushed
back.
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Another example that I considered was the
Van der Pol oscillator another two
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dimensional differential equation.
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This is interesting because it has what's
called a limit cycle.
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The limit cycle itself is this repeating
shape here and we can see that multiple
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orbits are drawn towards it.
So it's an attractor.
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Here's an orbit from outside that gets
pulled in to the cycle.
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Here's one from inside that gets pulled
in to the cycle.
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In two dimensional differential equations
we can have oscillating solutions and we
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can have cycles that are attractors; these
are known as limit cycles.
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We talked some about of the consequences
of determinism in the phase plane.
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So the dynamical system is deterministic.
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It's just a rule that tells how R and F,
or x and y, whatever they are,
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change in time.
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Another way of saying that is that
the initial condition and the differential
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equation, the rule, specifies a unique
path through the phase plane.
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If I tell you the starting point and the
rule there's one and only one solution
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to the equation.
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As a result of this curves on
the phase plane cannot cross.
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If they did cross it would violate
determinism. It would mean, that from
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that point, there would be two possible
curves and that violates this notion
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of determinism.
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So curves, solution curves, on the phase
plane cannot cross each other and that
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has some important consequences.
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The fact that curves cannot cross limits
the possible long-term behavior of 2D
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differential equations.
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So what can those behaviors be? Well there
can be stable and unstable fixed points.
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You could spiral in, in addition
to getting pulled in.
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You could have a fixed point and
you spiral towards it.
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Still fixed points, stable and unstable.
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Orbits could tend to infinity.
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There could be limit cycles like in the
Van der Pol oscillator which is a type
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of attracting cyclic periodic behavior.
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That's pretty much it. There's a famous
and important result known as the
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Poincaré-Bendixson theorem. Which says,
simplifying somewhat, that bounded,
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aperiodic orbits are not possible for two
dimensional differential equations.
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Of course that assumes that the right hand
sides of the differential equations
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aren't crazy and discontinuous or
something like that.
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So the main consequence of this is that we
can't see chaotic behavior in two
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dimensional differential
equations of this sort.
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We can't see aperiodic, bounded orbits.
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I should mention, by the way, that for
these 2D differential equations I've just
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been presenting solutions from Euler's
method but there's some very nice
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analytical techniques for 2D differential
equations but they're beyond the scope
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of this course. They need calculus and
typically a little bit of linear algebra.
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So in 1D we can characterize equilibria
points by whether or not the curve, the
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function f(x), the right hand side of the
differential equation, when we're
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drawing those phase lines if it crosses
the axis like this or like this.
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Basically what that means is that it
depends on the slope.
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Is the slope positive?
Is the slope negative?
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There's an analog in 2D but it has to do
with the eigenvalues of a 2x2 matrix
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instead of a simple slope.
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Anyway, this is in most
differential equation textbooks.
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Certainly those that take a modern
approach or a dynamical systems
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approach which is more and more
textbooks these days.
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If you want to know more it won't be hard
to find additional references.
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So that was two dimensional
differential equations.
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Then we looked at two
dimensional iterated functions.
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These are like the logistic equation,
an iterated function.
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Time moves in jumps, the population moves
in jumps. But now we just have two things
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we're keeping track of: x and y.
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This says the next value of x is this
function of the current value of y and x.
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The next value of y in this case is
a function only of x.
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This is an example of known as a Hénon
map; a commonly studied example.
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The story for 2D iterated functions is
pretty similar to that of 1D.
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There can fixed points, there can be
cyclic behavior and there can be aperiodic
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behavior, there can be chaos.
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What's new is that we can plot these x and
y points on a 2D grid like I've done here.
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That's another way to see the sorts
patterns that the orbits are creating.
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We'll spend a bunch of time on that in
the next unit.
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So just a little bit more about two
dimensional iterated functions.
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Mathematically they're very similar to 1D
iterated functions. Orbits can be periodic
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and we can also have chaotic behavior.
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Just a reminder that chaotic behavior is
bounded, aperiodic orbits that have
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sensitive dependence
on initial conditions.
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Lastly, we looked at three dimensional
differential equations.
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The example I used was the Lorenz
equations.
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Here are three differential equations.
The variables are x, y, and z.
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This is a very simplified model of
atmospheric convection.
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The physical origins of this model aren't
important for what we want to do here.
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There are three parameters. They are Greek
letters: σ (sigma), ρ (rho), β (beta).
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So this is a rule for how three things
vary in time instead of two.
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If we use Euler's method, or the like, to
get a solution for this we'll get three
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solution curves; x vs. time,
y vs. time, z vs. time.
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We can plot those in time,
and in this case we see
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periodic wiggling in all of these.
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Then it can be interesting and fun
to plot x, y, and z against each other.
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So this is like a phase plane but because
there is a z now it's phase space, it's
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three dimensional. That makes things a
lot more interesting.
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So here's what those periodic solutions
look like in phase space.
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So it's oscillating up and down in the z
direction, sideways in the x, and sideways
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in the y and it makes this curly, loopy
shape.
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It looks like these lines cross but they
don't really because this is three
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dimensional space so this line is above
that one. So just like this finger is
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above that one. It looks like they cross
but they don't.
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They aren't really because
it's a three dimensional space.
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So curves in phase space cannot intersect,
just like they can't intersect on the
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phase plane. But because space is three
dimensional curves can go over or under
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each other. That means for three
dimensional differential equations the
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solution curves can sort of wind around
each other and weave around each other
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without actually crossing. So we'll see
much more complicated behavior in 3D
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differential equations than in 2D.
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In particular, three dimensional
differential equations can be chaotic.
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We'll explore chaos in these 3D
differential equations and 2D iterated
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functions in the next unit.
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Those explorations will mostly be in phase
space so we'll be looking at what things
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look like in phase space and we'll be in
for some pretty fun surprises.
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So this brings us to the end of Unit 7.
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In this unit we've looked at dynamical
systems in two and three dimensions.
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In a certain sense the story is the same
as in one dimension.
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A dynamical system, remember, is just a
deterministic rule that describes how a
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quantity changes in time.
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For two and three dimensional systems we
are just keeping track of two or three
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quantities instead of one.
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Mechanics of finding those solutions is
essentially the same no matter how many
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variables we're trying to keep track of.
We just follow the rule. That's what
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the dynamical system tells us to do.
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However, visualizing solutions for higher
dimensional systems is a little more
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interesting and fun because, well, two and
three dimensions are a little more
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interesting and fun than one dimension.
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In particular, for differential equations
in higher dimensions we see much more
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complex and interesting behavior.
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That'll be the topic of the next unit,
Unit 8. Which is on strange attractors.
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We'll see these incredible structures that
combine order and disorder in some
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really interesting and surprising ways.
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We'll see you next week in Unit 8.