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In this video, I'll compare and contrast
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differential equations
and iterated functions.
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These are the two main types
of dynamical systems
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that we'll study in this course,
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and, although they're very similar,
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they do have some different
mathematical properties,
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and comparing the logistic equation,
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as an iterated function
and differential equation
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can help make this clear,
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and highlight some important distinctions.
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So here on the left, is the logistic
differential equation.
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A differential equation
(the form we'll be studying)
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describes a function P
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in terms of its rate of change.
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So this says, we know
the rate of change of P
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if we know what P is.
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The population growth
depends on the population value
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in these two parameters.
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For an iterated function,
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it also describes a population growth,
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but here, f(P) is
the population next year,
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given the population P this year.
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So we get a series of population values
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by iterating this function.
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So I began when I derived
the logistic equation,
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(I used this form),
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but it's often simplified to this,
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the A kind of gets absorbed inside x.
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So this is what we worked with,
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but the starting point
for these two equations
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is the same on the right-hand side.
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What's different is, we interpret things
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differently on the left-hand side.
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So the right-hand side here
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is interpreted as the growth rate.
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The right-hand side here
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is interpreted as
the population next year.
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So solutions to these iterated functions
and differential equations
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have a different character.
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For differential equations,
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the solution is population
as a function of time,
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and that would look, as we've seen,
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maybe something like this.
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For an iterated function,
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we end up with a time series plot,
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and that might look
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something like this.
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So notice the difference
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between these two solutions.
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In both cases, the blue curve
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is the solution to the dynamical system.
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The dynamical system is just a rule
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that tells this blue thing what to do.
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But for the differential equation,
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the blue curve changes continuously.
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It's defined at all times,
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and it smoothly increases,
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say, from here to here,
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and it has to pass through all
intermediate values.
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For the iterated function,
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the time moves in jumps.
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It has an initial value at time 0,
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then time 1,
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then time 2,
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and the value of the population
also moves in jumps.
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It goes from this value
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to this value,
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and even though we connect those dots,
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it doesn't slide through
all values in between.
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It jumps from here at time 0
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to here at time 1
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without going through
the intermediate values.
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In this one, the differential equation,
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time and the population are continuous.
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Time and population are continuous.
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But for the logistic equation
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and all iterated functions,
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the time and the population
or whatever we're measuring
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moves in jumps.
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So, again, for the logistic equation
and the iterated function,
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time and population moves in jumps.
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And this difference here,
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together with the fact that
these equations are deterministic,
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gives rise to very different ranges
of possible behaviors.
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So we've seen for the iterated function
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in Unit 3
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that it's capable of producing
cycles and chaos.
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So cycles and chaos are possible.
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Of course not all iterated functions
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will show a cycle or will show chaos
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and remember chaos is
an aperiodic bounded orbit
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that also has sensitive dependence
on initial conditions.
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For a differential equation, however,
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cycles and chaos are not possible.
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So let's think about why this is so.
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So suppose a cycle was possible.
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If that was the case,
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I would have a solution curve
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that looked something like that.
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It goes up and down.
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We can eliminate this possibility
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by appealing to the determinism
of this equation.
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This equation says that the derivative,
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the growth rate, the rate of change
of the population, depends
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only on the population.
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(And r and K, but we're imagining
those are fixed.)
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So let's think about this blue curve here
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that oscillates up and down.
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I'm going to draw, just arbitrarily,
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a dashed line through here.
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And notice what happens.
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Here, I have a particular p value,
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the p value is at this dashed line,
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and the population is increasing
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so the derivative is positive.
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The derivative is positive
for this p value.
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Over here, when the population
is going back down,
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the population is decreasing,
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so the derivative is negative.
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So that means at these two points,
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here and here,
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they're different derivatives.
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So at the first purple arrow
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the function is increasing :
positive derivative.
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At the second arrow
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the function is decreasing:
negative derivative.
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But the problem is that
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they have the same p value
as on the y axis here.
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And the p value is the same.
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If this was true, this would say
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different derivatives at the same p value.
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But that's impossible
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because the differential equation says
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the derivative is a function
of only the p value.
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Another way of saying that is
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a given p value only has one
derivative associated with it.
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If you know the population p
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then that determines the derivative.
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Here, if you know the population p
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that does not determine the derivative,
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because you have different derivatives
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at the same p value.
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So the conclusion, then, is
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that cycles are not possible,
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and chaos isn't possible as well.
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Any behavior that goes up and down
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(it doesn't have to be a regular cycle)
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we can eliminate by this argument.
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As we said in Unit 2
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the range of behaviors for one dimensional
differential equations
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are kind of boring.
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The function can increase to a fixed point
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decrease to a fixed point,
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decrease to infinity,
increase to infinity,
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and that's all it can do.
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Iterated functions have a much richer
array of behavior,
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and that's because determinism
doesn't constrain them
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in the same way, so that
it doesn't forbid cycles.
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So cycles are possible
in iterated functions
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and chaos, aperiodic behavior,
is possible as well.
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In the next sub-unit
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we'll leave iterated functions
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behind for a little bit
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and we'll look again at the logistic
differential equation
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and I'll add a term to it
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and we'll start
investigating bifurcations.