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