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In this lecture we will first focus on existence
and uniqueness of solutions. Now, so far we
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have actually taken the existence and the
uniqueness of solutions to x dot = f(x) completely
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for granted. Consider the equation x dot = x(t)he
one third starting from x(0) = 0. The point
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x = 0 is a fixed point. So x(t) = 0 is a solution
for all t. We can also go ahead and find another
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solution.
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We start by separating the variables and integrating
to find, so evaluating the integral of x(t)he
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-1/3 and the integral of dt. We get 3/2x(t)he
2/3 = t + an arbitrary constant C1. So with
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the initial condition x(0) = 0, we get C1
= 0, so x(t) = 2/3 t to the power of 3/2 is
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another solution to this differential equation.
So we have situation where we actually have
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two solutions to the differential equations.
One is x(t) = 0 and the other is x(t) = 2/3
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of t to the power 3/2.
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Here are some notes: The first point is that
the geometric approach actually fails, when
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we do not have uniqueness of solutions to
the differential equation. If a phase point
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starts at the origin, then does it stay at
the origin for all time or does it move according
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to the solution, that we found that is x(t)
= 2/3 of t to the power 3/2. Now let us take
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slightly closer look at the vector filed.
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So, we now go ahead and plot x dot versus
x that is the plot of x dot versus x. And
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we find that the fixed point would be repelling
from both ends. So, note that the fixed point
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x star = 0 is in fact an unstable fixed point.
Ok let us set out a much more challenging
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exercise for you. Can you go ahead and show
that the initial value problem x dot = x(t)he
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1/3, x(0) = 0 actually has an infinite number
of solutions.
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Now let us go ahead and state the theorem
that provides sufficient conditions for the
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existence and the uniqueness of solutions
to x dot=f(x). So let us call this the existence
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and the uniqueness theorem, consider the initial
value problem x dot=f(x), where x(0) = x not
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and suppose that f(x) and f prime of x are
continuous on an open interval R of the x
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axis and suppose that x not is a point in
R.
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Then the initial value problem has a solution
x(t) on some interval -T to T about t =0 and
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the solution is unique. So, let us take a
minute to just absorb this theorem. What does
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it really tell us and how can we translate
it into more plain English. So essentially
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what the theorem says is that is if f(x) is
smooth enough, then the solutions will exist
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and they will be unique.
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But there is actually no guarantee that the
solutions will exists for ever. Now this is
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a really important point to remember, so we
talking about the existence and the uniqueness
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of the solution. But remember that at this
point of time, we may not able to guarantee
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that the solution will actually be exist for
ever, it may only exist for a short period
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of time.
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Now let us consider an example. Let us look
into the existence and the uniqueness of solutions
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to the initial value problem x dot = 1 + x
square where x(0) = x not. In this example
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f(x) is 1 + x square, so the function is continuous
and has a continuous derivative for all x.
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The theorem says that solutions would exist
and be unique for any initial condition x
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not. However, the theorem does not say that
the solutions will exist for all time. So
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let us consider x(0) = 0, we separate the
variables, so we get dx / 1+ x square.
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We integrate that and we integrate dt, this
gives us tan inverse of x = t + an arbitrary
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constant. With x(0) = 0. We get C1 = 0, so
x(t) = tan t turns out to be a solution for
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the initial value problem. But the solutions
will only be exist for t when t is actually
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sandwich between pi on two and - pi on two,
as x(t) will tend + or - infinity as t tends
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to + or - pi on two, for x(0) = 0, there is
in fact no solution outside the above the
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time interval. So essentially we have an example
here, where the solutions exist but they only
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exist for a certain time interval.
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Our next sub heading is the impossibility
of oscillations. By now we are quite familiar
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that fixed points dominate the dynamics of
x dot=f(x). So trajectories either approach
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a fixed point or they diverge to + or - infinity.
In fact these are the only options for a vector
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field on the real line. Trajectories are forced
to either increase or decrease monotonically
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or actually just remain constant.
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So geometrically speaking if you plot x dot
versus x, that is an arbitrary function f(x)
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and we can identify the stable and the unstable
fixed points. So geometrically speaking a
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phase point actually never reverses direction
and this is one key point to note.
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Here are some notes, we can regard a fixed
point as an equilibrium solution. So, the
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approach to equilibrium is always monotonic.
So, overshoot damped oscillations and undamped
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oscillations are not possible. So, there are
actually no periodic solutions to x dot = f(x)
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and this is a really key point, so we highlight
it. The reason in fact is topological, it
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reflects the fact that x dot = f(x) represents
a flow on a line.
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So flowing monotonically on a line will actually
never get you back to your starting point.
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If we were in fact dealing with a circle rather
than with a line then we can return to our
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starting place. So in that sense vector fields
on the circle can actually exhibits periodic
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solutions.
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Now there were two main points that you wanted
to make in this lecture. Point number one
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was about the existence and the uniqueness
of the solutions. Hence, often a tendency
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to actually take existence and uniqueness
little bit for granted, so to that end we
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produced two examples. In one examples, we
showed that you could have multiple solutions
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and in the second example we showed that while
the solution would exist it would not necessarily
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exist for all time ok.
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So, I think this is more a question of being
aware and being careful that actually have
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a differential equation. Please do spend a
till bit of time thinking both about existence
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and uniqueness. The second point that we made
was about the impossibility of oscillations.
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Now, essentially when you are dealing with
one dimensional flows. If you have flow on
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the line then you are not really going to
come back same place again.
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So, the trajectories would either go off to
plus minus infinity or would go towards a
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fixed point. Of course, if you dealing with
flows on a circle you could have oscillations,
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but the main point to remember here is that
you would not get oscillations in a one dimensional
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equation of the form x dot = f(x), as long
as it was flowing on the real line.