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The first bifurcation is the saddle note bifurcation.
This is the basic mechanism by which fixed
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points are either created or destroyed, essentially
as a system parameter is varied. The two fixed
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points moves towards each other collide and
then mutually annihilate. Prototypical example
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is x dot = r + x square where r is a parameter
which is greater than 0, is equal to 0 or
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less than 0. So, for r less than 0, if you
plot x dot versus x, that is the plot for
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x dot versus x, we find that we have two fixed
points.
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So, the system has two fixed points one is
stable and the other is unstable, for r is
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equal to 0. When we plot x dot versus x, we
find that we only have one fixed point. The
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fixed point is attracting from the left and
repelling from the right. So, the fixed points
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actually turn into a half stable fixed points
at x* = 0 and when r is greater than 0 plotting
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x dot versus x reviles that in fact we have
no fixed points.
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So, a bifurcation effectively takes place
at r = 0 as the vector fields for r greater
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than 0 and r less than 0 are qualitatively
different from each other.
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Now, we go ahead and plot something know as
bifurcation diagrams. So, if we show a stack
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of vector fields for discreet values of r
that is the stack of vector fields, when r
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greater than 0 there are no fixed points.
When r is equal to 0, there is half stable
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fixed point and when r less than 0, we have
two fixed points, one stable and other unstable
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and that is what happen when r varies, so
we have stable and unstable branches.
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So, a common way is to treat the parameter
r as playing the role of an independent variable
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in the limit of a continuous stack of vector
fields. What we get is the following, so we
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plot x versus r as the stable branch and that
is the unstable branch. So, we can also include
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arrows in the diagram, so including arrows
in the bifurcation diagram we get. So, this
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is called the bifurcation diagram for the
saddle node bifurcation. So, this is the first
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bifurcation that we have dealt with and diagram
on the left is referred to as its bifurcation
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diagram.
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Let us now conduct a linear stability analysis
of x dot = r - x square, we first identify
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the fixed points, so x dot = f(x) = r - x
square, yields(gives)x* = plus minus square
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root of r. For r greater than 0, we get two
fixed points and to establish linear stability,
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we get f prime of x* = -2 times x*. So x*
= plus the square root of r yields a stable
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fixed points x* = - square root of r is unstable.
For r less than 0, there are actually no fixed
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points, for r = 0, we get f prime of x* = 0
and so the linearized term actually vanishes.
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Let us consider an examples, consider the
first order system x dot = r - x -e to the
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-x, show that it under goes a saddle node
bifurcation as r varies, further finding the
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value of r at the bifurcation point. First,
we plot r - x and e to the -x on the same
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graph, where the line r - x intersects with
the curve e to the -x, we get r - x = e to
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the -x and so f(x) which is r - x e to the
-x = 0. So, intersections of the line and
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the curve actually correspond to the fixed
points of the system.
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Now let us go ahead and make some plots, that
is the line r - x that is the curve e to the
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-x. So we note that this system has two fixed
points, one stable and the other one is unstable.
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Note that the flow is to the right where r
- x is greater than e to the -x and so x dot
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is greater than 0. Now let us go ahead and
decrease the parameter r little bit, so that
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is the line r - x, that is curve e to the
-x and note that in this case we have one
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fixed point.
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The fixed point turns out to be actually a
half stable fixed point. So, at some critical
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value r = rc the line actually becomes tangent
to the curve. So that is the saddle node bifurcation
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point, for r less than rc that is r - x that
e to the -x and note that they do not actually
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intersect at all, so there are no fixed points.
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We still need to find the bifurcation point
rc. So, we impose the condition that the graphs
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of r - x and e to the -x intersect tangentially.
We require equality of the functions and their
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derivatives. So, e to the -x = r -x and d
dx of e to the -x is d dx of r - x which gives
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us -e to the -x = -1, which yields x = 0,
so substituting x = 0 into e to the -x = r
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- x gives r =1. So, the bifurcation point
is at r critical =1 and the bifurcation occurs
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at x =0.
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In this lecture, we introduced the saddle
node bifurcation. This is the basic mechanism
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through which fixed points can be created
or destroyed. We introduced prototypical examples
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for the saddle node. Now essentially what
it showed us was the following, we had a model,
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where we actually had two fixed points, one
of the fixed point was stable and one of the
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fixed points was unstable. And then a parameter
in the system changes as the parameter changes,
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both of these fixed points actually came close
to each other.
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And at particular point of time they actually
merged and became one half stable fixed point
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and as the parameter change even more, in
fact the fixed point in the system vanished.
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So, we had a scenario, where we had two fixed
points stable and an unstable. Then we had
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one fixed point which was half stable fixed
point and as the parameter varied all fixed
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points in the system actually vanished.
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So that is why fascinating because when you
modelling the real world, you have the real
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world of which you abstract about a simplified
model and that model will have some parameters
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in it. And so the lesson here is that as parameters
varies you would have fairly serious qualitative
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changes in the underline dynamics and we illustrated
that through the saddle node bifurcation in
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this lecture.