WEBVTT
Kind: captions
Language: en
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In this lecture, we talk about the pitchfork
bifurcation. This bifurcation is rather common
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in physical problems that have symmetry, consider
the beam buckling problem. We had a small
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weight on top of a beam as long as the weight
was small the beam was stable. So, when the
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load is small the beam is stable in the vertical
position. So, we have a stable fixed point
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corresponding to zero deflection. Now consider
the case where we have a larger weight that
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is placed, which forces the beam to buckle.
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So, we find that the beam buckles under the
weight and the beam may in fact buckle either
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to the left or to the right. The vertical
position is now unstable and two new symmetrical
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fixed points have been born. There are two
types of pitchfork bifurcations one is a supercritical
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and the other is a subcritical.
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Now consider a supercritical pitchfork bifurcation.
The normal form is x dot = rx - x cubed and
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this equation is in variant under the change
of variable x to -x, what that means, is that
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if we replace x by -x we get the equation
back again. So, let us plot the vector field
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for different values of r. So, if r <0, we
plot x dot versus x
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and we find that we have a stable fixed point.
The origin is the only fixed point and it
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is stable. Now consider r =0, plot x dot versus
x.
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We again have only one stable fixed point,
the origin is still stable, but note that
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the linearization actually vanishes. Now consider
r >0 and we plot x dot versus x and here is
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where we get some interesting dynamics. We
get two stable fixed points and an unstable
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fixed point. So we are stable at x dot is
equal to plus square root of r and stable
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at x* is equal to minus square root of r and
origin is unstable.
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So the origin is actually now unstable, two
new stable fixed points on either side of
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the origin have now emerged. Now we have the
interesting situation where for r <0, we had
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a stable fixed point. For r=0, the origin
was still stable, but if you look at the linearization,
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then the linearization would vanish at the
fixed point. And that the r>0, we found that
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the origin turned unstable and we had these
two-new stable fixed points which emerged
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on either side of the origin.
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Now let us consider the bifurcation diagram
for a supercritical pitchfork bifurcation.
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As is the norm, we go ahead and plot x versus
r
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that is stable and dashed line is unstable.
So, straight line is stable and that is another
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stable branch. The term supercritical, usually
means that the bifurcating solutions themselves
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are stable.
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Let us consider an example, analyse the system
x dot = rx - x cubed for r<0, equal to zero
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and greater than zero via a potential function
V of x. The potential for x dot = f of x is
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defined by f of x = - dv dx. So, we need to
solve -dv dx = rx -x cubed. So, we integrate
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to get V of x = - 1/2 of r x squared + 1/4
x to the four, where we have neglected the
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constant of integration.
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First consider r less than zero, we plot v
versus x. We get a quadratic and you see that
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the quadratic has a minimum at the origin
and so the origin is stable. And r = 0 again
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plotting v versus x, the minimum is at the
origin, so the origin is still stable. For
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r>0, we get something more complicated. We
have a local maximum at the origin implying
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an unstable fixed point and symmetric pair
of local minima implying stable fixed points.
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Now let us consider a subcritical pitchfork
bifurcation. Where x dot = rx - x cubed. The
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supercritical case the cubic term is stabilising
with x dot = rx + x cube. The cubic term is
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actually destabilising and we get a subcritical
pitchfork bifurcation. We now plot the bifurcation
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diagram for the subcritical pitchfork. So,
it is customary to plot x versus r, the dashed
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lines represent the unstable branches, so
that is also unstable.
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The non-zero fixed points x* is equal to plus
minus square root of minus r are unstable
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and the origin is stable for r less than zero
an unstable for r greater than zero. Now when
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r is greater than zero, one can show that
x of t would tend to plus or minus infinity
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in a finite amount of time, starting from
any initial conditions x of not, not equal
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to zero.
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In real systems the instability is actually
opposed by the stabilising effect of higher
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order terms. Assuming that the system is symmetric
under x to – x, the first stabilising term
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would be x to the 5. So, the economical example
for a subcritical pitchfork is x dot = rx
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+ x cube - x to the five. Now we plot the
bifurcation diagram for the subcritical pitchfork.
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So, we plot x versus r and it turns out be
quite interesting looking bifurcation diagram,
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as this customary the straight lines are stable
and the dotted lines are unstable branches.
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For small x the picture is qualitatively the
same, the origin is locally stable for r less
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than zero, two branches of unstable fixed
points actually bifurcate from the origin
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when r =0. Now due to the presence of the
x to the 5 term, the unstable branches actually
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turn around and become stable at r = rs and
these stable branches when exists for all
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r greater than rs.
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Now here is a notes about bifurcation diagram,
when rs is less than r is less than zero.
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We have the existence of two qualitatively
different stable states the origin and the
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large amplitudes fixed points. Depending on
the initial condition x not, one can get to
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the different stable states as t tends to
infinity. So, if you start the system at x*
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= 0 and slowly increase the parameter r it
remains stable at the origin till r = 0 and
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then will actually jump to one of the large
amplitudes branches.
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As r increases further the state moves out
along the large amplitude branch, so that
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is the jumps that will actually occur. If
we now decrease r the state remains on the
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stable branch even when r is less than zero
and it is only when r actually goes past rs
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does this state actually jump back to the
origin. So, the lack of reversibility as a
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parameter is varied is called hysteresis.
So, this lack of reversibility it can now
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been seen in the bifurcation diagram.
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The bifurcation at rs is a saddle node bifurcation,
stable and unstable fixed points are created
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as the parameter r is increased.
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We offer a few last comments a supercritical
pitchfork bifurcation is closely related to
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continuous or second order phase transitions
in statistical mechanics. In the engineering
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literature, it is called a safe bifurcation
because the non-zero fixed points are born
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at small amplitude. Now let us make some comments
about the subcritical pitchfork.
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Now because these are the two types you got
a subcritical and supercritical. So, let just
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go ahead and highlight them. So a subcritical
is related to discontinuous or first order
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phase transitions and in the engineering literature
is often seen as dangerous because of the
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jump from zero to large amplitude.
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This lecture was about a bifurcation called
a pitchfork bifurcation. Now there are lots
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of situation in the physical world, where
you have symmetry in them. Let us give an
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example. Now recall the beam buckling problem.
So, we had the beam and on top of the beam,
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you went placed the weight. When the weight
crossed a certain threshold, the beam actually
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buckled under the weight.
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So, the weight actually acts as the control
parameter which when it cross a certain threshold
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the beam actually buckle. Having said that
it was not clear that it buckled to left or
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it will buckle to the right, so this is very
simple example motivating some symmetry in
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real world systems. The pitchfork bifurcation
comes in two forms one is a supercritical
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pitchfork bifurcation and the other is a subcritical
pitchfork bifurcation.
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In the supercritical case, the normal form
for the supercritical is x dot = rx - x cube.
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So, what we did was, we had an example and
we actually looked at the bifurcation diagram,
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for this particular example and what we found
was that the parameter as it changed, fixed
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points were not destroyed. But the stability
of the fixed point actually can change ok.
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Then we went on to the subcritical case, in
the subcritical case you can start with the
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equation of the form x dot = rx + x cube,
so the positive x cube is destabilising. But
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in the real world what would happen is that
you would actually have a another high order
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term, which would act as the stabilising force.
So, the equations would be of the form x dot
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= rx + x cube - x to the 5 ok. Now when you
look at the bifurcation diagram of this particular
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system and there are a few interesting things
that actually show up.
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Number one is that you have a jump in the
bifurcation diagram. So, if the equilibrium
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is at a particular state and the parameter
actually crosses a certain threshold and the
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system can take a fairly large jump to another
branch. The second interesting thing that
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shows up is the potential lack of reversibility
as parameters vary.
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So was that basically means is that if a parameter
was at one place and then it moved to another
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place you had a certain change in the solutions
and then when it actually came back it did
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not actually go back to original state, but
actually went to some other state. So, this
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lack of reversibility is referred to as hysteresis.
So, the ability for the bifurcation diagram
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to exhibit a jump and to exhibit hysteresis
is what we found in the subcritical case.