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In this lecture, we focus on linear stability
analysis to determine the stability of fixed
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points. We had earlier used graphical methods.
We now consider linearizing about a particular
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fixed point, what really is linearization?
Well it is finding a linear approximation
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to a function at a given point. So, the linearization
of a function is the first order term of its
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Taylor series expansion around the given point.
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In terms of nonlinear systems linearization
allows us to study the local stability of
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an equilibrium point of a nonlinear differential
equations. And it allows us use tools for
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studying linear systems to analyse a nonlinear
system near a given special point.
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Now consider x dot = f(x) let x* be a fixed
point and let x(t) be eta of t plus x* where
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eta of t is a small perturbation away from
x*. We really want to know if the perturbation
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actually grows or decays. So we go ahead and
derive a differential equation for eta. So,
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eta of t = x of t minus x* so eta dot = dx
dt x(t)- x* which = x dot as x* is simply
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a constant.
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So eta dot = x dot which = f(x*) which = f(x*)
plus eta and so now we go ahead and using
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a Taylor series expansion we get f(x*) plus
eta = f(x*) plus eta times f prime of x* plus
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terms which are order eta square. Note that
this is the big “O” notation where this
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term actually denotes quadratically small
terms in eta.
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Note that f(x*)=0 since x* is a fixed point.
Hence eta dot = eta times f prime of x* plus
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terms that are order eta square. If f prime
of x* is not equal to 0 the order eta square
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terms are in fact negligible and we get the
following approximation eta dot = eta times
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f prime of x*. Now this is a linear equation
in eta which is called the linearization about
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x*. And here are some notes the perturbation
etas of t grows exponentially.
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If f prime of x* is > 0 and decay is f of
prime of f star less than 0. If f prime of
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x* =0 the order eta square terms cannot be
ignored and we will need some sort of nonlinear
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analysis for the equations. Now if you consider
the magnitude of f prime of x* then this magnitude
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plays a role of an exponential growth or decay
rate. It is reciprocal one upon f prime of
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x* is a characteristic time scale. Now this
determines the time required for x of t to
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vary significantly in the neighbourhood of
the fixed point x*.
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Now let us consider this equation x dot = sine
x. Now recall that from the geometric view
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of thinking essentially, what we said was
that when we have equation of form x dot = f(x*)
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then what we first do is just plot x dot versus
x. So for this particular equation where f(x*)
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is sine x. So what we see here is a plot of
x dot versus x where using the geometric view
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of thinking that we have outlined earlier
we were able to classify the stability of
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the fixed point noting that we are able to
do so without any formal analysis.
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And so now we go ahead and use the linear
stability analysis to determine the stability
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of the fixed points for the equation x dot
= sine x. Recall that the fixed points occur
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where f(x*) = sine x =0. So the fixed points
are at x* = K pi where K is an integer then
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f prime of x* = cos of x* which = cos of K
pi which = 1if K is even and -1if K is odd.
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So when K is even x* is unstable and when
K is odd then x* is stable, so this is actually
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in complete agreement with the results that
we obtained from the geometric view of thinking.
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Now, note that with geometric view of thinking,
we had no analytic basis really to establish
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the stability of the fixed points and it was
purely geometric. But now we actually have
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an analytical and algebraic technique to actually
establish the stability or the instability
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of the fixed points.
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Now let us consider examples that arises in
population growth consider the equation dn
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dt = r N times 1- N by K where N of t is the
population at time t, r > 0 is the growth
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rate and K is carrying capacity of the population.
Our objective is that using linear stability
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analysis, we wish to classify the fixed points
of the model and we also wished to find characteristic
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time scale of the system. Now f(N) = r N times
1-N by k. So the fixed points are N*=0 and
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N*= K evaluating f prime of N we get r -2rN
/ K so f prime of N at N*=0 is r.
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So N*= 0 is actually a unstable fixed point.
And f prime of N evaluated at N*= K is –r,
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so N*= K turns out to be a stable fixed point.
In both of the above cases the characteristic
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time scale turns out be the same, so one upon
the absolute value of f prime of N*= one upon
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r for both the fixed points. Now let us go
ahead and plot N dot versus N by required
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familiar with making such plots. So that is
the plot for N dot versus N, 0 and K the two
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fixed points have been highlighted and K represents
the stable fixed point and 0 represents the
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unstable fixed point.
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So what can we say about the stability of
a fixed point when f prime of x* actually
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is 0 as a matter of fact in general we cannot
say anything stability has to be worked out
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and determined on a case by case basis. Now
let us go ahead and consider some examples,
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x dot = - x cube, x dot = x cube and x dot
= x square in all the cases x* = 0 and f prime
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of x* =0. Now let us go ahead and consider
stability in each of these cases.
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So we plot x dot versus x and what we find
is that what we have attracting stable fixed
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point and similarly plotting x dot versus
x in this particular case we find that we
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have an unstable fixed point. So the last
case actually presents us rather interesting
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situation. We go ahead and plot x dot versus
x and that is the plot. So we find that the
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fixed point is attracting from the left and
it turns out be repelling from the right.
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So we get the situation where the fixed point
turns out to be stable on the one side but
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unstable on the other side. So such a situation
is referred to as a half stable fixed point
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attracting from one end and repelling from
the other.
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Ok so let us do the quick recap of this lecture.
This lecture was essentially about linear
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stability analysis. We still dealing with
the equations of form the x dot = f(x*) and
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essentially the idea was that you identify
the equilibrium point. You take a Taylor series
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expansion of the original nonlinear system
around the equilibrium point. You retain the
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linear terms and all higher order terms are
actually thrown out. So you are essentially
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left with linearized version of the original
nonlinear system.
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And now you can go ahead and apply all the
tools and methods that you learned for linear
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system analysis to this particular equation.
So that is a big advantage with this advantage
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is that you actually thrown out all the nonlinear
terms out ok. So if you thrown all the nonlinear
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terms out all the fun all the interesting
dynamics are essentially been thrown out,
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it is a good start, but it is a only a start,
so then what we did was we took a few examples
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x dot = sine x. We took another example, where
which was motivated from a population dynamics.
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And then we essentially showed that if you
conduct a geometric reasoning around nonlinear
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system which is by plotting x dot versus x
or if you did a linear stability analysis
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then you got essentially the same results.
Except that we left you with some food for
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fork with last example. The last example was
rather interesting one, now essentially what
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you actually had was that you had was fixed
point, which it turned out be attracting from
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one end but repelling from the other end.
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So the fixed point could not really be classified
has an stable fixed point or an unstable fixed
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point and the terminology we use was a half
stable fixed point right and I think we were
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just we leave this lecture with that particular
food for fork with you that you could also
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end up you know fixed point would essentially
not be just stable or unstable. But you can
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an actually have this sort of hybrid cases
which arise, which is half stable and half
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unstable.