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Now welcome back to one dimensional flows,
where we are still dealing with flows on the
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line. The focus of this lecture will be on
fixed points and stability. Now the geometric
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way of thinking can readably be applied to
any one dimensional system x dot is equal
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to f of x. Now essentially you just need to
draw the graph of f of x and sketch the vector
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field on the real line.
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So, just as an example, x dot verses x and
you plot any arbitrary function f of x may
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be, highlight the fixed points of the system
and draw the direction of the flows. Now what
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we do is you place an imaginary particle which
is called a phase point at x0 and see how
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it is actually carried by the flow. Now, as
time moves on the phase point moves along
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the x axis according to some function x(t).
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Now this function is called the trajectory
based at x0 and represents the solution of
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the ordinary differential equations starting
from the initial condition x0. The above picture
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which shows all the qualitatively different
trajectories of the system is called a phase
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portrait.
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The fixed points x* defined by f of x* is
equal to zero correspond to stagnation points
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of the flow. The solid dot is a stable fixed
point and the local flow is towards the stable
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fixed point. The open dot is an unstable fixed
point and the local flow is away from it.
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In terms of the ordinary differential equation
fixed points represents equilibrium solutions,
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if initially x is equal to x* then x(t) is
equal to x* for all t and equilibrium is stable
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if all sufficiently small disturbances away
from it actually dumped out in time. Unstable
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equilibrium in which disturbances grow in
time are represented by unstable fixed points.
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Now let us consider some examples, we talked
about some theory, let us look at some examples
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now. So find the fixed points for x dot is
equal to x square -1 and classify the stability
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of those fixed points. So x dot is equal to
f of x, which is equal to x square -1, which
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is equal to x -1 times x + 1. So to find all
the fixed points, we set f of x* is equal
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to zero and solve for x*.
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This gives x* is equal to + 1 and -1 the algebra
is straight forward. Now we need to determine
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if these fixed points are actually stable
or unstable. So we have to plot x square -1
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and sketch the resulting vector field. So
we go ahead and plot x dot versus x and make
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a plot of f of x is equal to x square -1,
highlight the direction of the flows and highlight
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the fixed points. The flow is to the right
where x square -1 is greater than zero and
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to the left where x square -1 is less than
zero.
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Thus x* is equal to -1 is a stable fixed point
and x* is equal to one is an unstable fixed
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point. Now we go ahead and highlight the stable
fixed point and the unstable fixed point in
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the diagram. Now here is a question for you
can you actually find an explicit analytical
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solution to the above ordinary differential
equation? Now note that we have we are able
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classify stability of fixed points without
actually knowing the analytical solution.
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But do go ahead find, try and see an explicit
analytic solution to the above ODEs.
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Now here are some notes the definition of
stable equilibrium is actually based on the
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notion of small disturbances. So small disturbances
to x* is equal to -1, will actually decay
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a large disturbance, where x finds itself
to the right of x = 1 will actually not decay.
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In fact the phase point will get repelled
towards plus infinity to get this point across
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x* is equal to -1 is locally stable, but not
globally stable. In the sense, that we will
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not have convergence from any arbitrary initial
conditions.
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Now let us consider another example, consider
the equation x dot is equal to x - cos x,
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sketch the phase portrait and determine the
stability of the fixed points. Now let us
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plot y = x and y = cos x and that is the curve
for y = x and that is simple minded curve
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for y = cos x. Now note that they intersect
only at one point. The intersection is a fixed
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point since x* is equal to cos x* and therefore
f of x* is equal to zero.
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When the line is above the cosecant curve,
when x is greater than cos x and x dot is
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greater than zero and so the flow is to the
right when the line is below the cosine curve
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and x dot is less than zero and the flow is
to the left. Now, we go ahead and actually
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identify the fixed point x*, now x* is the
only fixed point and it is unstable. We do
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not have a formula for x*, but we were still
able to classify its stability.
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Now let us look at our final example and consider
an electrical circuit, so let us go ahead
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and actually make a diagram of our electrical
circuit. We have resistor R and capacitor
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C which are in series with a battery of constant
dc voltage V not. Now suppose that switch
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is closed at t is equal to zero and that there
is no charge initially on the capacitor. Let
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Q of t denote the charge on the capacitor
at the t greater than or equal to zero.
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Now let us try and understand some aspects
of this particular system. Now, this type
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of circuit is actually governed by linear
differential equations and can actually be
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solved analytically. However, we work with
a geometric approach to try and get some intuition
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about the about the above circuit.
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So now, if you go around the circuit, the
total voltage drop must equal zero. Which
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gives us -V not +RI + Q/C = 0. Where, I is
the current flowing through the resistor.
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The current causes the charge to accumulate
on the capacitor at a rate Q dot is equal
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to I, this gives us -V not + RQ dot + Q/C
= 0. Rewriting this we get Q dot is equal
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to f is the function Q = Vnot / R - Q / RC.
Now the graph of the f of Q is just a straight
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line with negative slope.
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Now that is the plot of Q dot versus Q and
we plot f of Q on this plot the vector field
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as a fixed point where f of Q = 0 at Q star
= CVnot. So highlight the fixed point the
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flow is to the right where f of Q is greater
than zero and the flow is to the left where
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f of Q is less than zero. So the flow is towards
Q star and hence we have stable fixed point.
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Now let us plot the Q of t versus t, start
a phase point at the origin of the figure
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and let us imagine how it would move the flow
would carry the phase point monotonically
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towards the fixed point Q star. The speed
Q dot decreases linearly as it approaches
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Q star. So Q of t is increasing and concave
down. Now let us go ahead and plot Q of t
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versus t and highlight the equilibrium value
Q star and denote the stable fixed point and
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that is the curve for Q of t versus t. Now
in fact the fixed point is approached from
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all initial conditions and so in that sense
it is actually globally stable.
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Ok so let us have some final comments and
remarks about this lecture. Now this lecture
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was centred around fixed points and stability
and we started dealing with stability in little
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bit more detail because stability is key concept
in study of control and nonlinear dynamics.
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Now we have these special points referred
to as fixed points and we were able to classify
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whether fixed points was stable or they were
unstable.
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And we were able to do this without any formal
algebraic technique. But there was one point
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of subtlety that arouse and that was of local
versus global stability. Now let just take
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an example, for example we got this pen of
mine, and let assume that the tip of the pen
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is a stable fixed point. So if I found myself
to be a silent neighbourhood of this stable
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fixed point. Then there is decent chance that
I would actually get attracted towards that
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stable fixed point.
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On the other hand, if there was a large disturbance
and I found myself rather far away from this
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particular stable fixed point, then it is
not very clear that the trajectories would
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actually converge towards to the stable fixed
point. In fact one of our examples highlighted
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that, you could actually diverge away from
that stable fixed point, they could actually
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have large disturbance. So there is one subtle
point which came out in the lecture and that
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was of local versus global stability and that
should be one of the key take away from this
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lecture.