WEBVTT
Kind: captions
Language: en
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We are still dealing with two dimensional
flows and here we deal with the phase plane.
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We start our study with phase portrait, so
we now begin our study of two dimensional
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nonlinear systems. The general form of a vector
field on the phase plane is x1 dot = f1 x1
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x2 and x2 dot = f2 is the function of x1 x2,
where f1 and f2 are given function. So, writing
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in vector notation, we have x dot = f of x,
where x = x1 x2 and f of x = f1and f2 of x.
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Here x represents a point in the phase plane
and x dot is the velocity vector at that point.
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Note green represents vectors. So, the entire
phase plane is usually filled with trajectories
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as each point can play the role of an initial
condition. With nonlinear systems, there is
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no hope of finding the trajectories analytically.
So, we normally try to determine the qualitative
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behaviour of the solutions. So, our objective
is to find the systems phase portrait directly
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from the properties of f of x.
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We outline some salient features of any phase
portrait, these include fixed points, closed
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orbits, the arrangement of trajectories near
the fixed points and closed orbits and finally
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stability or instability of fixed points and
closed orbits. Now let us plot an example
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of a phase portrait, a is the fixed point,
b is the fixed point, c is the fixed point
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and then we have the closed orbit d. So, a,
b and c are fixed points. Fixed points satisfy
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f of x* = 0 and are study states or equilibrium
of the system.
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D is a close orbit, these are periodic solutions
ie solutions for which x of t + capital T
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= x of t for all t and for some capital T
greater than zero. So now what we do is we
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start filling out the phase portraits. See
we get rather a pretty looking picture. Now
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observe the trajectories near the fixed points
and the close orbits the flow pattern near
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a and c is similar, which is different from
that near b.
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We also need to ascertain the stability or
instability of the fixed points and the closed
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orbits, a b and c are unstable as nearby trajectories
tends to move away from them and d which is
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a close orbit is stable.
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So, we look at an example, consider the system
x dot = x + e to the -y and y dot = -y. So
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now we use qualitative arguments to obtain
information about the phase portrait. So,
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we first find the fixed points. So, we solve
x dot and y dot =0 and the only solutions
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is x* y* = -1 0. Now we try and determine
stability, note that ys of t is tends to zero
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as t tends to infinity as the solutions to
ys of t = -y is ys of t = y not e to the -t.
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Thus, e to the -y = -1 and so in the long
run the equation for x is simply x dot is
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= x +1. Now this equation has solutions that
are growing exponentially which tells us that
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the fixed point is unstable. Let us restrict
our attention to initial conditions on the
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x axis, when y not = 0 and ys of t =0 for
all time. Now the flow on the x axis is in
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fact governed by x dot = x +1 and thus the
fixed point is indeed unstable.
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Now in order to sketch the phase portrait,
we plot the null clients. These are defined
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as curves where either x dot = 0; y dot = 0.
The null clients indicate where the flow is
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purely horizontal or vertical. For example,
flow is horizontal where y dot = 0 and as
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y dot =- y, this happens on the line y = 0.
Along this line, the flow is to the right
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where x dot = x +1 greater than zero that
is where x is greater than -1.
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In a similar way, the flow is vertical where
x dot = x + e to the -y and on the upper part
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of the curve where y is greater than zero.
The flow is actually downward since y dot
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is less than zero. Armed with all this information,
let us plot the phase portrait, so that is
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familiar y versus x. We highlight the fixed
point; the null clientâ€™s partition the plane
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into regions where x dot and y dot have various
signs. So, the fixed points actually look
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like a nonlinear version of a saddle point.
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The topic of the discussion in this lecture
was phase portraits in two dimensional systems.
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So, we had equation of the form x1 dot = to
f1 is a function of x1 and x2 and x2 dot = f2
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that is another function f2 as function of
x1 and x2. And the objective is to plot x1
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versus x2. There are four salient features
that you need to keep in mind when plotting
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the phase portrait for the two-dimensional
systems.
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Number one identify all the fixed points,
number two identify all the close orbits,
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number three pay close attention to the arrangement
of the trajectories closed to the fixed points
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and close to the closed orbit and number four
highlight the stability and or the instability
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of the fixed points and the closed orbits.
And in this lecture, we have one particular
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example, but these are four salient features,
that you should look for in any phase portrait
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of a two-dimensional system.