WEBVTT
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Language: en
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We focus is on ruling out closed orbits. So,
let us assume that based on some numerical
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evidence or our own intuition, we think that
a particular system has no periodic solutions.
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So how can we establish that the system indeed
has no periodic solutions? We will briefly
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outline three methods for ruling out closed
orbits. Number one Gradient systems number
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two is Lyapunov functions and number three
is Dulac’s criteria.
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So, let us look at Gradient systems, suppose
the system can be written in the form x dot
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= -del v, for some continuously differentiable
single valued scalar function v of x. Such
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a system is called a Gradient system with
potential function v. Note suppose that x
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dot = f(x)y, y dot = gs of xy then x dot = -del
v implies f(x)y -dv dx and gs of xy is -dv
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dy. So, there is a theorem which states that
closed orbits are impossible in gradient systems.
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We observe that, in fact most two dimensional
systems actually do not turn out to be gradient
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systems. Recall that all vector fields on
the line are gradient systems. Let us consider
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an example, show that there are no closed
orbits for the system x dot = sine y and y
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dot = x cos y. The system is a gradient system
with potential function v of xy = -x sine
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y and we can readily verify that x dot = -dv
dx and y dot = -dv dy and so by the above
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theorem there are no closed orbits.
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Have we discussed about Lyapunov function,
consider a system x dot = f(x) with a fixed
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point at x*. Suppose that we can find a Lyapunov
function that is a continuously differentiable
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real valued function v(x)with the following
properties, 1. The effects is greater than
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zero for all x0 = x* and v(x*) =0, so v is
positive definite. 2. v dot is less than zero
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for all x0 = x*, so all trajectories flow
towards x*.
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Then x* is globally asymptotically stable,
in sense that for all initial condition x
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of t tends to x* as t tends to infinity. In
particular, the system has no closed orbits,
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there are conclusion is as follows; all trajectories
move monotonically down the graph of v(x)towards
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x*. So, let us visualise this through a figure,
so that is the equilibrium point x*, that
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is the graph of v(x)and the trajectories move
monotonically towards the equilibrium x*,
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the solutions actually do not get stuck anywhere.
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If they did get stuck then v would actually
stop changing, but by assumption v dot is
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less than zero, everywhere except at x*.
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So how can we actually construct a Lyapunov
function, unfortunately there is no systematic
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way to construct a Lyapunov function. So,
let us consider an example, by constructing
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a Lyapunov function, show that the system
x dot = -x + 4y and y dot = -x -y to the cube
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has no closed orbits. Consider the function
v of xy = x square + ay square where a is
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a parameter which we will choose later. Then
v dot = 2x x dot + 2ay y dot which is = 2x
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times -x +4y + 2ay times -x -y cube which
is = -2x square +8 -2a times xy -2ay to the
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4.
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So now we can choose a = 4 and the xy term
they will vanish and we are left with v dot
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= -2x square -8y to the 4, so we can easily
check that v >0 and v dot <0 for all xy not
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equal to 0 0. Hence v = x square +4y square
is indeed a Lyapunov function and so we do
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not have closed orbits. In fact, all the trajectories
approach the origin as t tends to infinity
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and so the system is in fact globally asymptotically
stable.
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(Refer Stable Time: 08:10)
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So now we discuss Dulac’s criterion, let
x dot = f(x) be a continuously differentiable
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vector field defined on a simply connected
subsect R of the plane. If there exists the
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continuously differentiable real value function
g(x) such that delta times g x dot has one
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sign throughout R, then there are no closed
orbits lying entirely in R. Unfortunately,
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there is no systematic way of finding g(x).
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Let us consider an example, show that the
system x dot = x times 2- x - y and y dot
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= y times 4x - x square -3 has no closed orbits
in the positive quadrant xy greater than zero.
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So, let us go ahead and pick g = 1 / xy, then
del dot g x dot = ddx g times x dot + ddy
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g times y dot = ddx times 2- x - y / y + ddy
of 4x -x square -3 / x = -1 upon y which is
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less than zero. Since the region xy>0 is simply
connected and g and f satisfy the smoothness
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conditions, so Dulac’s criteria tell us
that there are no closed orbits in the positive
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quadrant.
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Now we will have a model of the real world
is quite use full to know if we can actually
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rule out closed orbits. So, in this lecture,
we outlined three methods that can be useful
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for ruling out those topics, number one was
Gradient systems so you show your system is
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a Gradient system, the second is based on
the method of Lyapunov functions and third
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was based on Dulac’s criteria. Now all these
three are very, very powerful theoretical
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ideas.
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And when work they can be very powerful except
that the only issue is that find to get them
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to work in practice can be slightly tricky
because there is no real systematic procedure
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on for example you might be struck Lyapunov
function. So, it is nice to know that this
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method exists, that it is also nice keep in
mind that in practice, they sometimes can
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be little tricky to use and to actually show
for your real-world model none the less they
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are extremely powerful methods which you should
be aware off.