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We start with one dimensional flows and we
focus on flows of the line.
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Consider the following general nonlinear system,
X1 dot is equal to f1 is the function of X1
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to Xn all the way to Xn dot is equal to fn
is the function of X1 to Xn.
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Now let us make a statement about the above
system, the solutions can be visualised as
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trajectories flowing through an n dimensional
phase space with coordinates X1 all way to
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Xn.
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Now let us actually go bit easy and start
with the case n is equal to one, that is x
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dot is equal to f(x) where x(t) is the real
valued function of t and f (x) is a smooth
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real valued function of x.
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So this is an example of a one dimensional
system.
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Now here are some notes as the function f
does not explicitly depend on time the resulting
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equation is autonomous.
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Non-autonomous equations are the equations
of the form x dot is equal to f(x) t and are
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in general much harder to analysis.
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Now given a nonlinear equation x dot is equal
to f of x, normally we first try and look
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for an explicit analytical solution.
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This is usually very hard, but we can start
learning about the equation using geometric
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methods.
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Here is an example, x dot is equal to sine
x, which is nonlinear because of the sine
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x term.
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Now let us actually first consider a small
angle approximation, which is useful for trigonometric
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functions in the limit that the angle approaches
zero so sine x is approximated as x.
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So now consider dx dt is equal to x, which
is a linear equation.
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So to solve it we separate variables to get
dx by x is equal to dt.
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Note that we are assuming that x is not equal
to zero.
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However the X of t is equal to zero is also
a solution to this ordinary differential equation.
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So we integrate and then exponentiate to find
a solution X of t is equal to C1e to the t
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where C1is arbitrary constant.
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But what we really wanted to do was to learn
about the original nonlinear system.
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Now let us actually start with the nonlinear
equation x dot is equal to sine x.
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Now this equation actually does have an analytical
solution, but very few nonlinear equations
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actually admit exact solutions.
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So to solve the equation we separate the variables
and then integrate, so dt is equal to dx by
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sine x which gives us t is equal to integral
of cosecant of xdx evaluating this integral
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we get minus log of cosecant of x plus cotangent
of x plus a constant.
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We suggest that, the look this integral up,
we need to evaluate the constant C1, so let
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x is equal to X0 at t is equal to zero and
we get C1 is equal to log of cosecant of X0
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plus cotangent of X0.
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So the final solution turns out to be t is
equal to the cosecant of X0 plus cotangent
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of X0 divided by cosecant of X plus cotangent
of x.
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So here is an exercise can you solve for X
in terms of t you should do it on your own.
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Now we have an exact result but can it actually
help to build intuition about the original
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nonlinear system.
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Now, let us just recap what we have, we have
a nonlinear equation x dot is equal to sine
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x and using some analytic methods we were
able to obtain an exact analytic solution
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to this particular equation.
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Now what we really want is to build intuition
about the nonlinear system and be able to
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answer practical minded questions.
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For example suppose that the initial conditions
is X0 is equal to pi on four and can we describe
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the qualitative features of the solutions
X(t) for all t greater than zero.
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What happen as t tends infinity more generally,
suppose we pick any arbitrary initial condition
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X of 0, than what is the behaviour of X of
t as t tends to infinity.
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Now the explicit solution is exact, but not
extremely helpful to answer the above questions
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quickly.
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Now the nonlinear equation at hand is x dot
is equal to sine x, so let us actually plot
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x dot versus x.
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Here is slightly simple minded plot of x dot
versus x.
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Now think of X as the position of an imaginary
particle, that it is moving along the real
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line when x dot is the velocity of the particle
then x dot is equal to sine x represents the
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vector field on the line.
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Now, essentially it represents the velocity
vector x dot at each x.
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Now to plot the vector field we do the following
plot x dot versus x.
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Draw arrows on the x axis to indicate the
corresponding velocity vector at each x, the
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arrows should point to the right when x dot
is greater than zero and they should point
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to the left when x dot is less than zero.
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Now, please pay very close attention to the
plot of x dot versus x.
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We highlight the region of positive velocity
where the arrows point to the right and note
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the region of negative velocity when the arrows
when point to the left.
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Now let us actually offer some physical interpretation,
imagine that we have some fluid that is flowing
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steadily along the x axis with a velocity
that actually varies from place to place then
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this is essentially just happening according
to x dot is equal to sine x.
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Now the flow is to the right when x dot is
greater than zero and the flow is to the left
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when x dot is less than zero at points where
x dot is equal to zero there is actually no
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flow and such points are called fixed points
this are points there is no velocity.
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Now there are two kinds of fixed points, stable
fixed points also referred to as attractors
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and unstable fixed points referred to as repellers.
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Now look at the diagram the closed circles
are the stable fixed points and the open circles
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are the unstable fixed points.
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Note that on the stable fixed points the flow
is getting attracted towards them and on the
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unstable fixed points the flow is getting
repelled from there.
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What we really want to do now is get some
additional insight into this nonlinear system
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that we have now recall the earlier question
suppose that x0 is equal to pi on four then
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what really happens as t tends to infinity.
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Now from the above picture the particle starts
at x0 is equal to pi on four and moves to
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the right faster and faster until it cross
X is equal to pi on two then the particle
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slowly starts to slowdown and approaches the
stable fixed point at X is equal to pi.
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So the limit as t tends to infinity X(t) is
equal to pi if the particle actually as started
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at pi on four.
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Here are some notes, first the curve will
be concave up corresponding to the initial
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acceleration for X less than pi on two.
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Then the curve will be concave down highlighting
deceleration towards X is equal to pi.
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Now let us go ahead and plot X versus t.
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We first highlight X is equal to pi which
is our stable fixed point, highlight pi on
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two pi on four which was our initial condition
and that is the curve of X versus t.
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Observe that we did not use the analytical
solution to actually answer the above question.
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We can apply the same reasoning to any initial
conditions X0.
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If x dot is greater than zero initially the
particle moves to the right and asymptotically
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reaches the nearest stable fixed point.
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If x dot is less than zero initially, the
particle approaches the nearest stable fixed
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point from its left and if x dot is equal
to zero then x remains constant.
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The qualitative form of the solution for any
initial condition can actually be plotted
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using the rules that we have identified on
the left.
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Here is the snapshot of what the solution
should look like.
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Note that the trajectory are converging towards
x is equal to pi and x is equal to minus pi
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and the reason is because these are the stable
fixed points.
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Ok so now let us just rap up this lecture
with some concluding remark.
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We started our study of nonlinear system with
equations of the form x dot is equal to f
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of x.
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These were one dimensional flows and in particular,
we were rather keen to understand flows on
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the line.
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Now when you given a nonlinear equation the
first thing you normally try, is to try and
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get an explicit analytical solution.
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A first thing to remember is that explicit
analytical solutions are usually very, very
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difficult to get a nonlinear system ok.
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So we have a particular example we said let
x dot is equal to sine x.
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Now in this particular case we were able to
get an explicit analytic solution we separated
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the variables and we integrated and the integral
work out explicitly.
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And in this case, we were able get explicit
analytic solution.
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However, the form of the solution was not
very easy to understand it was not very easy
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to absorb its certainly was not easy to develop
intuition about the form of the solutions.
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So what we did was we went on to look at some
geometric reasoning.
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Now essentially, all we did was we plotted
x dot versus x, for this particular equation
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x dot is equal to sine x.
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Now as soon as we did that we found that there
were interesting cases that showed up when
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x dot was greater than zero, when x dot was
less than zero and when x dot was actually
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equal to zero, we encountered fixed points
for the first time.
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Now such fixed point we found either be stable
or they could be unstable.
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And now with this geometric form, we were
able to ask and start answering questions
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of the form that if you choose a particular
initial condition, what would be the solutions
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look like as t tends to infinity.
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So in particular, we just picked X0 is equal
to pi on four as initial condition and ask
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what happens as time carried on.
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So the lesson learnt from there was that geometric
reasoning can certainly complement analytical
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solutions within the case.
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That we can find analytic solutions and in
the case, that we actually cannot find the
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analytic solutions is a good way to actually
start yah.
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And but we have to be very careful that geometric
reasoning may not be able to give all the
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answers we want.
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For example, if we were ask after a particular
quantitative question of the form, what is
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the time at which x dot is equal to sine x
has the greatest speed then geometric reasoning
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will not be in a position to answer such form
of quantitative questions.