We are close to the end of the course.
Through the whole set of lectures, we have
learnt that there are some very peculiar specialties
of chaotic systems.
Most important of them being that tiny perturbations
in the state can lead to very different results.
Tiny perturbations in the parameters can again
lead to very different results.
Things are very sensitive to perturbations
in the state, to perturbations in the parameter.
Naturally this often gives raise to the intuitive
feeling that these guys are the very difficult
one to follow.
In fact that has been the belief ever since
the advent of this subject that here are some
systems which are chaotic by themselves and
if that happens then you have difficulty because
it has certain characteristic features but
nevertheless ultimately when you want to control
it, it becomes very difficult because slide
change here and there will lead to completely
different results.
But over the last 12 years or so, there have
been some very important developments that
have proved this belief to be wrong.
In order to illustrate what this is, I will
get into a step by step.
Another important specialty of chaotic system
we have learnt is that while the state moves
in the attractor chaotically that means the
state never repeats itself.
There are infinite number of unstable periodic
orbits embedded in the attractor and while
going through this kind of erratic motion,
this state often comes very close to one of
those unstable periodic orbits.
It cannot get locked there because it is unstable.
Slight perturbations, slight difference from
the unstable periodic orbit will lead it to
elsewhere but nevertheless the point is that
there are an infinite number of unstable periodic
orbits embedded in that attractor.
It may so happen that one of those unstable
periodic orbits represent a very desirable
behavior of the system.
For example there is an engineering system
where the desirable behavior is represented
by one of the unstable periodic orbits but
it’s unstable.
Therefore if you really release it there,
it will not remain there.
It will go away elsewhere but if you can identify
one of those unstable periodic orbits as representing
a desirable behavior, for example there have
been the situations where there was a laser.
You see people want to improve the power throughput
in the laser guard and that depends on which
periodic orbit you are actually working in.
They were shown to the situations where the
power can be doubled, if you lock into one
of those unstable periodic orbits and there
are similar situations in other areas also.
The problem then becomes, can we control the
unstable periodic orbits embedded within a
chaotic attractor?
The reason that I showed that this is especially
advantageous is that for a non-chaotic system,
the kind of system that you have all come
across in regular control theory courses are
where if you want to bring about some change,
you have to put in some control action.
In order to get a desirable behavior which
may be quite different from what your presently
operating behavior it is.
That means if you want to get into a large
change in the resulting behavior, you have
to put in a large control action.
That means the kind of system that you have
already come across, not in this course elsewhere.
There large control action is required in
order to bring forth some kind of a large
change in the character of the system, small
control action will only result in a small
change.
That’s what but here in the kind of system
that we have been discussing, there exist
a possibility that tiny change in the parameter
or the state might lead to a very large change
in the resulting behavior because there is
sensitive dependence on initial conditions,
there is sensitive dependence on parameters
and so there exists the possibility that a
very tiny nudge, a tiny perturbation, tiny
very directed control action might lead to
very different system behavior and that very
different system behavior might be the one
that is desirable.
You now have the possibility of enabling a
control action by very tiny perturbations.
How can we do that?
That is the subject matter of today’s talk.
The point is this could be handled in the
plane of continuous time dynamical system
but as you know mostly such things are handled
more easily, the mathematics become much simpler
if we treat this problems in discrete time.
We are essentially talking in terms of the
Poincare section and what happens there.
If you consider what happens there then you
will find there is a chaotic orbit and if
you place a Poincare section there are an
infinite number of points through which this
piercings happen.
But out of that, one represents an unstable
periodic orbit and that on the Poincare plane
is an unstable fixed point of that Poincare
map.
Suppose that Poincare map is represented as
zn+1 is equal to some function of zn and the
parameter, z is the state.
We are now bringing the problem down to the
state of discrete time representation and
this is the discrete time representation of
the system suppose.
Then in the neighborhood of that unstable
periodic orbit, the one that you are trying
to stabilize.
Suppose that unstable periodic orbit is represented
by zstar then in the neighborhood of that
suppose here is the unstable periodic orbit
and so this is the zstar and here is say zn.
Presently the deviation is this much.
In the next iterate suppose it mapped to this
point, see it went like this.
As a result the deviation is this much.
Then this deviation can be expressed as a
function of this deviation.
In fact in the local linear neighborhood this
can be represented as a linear mapping.
We can write that local linear approximation
as zn+1 minus zstar that means the resulting
deviation is equal to, we are now writing
in terms of the linear so there would be A
matrix multiplied zn minus zstar.
This will be the relationship as dependent
on the state.
Now see the original thing was also dependent
on the parameter.
So you can also write another parameter dependence
factor as B (p minus p0) where p0 is the parameter
for which you had obtained zstar.
This representation is nothing, very simple.
It is essentially, if you consider no change
in the parameter then this part is 0.
Then you are saying that the deviation in
the next iterate is nothing but a matrix time
the deviation in the previous iterate.
That means you are essentially representing
as a linear equation.
Similarly if you suppress this, that means
this remains constant then the change in this
brought about by a change in the parameter
is given by the matrix B. Here you have a
local linear representation.
Now what will you do?
You will essentially observe how much is this
deviation.
That means you will observe in any particular
iterate, how far have I fallen away from the
state where you want to be.
That means this is the state where I want
to be and this is where I have fallen.
I will measure this and depending on this,
I will change the parameter by a small amount,
very tiny amount but nevertheless I will change
the parameter depending on this.
We can write this again like a linear relationship
as the p in the nth iteration that means I
am assuming that it is possible to change
the parameter at every iteration by small
amount but it is possible to change by a small
amount.
So p, the parameter in the nth iterate minus
the p nominal value that means how much perturbation
I am giving in the parameter that should be
dependent on some constant matrix times the
deviation in the state.
If I have deviated so much, I will give so
much parameter perturbation.
Now here there has to be a transpose because
obviously you need to bring it to a one dimensional
state.
Then only it is represented as minus k transpose
that means if the deviation is positive, you
give a negative deviation in the parameter.
This is again a very linear way of looking
at it.
I will show that also works because see in
the neighborhood of that unstable fixed point,
you can always locally linearize it and that
behavior is essentially this behavior.
We can write it like this.
What is the dimension of this?
The dimension of z that means if it is n dimensional,
this would be n dimensional stuff.
Naturally this has to be an n dimensional,
n to one dimension so that their product gives
a one dimensional.
Now if I substitute it here, what do we have?
We have zn+1 minus zn, let us write it as
delta zn+1.
See we are substituting it here.
This zn minus zstar which is nothing but delta
zn or deviation in the nth iterate that remains
common.
So what do we have here?
You have A minus B k transpose times delta
zn.
If the initial deviation were this much, this
will be the final deviation after one iterate,
if you are giving a parameter perturbation
of this extent.
Now do you see what is the condition for stability?
Simple, this matrix here must have eigenvalues
all within the unit circle that’s it.
As simple as that.
Just ensure that this matrix has eigenvalues
inside the unit circle that immediately guarantees
that in successive iterates, this deviation
will die down to zero.
Now normally the way it is done is obviously
if the zn is large, see this relationship.
If the zn is large deviation from the zn star
then what will happen?
This number is large, as a result multiplied
with the k transpose it will give a large
perturbation and mostly we don’t want it.
Because the main advantage of the chaotic
system is that we can do it with a small perturbation.
So what we do is if the deviation is large,
we simply wait.
No, this is not the right time to apply the
perturbation.
Wait, because we know that if the system is
chaotic, it’s also ergotic means that if
we wait sometime then sometime or other, the
state will fall in a close neighborhood of
zstar and when it does, apply the perturbation.
You simply wait till this term becomes tiny
enough so that if your controller says that
my control will be only this much and no further.
Then simply wait till it comes within that
range and then apply.
You normally see that there is a chaotic system
going on, it moves chaotically and the moment
it falls within that small neighborhood, immediately
you apply the control action and there it
is.
It immediately gets locked to the unstable
period orbit.
Now this getting locked to the unstable periodic
orbit or getting controlled into the unstable
periodic orbit, you might visualize as something
like this.
See you can make a stick stand on your finger.
So that’s inverted pendulum position, you
know we have already said that is a saddle,
that is the unstable fixed point, unstable
equilibrium point.
You still can move it tiny bits and keep it
vertical, you don’t really need to go around
moving a large amount.
You can do it by small amount.
Can you not?
The way you can, here you were doing exactly
that.
Under what condition we will need to move
it by large amount?
If the position of the stick is like this
then you will have to move it like this so
that it becomes vertical.
But if it is very close to the vertical position,
you can always move it by very tiny amount
to keep it vertical that’s exactly what
we are doing.
A is essentially the Jacobian matrix and B
is this one as differentiated with respect
to p.
So here is a zn+1 as a function of zn and
p, if you differentiate it with respect to
zn, you get the Jacobian matrix which is A.
If you differentiate with respect to p you
get, which is B. But you might ask in that
case this functional form has to be known
otherwise how do I differentiate.
I will come to that issue a little later.
This algorithm has been applied to many different
situations but the most, the one that goes
into scientific facular is the paper where
the scientists created artificial fibrillation
in a frog heart.
You know they dissected the heart that was
still alive, heart of a frog and they induced
artificial fibrillation, the way a man dies
before that there is a fibrillation.
That’s why you put defibrillates and all
that.
They induced this and then they apply this
small tiny nudges and they were able to stabilize
the heart for a long time.
So that was a nature paper that has become
sort of a turning point in the application
of this theory.
It has been applied to lasers, it has been
applied to many different areas.
Let us try to understand this scheme somewhat
intuitively.
What are we doing?
Suppose the fixed point zstar is a saddle
fixed point.
Here is your zstar this particular position,
a saddle means there would be the manifolds
which in this case will be a stable manifold
and an unstable manifold.
Suppose this is the unstable manifold and
this is the stable manifold.
Here is your zstar and suppose at any particular
point of time it lands here.
What are you trying to do?
You are trying to move it back here but notice,
here you have the advantage of having a stable
manifold means that if you can somehow nudging
to this point then you don’t have to do
anything.
Automatically it will run into this.
Essentially the point is true is to push it
here.
Now what are we doing?
We are changing the parameter.
A parameter change means for that changed
parameter, if you now calculate the fixed
point, it will be a different fixed point.
Fixed points position will be different.
Now suppose the fixed point’s position now
is somewhere here.
It has moved that means this is zstar for
p0 and this is the zstar for p0 plus delta
p.
So it is moved here.
As it is moved here, it will again have the
stable manifold and the unstable manifold.
This is the unstable manifold and this is
the stable manifold.
What will be the character of the unstable
manifold?
What will it do to a particular orbit sitting
here?
In the next iterate how will it move?
It will move away more or less in the direction
of the unstable manifold and it will move
towards this in the direction of the stable
manifold.
As a result in the next iterate, it will fall
somewhere closer here because it will move
towards this and towards that and by proper
choice you might make it fall just here.
That means exactly on the stable manifold
of the earlier fixed point and then withdraw
that perturbation.
The fixed point comes back here and now you
have the point exactly on the stable manifold.
Just wait, it will automatically get there.
What have you done?
You have just applied a perturbation at a
particular instant and then left it.
Only once you have applied the perturbation
and that’s it and then withdrew the perturbation.
Not that you are keeping on the perturbation
but the action of this specific geometry of
the system ensures that the iterates slowly
converge on to the fixed point.
It’s not difficult to see that if you want
to bring it exactly on the stable manifold
here, all that you need to do is to ensure
that one of the eigenvalues here is 0 and
the other one is what it was without the perturbation.
You can easily calculate the K matrix because
the A and B are known.
Is that clear?
Here you have equation, its one of the eigenvalue
should be 0, the other one should be as it
was without the perturbation then it will
simply be landing here.
It’s extremely simple to obtain the K matrix.
Of course all the time you might not need
to say it exactly equal to 0 because here
what you are doing?
You are exactly putting it on to the stable
manifold.
That doesn’t always mean that it will remain
there because the system has some noise.
Though I am saying that if you land it here,
it will automatically come to this.
It doesn’t really, this argument holds in
absence of noise.
If there is noise then obviously it goes again
out.
The moment it goes out, you wait for some
time but you again apply the perturbation.
So that this algorithm actually rests on repeated
application of the perturbation depending
on the amplitude of the noise.
This method was invented by Ott, Grebogi and
Yorke and that is why this method is called
OGY algorithm or OGY method.
The question he asked was what about A and
B, how do you know them?
Obviously if have the system equations given
like this, obtaining A and B are trivial but
for a realistic system experiment is running
you don’t really know A and B. You would
like to somewhat estimate A and B. How to
estimate A and B?
What is happening?
You have the system running, actually it is
a continuous time system and you are placing
a Poincare section, you are observing the
points on the Poincare section.
First what would be the character of the orbit
when it comes in the neighborhood of that
particular fixed point because it is ergotic
so it will go on moving everywhere and it
will come arbitrary close to the fixed point
also.
If it does what will be the character of the
next iterate?
It will again fall close to that because it
is close to a fixed point.
If you have a time series on the Poincare
section that means this point to this point
to this point to this point and if you have
a file containing the time series, if you
go on scanning the file you will find two
lines that are very close to each other.
If you do detect then you know that here the
pointer is a close neighborhood of that unstable
fixed point.
Take these two values.
Again if you keep on scanning it, again sometime
later it will come very close to that.
That means you will again get a pair that
are very close to each other, take those values.
Similarly by scanning the whole file containing
the data of the system, you will get a large
number of such pairs.
The pairs that fell close to each other.
That means you have got now a collection of
points that fell close to each other.
Now you are trying to estimate the values
of A and B formula.
What do you have?
You have got a point that map to another point.
You have got another point that map to another
point, you have got third point that map to
third point.
All these data you have.
Essentially you have zn mapping to zn+1 and
as we have seen that you can represent it
as a linear function like zn+1 is equal to
A plus, it’s like a affine transformation.
You can represent it like this, these hats
I am putting because these are the values
that we need to estimate.
Essentially since we do not know, if we had
known the fixed point then we would say that
now let that fixed point be my origin and
we will count only the deviation from that
origin, from that fixed point.
But here we do not know the location of the
fixed point.
We only know that this point map to this point
but we only know intuitively that the fixed
point must be somewhere in that neighborhood.
We do not know the fixed point.
If we do not know the fixed point, we are
working on the original coordinate system.
We are unable to move the origin to the fixed
point and that is why we need to consider
this C.
This becomes an affine transformation.
Once you have a large number of these values
zn+1 and zn, you can do a least square fit
to obtain the values of A matrix and C matrix.
You can do a least square fit to obtain these
values A matrix and C matrix.
Once you have obtained so from here 
fit to A matrix and C matrix, you obtained.
Now the question is can you locate the fixed
point?
Yes, you can because once you have obtained
it, you will say the zstar is equal to A hat
zstar plus C. How would you obtain zstar?
I minus A into… this has to come to this
side, inverse so that will be the zstar.
We have located the fixed point.
So zstar, the fixed point is we did not know
where it is.
But simply from observing the data, we can
locate the fixed point.
But for that we need to first obtain the A
hat and C hat and then just do this, obtain
the position of the fixed point.
That means we have been able to estimate A
matrix, this is the same as the matrix A that
appears here.
Estimated A matrix from the data but what
about B?
Now B is obtained, as you can see B is where
you are asking the question if I change the
parameter how much will my fixed point move?
That is the essential question you are asking.
All you need to do is to give some perturbation
to the parameter and redo this procedure.
As a result of which the zstar will change
and then you ask how much did my zstar change
due to a unit change in my parameter that
is the B.
B is nothing but zstar of p plus delta p that
means zstar as a function of p plus delta
p minus zstar at p, this whole thing divided
by delta p.
So estimated B is this.
You see, you did not really require the system
equations.
You simply observe the evolution or from there
you could extract all this information’s,
so you could extract all these information’s.
Not only that, I have already told you that
there are certain situations where you have
only one train of data that means some experiment
is going on and you have access to only one
state variable and not every state variable.
You don’t even know what is the complication
of the system, how many state variables are
there.
You may not know and you have got just one
variable that has been measured.
In that case what you will do?
You will do the delay coordinate embedding.
That means you create additional state variables
by delay coordinate, you do the same thing.
Then that delay coordinate system, you place
a Poincare section, you can do that.
You can thereby obtain these points from which
you will do the estimations.
You can do that and from there, you can estimate
A and B and C. Yes, that is also possible.
It has been demonstrated that this whole thing
was even if you have access to just one state
variable and then you can decide how much
should the perturbation B that you apply in
the parameter.
You apply the perturbation in one instant
and simply wait.
You got the point?
Still there is a problem.
The reason that people were so very excited
about it is essentially the cardiac problem
of humans.
It is known that as a man goes close to death
because of cardiac failure, essentially the
dynamics which is a periodic dynamics that
changes to various high periodic orbits and
to chaos and naturally the problem becomes
how to control that chaos.
Presently do you know what is done?
Presently they implant what is known as a
defibrillator, this big device.
Some of your fathers may have already, I know
people who have and it is extremely painful
when it really strikes because whenever it
goes into a… that stability is lost, it
is going into a high periodic orbit that means
you see erratic oscillation of the heart.
Then the defibrillator works.
What it does it gives an enormous shock that
means it just gives a big nudge.
So that it gets into again the regular periodic
orbit, it does and the man survives.
But due to that shock, there is often death
of cardiac tissues.
That means the life is prolonged but not very
long and also that particular event is very
painful for the patient.
I mean, I have met people who have undergone
that.
It’s like somebody dropping a whole chest
on the chest of the person, he feels like
that.
The idea is that can you then instead of giving
one big shock, can you give tiny nudges to
get the orbit back to.
It has been successful in non-human hearts.
Yes, people as I told you that it has been
done for the frog heart.
Permission has not been obtained to do it
on human heart as yet.
That means it is done on by dissecting, opening
the heart, keeping it alive and doing that.
It has so far been unsuccessful that means
you do have one chain of data coming, from
there you do delay coordinate embedding, from
there you do estimate these values and depending
on that you give very small amount of electrical
pulse.
Instead of any other thing, electrical pulse
is the most convenient thing to apply here.
Keep a tiny electrical pulse and that does
it.
This has been shown, it does stabilize.
But the question is when I said that if the
state is far off from the equilibrium point,
from the fixed point you simply wait till
comes back.
For a patient will you wait?
Will you wait long enough, let it come back
it is ergotic.
It will sooner or later come back and the
fellow might die before that.
Obviously the question comes that how can
we quickly bring a state to a desirable state.
That means instead of waiting, I want to bring
a state quickly to a desirable state.
Is it possible?
Yes, it is possible only in chaotic systems
because in a non-chaotic system, you will
again have to give a large change, large perturbation
in order to move a state from one point to
the other.
While in a chaotic system the advantage is
that slight tiny perturbation can result in
a large change.
The question is how can we make a tiny perturbation
so that within a very short time, I will get
where I want.
That is the problem then.
Let us illustrate that.
Let’s start targeting.
so far what I was discussing is known as control
of chaos.
By the way before going to targeting, let
me give you the idea of another algorithm
that has been very widely used.
That is supposing you have got a chaotic system
whose data is coming and here is the plant.
It is just chaotic and you 
make a feedback loop.
Here is the input and here there is a feedback
loop.
Here is the plant that is now behaving chaotically
and you want to control it into one of the
unstable periodic orbits.
What is this fellow?
This is nothing but a delay.
What are you doing?
You are taking the output, giving it a delay
and adding it to the system behavior.
What will be the result?
This is the error that is going into the plant.
If the error is non-zero then it will lead
to some kind of a control action.
It will be zero only when the delay is such
that it is exactly the same delay or a same
period as the periodic orbit that you want
to stabilize.
Suppose you have got a delay and you have
got a means by which you can change the delay.
Then what will happen?
So long as its period or delay is not the
same as the period of the unstable periodic
orbit, nothing will happen.
But the moment it becomes the same as the
unstable periodic orbit, immediately this
fellow will go to zero and you get a locking
on to… this algorithm was invented by a
person called Pyragas and that is why it is
called as the Pyragas statement and this has
also been applied to many chaotic systems.
Now let us come back to the issue of targeting.
In targeting essentially what are you trying
to do?
We want to reach that state, some desirable
state in the least possible time and again
we will use an algorithm which has no equivalent
in non-chaotic system.
These are very dependent on the sensitive
dependence on initial condition.
Let us illustrate that with the logistic map.
You have got the logistic map here, so this
is xn and xn+1.
Suppose you are here and you want to reach
here.
One logic would say that start from here and
keep allow it to oscillate for long time.
Sooner or later it will come here.
Yes, that’s true but then we said, we don’t
want to wait till then.
The equation here is as you know xn+1 is equal
to mu xn (1 minus xn).
Here is the parameter.
Assume that you can vary the parameter as
your control action by tiny amount in any
iterate.
Now if you have the variation, suppose it
is now for a chaotic behavior you would need
something like 3.9.
The 4 is fully chaotic but suppose it is 3.9
which will give you chaotic behavior.
Normally it is 3.9 but suppose you have scope
to vary it in the range 3.8 less than mu 4.0.
That means this way 0.1 and that way zero
point.
That is an extreme amount that you can vary.
Then what will happen?
Supposing this was here and this would map
to this point.
If you had used 3.9 then it would map to some
other point.
If you use 4.0, it would map to some other
point because for 3.9 the graph would be slightly
different like this.
For 4.0, the graph would be slightly different
like this.
In one case it maps here another case it maps
here.
There will be a range over which it maps,
it has a capability of mapping for this range
of mu.
If you have the option of varying mu over
this range, you have the capability you have
the option of reaching this range in the next
iterate.
If you now withdraw this perturbation then
this range will map to some range.
How will you obtain it?
This range you bring to the 45 degree line
and you bring here, so you have got this range.
In the next iterate this range will map to
this range.
Do you notice that because of the stretching
behavior, this range is slowly increasing.
So after sometime this range will increase
and within couple of iterates, it will include
the point that you want to reach.
So what was our logic?
I will apply a small nudge now, directed in
such a way that three iterates later I will
be where I want to be.
Do you see that this is possible now?
What do you need to do?
You need to find out that if I give the parameter
perturbation now, what is the range of values
I will reach in the next iterate?
Start from that range of values which is the
range of value that I will reach in the next
iterate, start from that range of values and
find out the range of values in the iterate
after that.
Very soon you will find that the target is
included within the range.
The moment you have found that the target
is included, you have found how many iterates
did I meet in order to reach that.
Now notice the argument.
One, in first iterate give the perturbation.
Two, find the range 
of values in the subsequent iterates.
Suppose we have found after the third iterate,
the target is contained in that set.
Then it is only pressing the calculator.
What will you do?
You will start from that target point and
back track.
Starting from there, you will back track that
means xn+1 you know, you calculate xn, back
track and finally at the first iterate then
you will land up in equation like this where
you need to know mu.
You will be able to calculate that.
It only needs pressing a calculator not even
a computer.
You understood?
So three is back track, starting from the
target which tells you that in the first iterate
if I give only that much of perturbation say
from 3.9, it needs to be made 3.8 may be.
In the first iterate you change the parameter
to 3.83 and then bring it back to 3.9 and
let it done.
Automatically it will come there.
You understood the point?
How could this be possible?
This was possible only because of the sensitive
dependents on initial condition.
No other reason.
It was possible because of the sensitive dependence
on initial condition.
Here we are using the sensitive dependents
on initial condition.
It’s not difficult to see how to apply this.
This I illustrated with a map.
How will you apply it to say 2 D map?
Suppose in the henon system, the orbit is
something like this.
Suppose you are here and you want to reach
this point.
How will you do it?
Give the parameter perturbation in the first
iterate that means there is a range of parameter.
If you give that range of parameter, this
point maps to say this point, this range.
Now withdraw that parameter perturbation.
This range will map to say this range and
then you will find that this fellow is included
which means that it is possible to reach from
here to there with only two iterates, two
jumps.
Then you calculate how much was the exact
change necessary in order to get from this
to this point, to this point.
Easily be done.
In case of continuous time system, essentially
the logic is the same.
You start from a particular state, you are
trying to target that state.
Now if this state is my target then how would
I go?
From here I will apply the tiny perturbation
for some span of time.
In this case one iterate, in that case some
span of time say 5 seconds.
I say that I will apply a tiny perturbation
for 5 seconds, so apply the tiny perturbation
for 5 seconds, you can easily find out by
solving the equations.
You can easily find out the ball in the state
space that will be reachable and then let
that ball evolve.
After sometime you will find that the target
is included in that ball.
Find out the time that was necessary in order
to reach from here to there?
Then back track, you can find out the exact
amount of perturbation that would be necessary
in order to reach from here to there.
This logic has been applied in a very unlikely
scenario I suppose it was 1988.
There was a space craft, not space star.
It was basically a satellite that was almost
nearing the end of its time.
There was a comet coming then NASA scientists
decided that this particular space craft would
be used to observe that comet.
They wanted to have a commentary encounter
but the distance was something like 50 million
miles and the fuel was almost exhausted.
There are only tiny amount of hydrazine fuel
left and it could give only tiny nudges and
you wanted to reach 50 million miles.
How is that possible?
They calculated because the three body system
earth, moon and the satellite is a chaotic
system therefore there is sensitive dependence
on initial condition and therefore it should
be possible to hold the space craft over such
a large distance just by using a tiny perturbations.
They calculated that, they did that and they
reached there and that was the first planetary
encounter.
It essentially involved five rounds around
the moon.
The actual orbit was rather complicated but
they had to give only tiny perturbations.
So that the gravitude steer the system to
that point.
These are typical applications of this logic,
happens only in chaotic system.
You cannot have this kind of things happening
in non-chaotic systems.
We are very close to the end of the course,
essentially through this course we have learned
some very typical features of nonlinear systems.
We have learned that not all nonlinear systems
are chaotic but all chaotic systems must be
nonlinear systems.
Normally in regular control theory course
or whatever course you have learnt in engineering,
they look at only the linear system behavior.
So all these possibilities are essentially
left out of the ambit of what you learn.
That sufficed more or less for a 19 century
engineer or 20th century engineering.
For a 21st century engineer that often does
not sufficed because firstly most of the things
that one has to deal with are nonlinear.
Secondly earlier we use only the linear theory
in order to design control systems.
Now no longer.
Probably you know the fighter air crafter
all open loop unstable.
They make it open loop unstable because otherwise
there is no maneuverability and then you need
to stabilize with your hand.
So here is the control that we exercise on
a system that is chronically unstable.
Likewise many other systems are designed in
the same way to have maneuverability.
You can easily see that chaotic systems have
offered additional advantage in maneuverability
because in a chaotic system there are enormous
number of unstable periodic orbit involved
and you can switch from one to the other.
You do not need to design say 100 different
systems for 100 different works.
Just one system, stabilize a particular periodic
orbit you have the behavior that you want.
The versatility, the width of the different
types of behavior that are possible, these
are offering many advantages to chaotic systems
so that now it is even thinkable to design
systems chaotically, to be used in engineering
applications.
That was say 5 years back nobody was thinking
about that.
By the time you guys become full-fledged engineers,
you will find more applications and then the
things learned in this course might prove
to be useful.
I suppose that will be enough for this course.
Thank you.
