1
00:00:00,930 --> 00:00:06,960
Fine so we are still dealing with one dimensional
flows, but now we focus on an area called
2
00:00:06,960 --> 00:00:08,460
bifurcations.
3
00:00:08,460 --> 00:00:17,420
This lecture is going to be a brief introduction
to the area of bifurcation theory.
4
00:00:17,420 --> 00:00:25,830
Now dynamics of vector field on the line are
actually not so exciting.
5
00:00:25,830 --> 00:00:35,300
The solutions either go ahead and settle down
towards equilibrium or they actually go off
6
00:00:35,300 --> 00:00:38,160
to plus or minus infinity.
7
00:00:38,160 --> 00:00:42,510
So in that sense they are not extremely exciting
dynamics.
8
00:00:42,510 --> 00:00:49,030
So, what really is exciting about one dimensional
systems?
9
00:00:49,030 --> 00:00:56,710
The answer turns out to be its dependence
on parameters.
10
00:00:56,710 --> 00:01:06,590
The qualitative structure of the flow can
actually change quite dramatically as system
11
00:01:06,590 --> 00:01:20,260
parameters are varied, for example fixed points
can be created or destroyed and the stability
12
00:01:20,260 --> 00:01:27,220
of the fixed point itself may change has system
parameters are varied.
13
00:01:27,220 --> 00:01:37,140
So, the qualitative change in dynamics is
what we referred to as bifurcations and the
14
00:01:37,140 --> 00:01:46,509
parameter values at which such bifurcations
occur are called bifurcation points.
15
00:01:46,509 --> 00:01:52,840
So, we are really delving into this area of
applied mathematics called bifurcation theory.
16
00:01:52,840 --> 00:02:05,100
It is really the study of changes in the qualitative
or topological structure of dynamical systems.
17
00:02:05,100 --> 00:02:17,020
So, a bifurcation occurs when a small smooth
change to some parameter values referred to
18
00:02:17,020 --> 00:02:28,549
as bifurcation parameters of a differential
equation causes a sudden qualitative or topological
19
00:02:28,549 --> 00:02:31,120
change in its behaviour.
20
00:02:31,120 --> 00:02:41,700
Now interestingly the name bifurcation apparently
was first introduced by Henry Poincare in
21
00:02:41,700 --> 00:02:43,790
year 1885.
22
00:02:43,790 --> 00:02:53,909
So, bifurcations essentially provide us with
a way to understand transitions and instabilities
23
00:02:53,909 --> 00:03:00,579
as some system control parameter actually
varies.
24
00:03:00,579 --> 00:03:06,330
Consider a simple example bulking of a beam.
25
00:03:06,330 --> 00:03:14,799
So, you have a beam, the beam has a certain
weight which is placed on top of it and if
26
00:03:14,799 --> 00:03:21,699
you go ahead and increase the weight beyond
certain critical value the beam actually buckles.
27
00:03:21,699 --> 00:03:28,559
So, what we find is that beam as buckled under
the weight, when the weight crosses certain
28
00:03:28,559 --> 00:03:29,559
value.
29
00:03:29,559 --> 00:03:41,370
So, with a small weight the beam remains vertical
and can comfortably support the load.
30
00:03:41,370 --> 00:03:52,799
As the weight increases the beam then buckles
under the weight and we observe a qualitative
31
00:03:52,799 --> 00:03:54,959
change in the system.
32
00:03:54,959 --> 00:04:02,840
So, in this example the weight act as the
control parameter and the deflection of the
33
00:04:02,840 --> 00:04:08,540
beam from the vertical is the dynamical variable.
34
00:04:08,540 --> 00:04:15,419
So, this is our first example may be observed
bifurcation phenomena happening (()) (04:20),
35
00:04:15,419 --> 00:04:24,660
where there is a qualitative change in the
system when a certain parameter varies.
36
00:04:24,660 --> 00:04:29,539
The dynamics of the vector field on real line
are not very exciting.
37
00:04:29,539 --> 00:04:35,900
One of two things can actually happen, number
one is that the solution can actually blow
38
00:04:35,900 --> 00:04:42,879
of to infinity or solutions which actually
converge to some fixed point, so only these
39
00:04:42,879 --> 00:04:44,330
two things that really happening.
40
00:04:44,330 --> 00:04:50,680
You say what so exciting about dynamics or
of vector filed on real line?
41
00:04:50,680 --> 00:04:54,680
And the answer turns out to be its dependence
on parameters.
42
00:04:54,680 --> 00:05:00,740
When you construct a model of the real world,
almost every model that you find will have
43
00:05:00,740 --> 00:05:05,969
some parameter or the other in fact usually
they too many parameters.
44
00:05:05,969 --> 00:05:13,740
So the question we are interested in, is that
how does changes in the parameter of the underline
45
00:05:13,740 --> 00:05:19,960
model actually induce a qualitative change
in the dynamics and that can happen in one
46
00:05:19,960 --> 00:05:21,380
or two ways.
47
00:05:21,380 --> 00:05:29,100
Number one, is that either fixed points can
be created or destroyed as parameter varies
48
00:05:29,100 --> 00:05:35,280
or the stability of the fixed points themselves
can change.
49
00:05:35,280 --> 00:05:40,220
These are two important qualitative changes
that can actually happen in the system as
50
00:05:40,220 --> 00:05:42,659
a parameter actually varies.
51
00:05:42,659 --> 00:05:46,729
And we give a very simple example, you can
just get a motivation and get an intuition
52
00:05:46,729 --> 00:05:53,740
in order and that is that imagine you have
beam on top the beam you have a certain weight
53
00:05:53,740 --> 00:06:00,189
and the beam can actually hold the weight
without actually buckling.
54
00:06:00,189 --> 00:06:05,650
But now what we do is we slowly increase the
weight, when the weight increases a certain
55
00:06:05,650 --> 00:06:06,820
threshold.
56
00:06:06,820 --> 00:06:10,110
You will actually find that the beam buckles.
57
00:06:10,110 --> 00:06:17,460
So, this transitions from the beam being straight
when the weight is small to the beam buckling
58
00:06:17,460 --> 00:06:18,969
when the weight is big.
59
00:06:18,969 --> 00:06:22,569
Actually, is a qualitative change in the system
dynamics.
60
00:06:22,569 --> 00:06:29,539
So, the weight actually acts as a control
parameter and as that parameter varies, there
61
00:06:29,539 --> 00:06:32,770
is a qualitative change in the system dynamics.
62
00:06:32,770 --> 00:06:39,120
So, this is just you know brief motivation
to bifurcation theory and why we would be
63
00:06:39,120 --> 00:06:45,430
interested in bifurcation in the system and
in particular the role that parameters will
64
00:06:45,430 --> 00:06:48,930
play inducing bifurcation phenomena of certain
type.