1
00:00:12,905 --> 00:00:13,974
We've been talking about how
2
00:00:13,974 --> 00:00:15,520
chemical dynamics can naturally lead
3
00:00:15,520 --> 00:00:17,244
to cycling behavior and chaos,
4
00:00:17,244 --> 00:00:20,566
and here we want to make that
a little bit more explicit.
5
00:00:20,566 --> 00:00:22,259
A classic model that's been used
6
00:00:22,259 --> 00:00:24,143
to look at these types of phenomenas
7
00:00:24,143 --> 00:00:25,724
is known as the Brusselator.
8
00:00:25,724 --> 00:00:27,903
It's described by this abstract set
9
00:00:27,903 --> 00:00:30,405
of chemical reactions listed here
10
00:00:30,405 --> 00:00:32,220
in the middle of the screen.
11
00:00:32,220 --> 00:00:33,707
Now, these chemical reactions
12
00:00:33,707 --> 00:00:37,395
can be converted into a set
of two differential equations
13
00:00:37,395 --> 00:00:39,309
assuming that we hold the concentrations
14
00:00:39,309 --> 00:00:42,017
of A and B constant in the system.
15
00:00:42,017 --> 00:00:44,163
And in this case, we've added
16
00:00:44,163 --> 00:00:46,605
these different reaction rates K1 and K2
17
00:00:46,605 --> 00:00:48,627
which describe the reaction rates
18
00:00:48,627 --> 00:00:51,985
associated with the arrows above.
19
00:00:51,985 --> 00:00:55,401
The system can be easily
non-dimensionalized
20
00:00:55,401 --> 00:00:59,290
to a slightly simpler system
where we've combined parameters
21
00:00:59,290 --> 00:01:03,441
into effective parameters
little a, little b, and so forth.
22
00:01:04,734 --> 00:01:07,339
This system has a simple steady state at
23
00:01:07,339 --> 00:01:11,809
x = a (this effective parameter), and
24
00:01:11,809 --> 00:01:16,297
y = b / a
(the ratio of effective parameters).
25
00:01:17,153 --> 00:01:20,673
Furthermore, what's very interesting
about this system is that,
26
00:01:20,692 --> 00:01:23,014
we're intersted in not only what the steady state is,
27
00:01:23,014 --> 00:01:26,413
but in the dynamics
leading to that steady state
28
00:01:26,413 --> 00:01:29,158
and whether the steady state is stable.
29
00:01:29,158 --> 00:01:31,390
Using techniques known as
30
00:01:31,564 --> 00:01:34,371
stability analysis which
is an entire field
31
00:01:34,371 --> 00:01:36,965
it's possible to show
that this steady state
32
00:01:36,965 --> 00:01:41,677
will be unstable for b > a^2 + 1.
33
00:01:41,677 --> 00:01:44,005
So let's take a look
at what that looks like.
34
00:01:44,917 --> 00:01:48,729
If b is safely smaller than a^2 + 1,
35
00:01:48,729 --> 00:01:50,972
then you can start off at some initial
36
00:01:50,972 --> 00:01:53,300
concentrations for x and y
37
00:01:53,300 --> 00:01:55,519
and these two chemical species will have
38
00:01:55,519 --> 00:01:58,062
some transient behavior
that eventually leads to
39
00:01:58,062 --> 00:02:00,422
these two steady states that we described
40
00:02:00,658 --> 00:02:02,527
in the previous slides.
41
00:02:03,564 --> 00:02:07,063
Now, if we make b
just smaller than a^2 + 1,
42
00:02:07,063 --> 00:02:09,877
so we make it very close to a^2 + 1,
43
00:02:09,877 --> 00:02:12,296
we see that you go the same steady states
44
00:02:12,296 --> 00:02:14,296
but now, in the transient behavior
45
00:02:14,296 --> 00:02:16,415
we pick up these oscillations that
46
00:02:16,415 --> 00:02:17,917
over time, ring down,
47
00:02:17,917 --> 00:02:19,640
and it takes us a long time
48
00:02:19,640 --> 00:02:21,904
to go to the actual steady state itself.
49
00:02:21,904 --> 00:02:24,681
And if we make b just greater than a^2 + 1
50
00:02:24,681 --> 00:02:27,435
so this is the case where
things should be unstable,
51
00:02:27,435 --> 00:02:31,096
we now see that we have
oscillations that go on forever.
52
00:02:31,096 --> 00:02:33,522
So this system naturally
goes through an oscillation;
53
00:02:33,522 --> 00:02:37,749
naturally returns to
specific states periodically,
54
00:02:37,749 --> 00:02:40,704
and in this case, how fast that's occuring
55
00:02:40,704 --> 00:02:44,199
how frequently these states
are passed through,
56
00:02:44,199 --> 00:02:46,608
that rate is increasing in time.
57
00:02:47,894 --> 00:02:50,065
Now another way that we can look
58
00:02:50,065 --> 00:02:53,106
at these steady states
in the set of dynamics,
59
00:02:53,178 --> 00:02:56,588
is in a phase diagram of x against y.
60
00:02:56,834 --> 00:02:59,124
So this is looking at the paired values
61
00:02:59,214 --> 00:03:01,607
of these two concetrations, x and y.
62
00:03:01,908 --> 00:03:04,241
So time isn't explicit except in animation
63
00:03:04,241 --> 00:03:06,895
and in this first case where b
64
00:03:06,895 --> 00:03:09,910
is much less, or significantly less
65
00:03:09,910 --> 00:03:12,587
than a^2 + 1, then we'll see that the
66
00:03:12,587 --> 00:03:14,891
dynamics have the simple case where you
67
00:03:14,891 --> 00:03:17,991
start off from any point in the space
68
00:03:17,991 --> 00:03:21,253
and you simply go to this one steady state.
69
00:03:21,361 --> 00:03:23,373
There's a little bit of transient behavior
70
00:03:23,373 --> 00:03:26,093
but we go to a single point fixed system.
71
00:03:26,604 --> 00:03:30,864
This case where b is just less than a^2 +1
72
00:03:31,155 --> 00:03:34,339
where we had this ring down behavior to a
73
00:03:34,339 --> 00:03:36,296
steady state in oscillations
to the transient,
74
00:03:36,296 --> 00:03:39,098
will look something like this
in the base plot.
75
00:03:39,360 --> 00:03:42,264
So you can see that we spiral into
76
00:03:42,264 --> 00:03:44,793
the steady state that we expect to find.
77
00:03:44,793 --> 00:03:46,082
So there's a very interesting
78
00:03:46,561 --> 00:03:48,891
oscillatory transient dynamics.
79
00:03:48,891 --> 00:03:53,876
And if we go above a^2 +1
80
00:03:53,876 --> 00:03:56,220
where we expect the system to be unstable,
81
00:03:56,220 --> 00:03:58,725
then we get this limit cycle.
82
00:03:58,725 --> 00:04:03,197
So we are constantly gonna pass through
83
00:04:03,197 --> 00:04:05,876
this set of paired x and y values
84
00:04:06,178 --> 00:04:09,268
periodically with an increasing rate.
85
00:04:10,843 --> 00:04:12,862
What's interesting about these dynamics
86
00:04:12,862 --> 00:04:14,687
is that they're fully deterministic.
87
00:04:14,687 --> 00:04:18,157
So we haven't put any
stochasticity in the system.
88
00:04:18,157 --> 00:04:22,912
We have this completely deterministic way
89
00:04:22,912 --> 00:04:25,265
of getting oscillations in time, and so
90
00:04:25,265 --> 00:04:27,261
it says that just the inherent dynamics
91
00:04:27,261 --> 00:04:30,531
of chemical dynamics, even without noise,
92
00:04:30,531 --> 00:04:33,886
can lead to really interesting
oscillatory behavior.
93
00:04:34,416 --> 00:04:36,460
And that oscillatory behavior can be
94
00:04:36,460 --> 00:04:39,104
even more complicated than
what we've been describing.
95
00:04:39,104 --> 00:04:42,074
So a classic system for looking at chaos
96
00:04:42,074 --> 00:04:43,643
is the Lorenz System.
97
00:04:43,643 --> 00:04:46,336
And so now, we have a coupled value,
98
00:04:46,336 --> 00:04:51,170
or we have a triplet of values
of x, y, and z concentrations,
99
00:04:51,170 --> 00:04:54,448
each related to these
differential equations
100
00:04:54,448 --> 00:04:56,763
that I'm showing here in
non-dimensional form.
101
00:04:56,763 --> 00:04:58,583
This was originally discovered
102
00:04:58,583 --> 00:05:01,518
in atmospheric dynamics by Ed Lorenz
103
00:05:01,518 --> 00:05:03,455
and can be mapped onto
104
00:05:03,455 --> 00:05:07,290
a variety of systems including
certain chemical dynamics.
105
00:05:07,290 --> 00:05:10,133
Now what's interesting about this case
106
00:05:10,133 --> 00:05:15,384
is that this system gives rise also
to fixed steady states,
107
00:05:15,384 --> 00:05:19,395
so where x, y, and z
will go to a single value,
108
00:05:19,395 --> 00:05:21,699
but for other parameter values
in the system,
109
00:05:21,699 --> 00:05:23,925
we'll find that you go
to chaotic dynamics.
110
00:05:25,094 --> 00:05:28,514
And here is what those
chaotic dynamics looks like.
111
00:05:28,514 --> 00:05:30,874
So we're only looking at x, y.
112
00:05:30,883 --> 00:05:33,051
So there's a set of z values
113
00:05:33,051 --> 00:05:34,461
associated with all of these,
114
00:05:34,578 --> 00:05:37,017
and we're just looking at
115
00:05:37,017 --> 00:05:39,236
the projection into the x, y space.
116
00:05:39,236 --> 00:05:42,479
In doing that, we see that we have this
117
00:05:42,479 --> 00:05:45,181
very complicated oscillation where
118
00:05:45,181 --> 00:05:47,387
you not only oscillate over a set
119
00:05:47,387 --> 00:05:49,801
of complicated values in x and y,
120
00:05:50,170 --> 00:05:52,780
but you also transition between
121
00:05:52,780 --> 00:05:56,774
these two basins of attraction.