1
00:00:02,083 --> 00:00:04,754
This first question
is primarily vocabulary
2
00:00:04,768 --> 00:00:06,166
and asks you to pick
3
00:00:06,170 --> 00:00:07,419
which term does not correspond
4
00:00:07,419 --> 00:00:09,773
to the same representation
as the other terms.
5
00:00:09,799 --> 00:00:13,304
These are the two different
representations of nonlinear dynamics
6
00:00:13,317 --> 00:00:15,776
that are in question in this quiz problem.
7
00:00:15,796 --> 00:00:19,585
This type of plot is known
as the cobweb diagram correlation plot,
8
00:00:19,596 --> 00:00:21,547
or return map.
9
00:00:21,578 --> 00:00:24,835
This type of plot is referred to
as the time-domain plot,
10
00:00:24,844 --> 00:00:27,013
as the x-axis is time.
11
00:00:27,216 --> 00:00:29,812
So time-domain plot is the term
12
00:00:29,826 --> 00:00:32,112
that does not correspond
to the same representation
13
00:00:32,134 --> 00:00:34,118
as the other terms in the list.
14
00:00:34,150 --> 00:00:36,905
Question 2a asks us
to use the cobweb applet
15
00:00:36,920 --> 00:00:38,642
to add one iterate at a time
16
00:00:38,661 --> 00:00:41,947
to a cobweb diagram starting
at initial condition 0.5
17
00:00:41,956 --> 00:00:43,934
using a parameter of 2.7.
18
00:00:43,953 --> 00:00:46,983
From this, we want to analyze
what kind of dynamics comes out
19
00:00:47,000 --> 00:00:49,688
and compare this with our current
knowledge of the time series plot
20
00:00:49,704 --> 00:00:51,217
to determine the dynamics.
21
00:00:51,238 --> 00:00:53,709
As we slowly add
single iterates to this map,
22
00:00:53,729 --> 00:00:55,336
what we see in the cobweb diagram
23
00:00:55,361 --> 00:01:01,095
is the red lines slowly spiraling in
to the intersection of the line x+1 = x
24
00:01:01,131 --> 00:01:03,101
and the parabola
defined by the logistic map.
25
00:01:03,132 --> 00:01:05,303
As we also consider
what's happening in the time domain,
26
00:01:05,324 --> 00:01:07,474
we see that we are converging
onto a single point.
27
00:01:07,518 --> 00:01:10,622
By examining both the cobweb diagram
and the time-domain plot,
28
00:01:10,667 --> 00:01:12,926
we can see that these are
fixed point dynamics.
29
00:01:12,955 --> 00:01:15,067
Part b asks us to use a correlation plot
30
00:01:15,080 --> 00:01:17,377
to analyze the stability
of these dynamics
31
00:01:17,398 --> 00:01:21,818
by choosing several initial conditions
between 0.1 and 0.9.
32
00:01:21,844 --> 00:01:23,678
To do a basic analysis of stability,
33
00:01:23,690 --> 00:01:26,052
let's apply several different
initial conditions
34
00:01:26,112 --> 00:01:35,040
to the parameter r = 2.7:
0.1, 0.2, 0.3, 0.4, 0.6, 0.7, 0.8.
35
00:01:35,060 --> 00:01:37,154
So it does appear
that these fixed point dynamics
36
00:01:37,182 --> 00:01:39,916
are indeed stable for r = 2.7.
37
00:01:39,946 --> 00:01:42,094
For this problem, we set r equal to 3
38
00:01:42,117 --> 00:01:44,455
and the number of initial iterates to 5,
39
00:01:44,470 --> 00:01:46,796
and have an initial condition
of x₀ equal to 0.5.
40
00:01:46,854 --> 00:01:48,792
Let's click start animation
41
00:01:48,802 --> 00:01:51,751
and see what shape
emerges in the red line.
42
00:01:53,550 --> 00:01:55,044
So as you can see, it does appear
43
00:01:55,053 --> 00:01:57,865
that the red lines
are converging to something.
44
00:02:01,456 --> 00:02:04,137
At this point, it seems like the red lines
have stopped changing
45
00:02:04,148 --> 00:02:05,889
and have converged to a small square.
46
00:02:05,926 --> 00:02:08,359
As an aside, a small square
on the cobweb plot
47
00:02:08,366 --> 00:02:11,685
is associated with a two-cycle
in the logistic map dynamics.
48
00:02:12,880 --> 00:02:17,576
The next three questions ask to match
cobweb plots in figure 1a
49
00:02:17,617 --> 00:02:19,813
to their corresponding dynamics.
50
00:02:19,838 --> 00:02:24,234
Part a asks what type of dynamics
correspond to those seen in figure 1a
51
00:02:24,256 --> 00:02:28,624
This type of cobweb plot
is associated with chaotic dynamics.
52
00:02:29,481 --> 00:02:33,844
For part b, we are told in figure 1c
that the transient has not been removed,
53
00:02:33,897 --> 00:02:36,422
and it appears
that the trajectory has converged.
54
00:02:36,455 --> 00:02:39,516
As we saw in question 2a,
this type of cobweb plot
55
00:02:39,529 --> 00:02:42,299
is associated with a fixed point dynamic.
56
00:02:42,363 --> 00:02:45,767
For part c, we're told that figure 1d
has had the transient removed
57
00:02:45,779 --> 00:02:47,347
and asked to classify the dynamic.
58
00:02:47,387 --> 00:02:51,881
This type of cobweb plot
is associated with high-period attractors.
59
00:02:53,841 --> 00:02:56,700
This next series of questions is intended
to help you understand
60
00:02:56,712 --> 00:02:59,754
the connection between time-domain plots
and cobweb plots.
61
00:02:59,771 --> 00:03:03,116
The dynamics in figure 2a
are the same as which in figure 1.
62
00:03:03,131 --> 00:03:07,845
As you can see, the dynamics in figure 2a
are the same as in figure 1b.
63
00:03:07,869 --> 00:03:10,590
One thing that might help
you understand this or see this better
64
00:03:10,634 --> 00:03:13,916
is to notice the small square
that forms in figure 1b
65
00:03:13,949 --> 00:03:18,242
and see what it corresponds to
in figure 2a.
66
00:03:21,116 --> 00:03:24,432
The dynamics in figure 2b
are a fixed point dynamic.
67
00:03:24,461 --> 00:03:26,281
As we've seen throughout this quiz,
68
00:03:26,300 --> 00:03:30,699
this corresponds to the dynamics
seen in figure 1c.
69
00:03:32,484 --> 00:03:35,153
The dynamics seen in figure 2c
in the time-domain plot
70
00:03:35,153 --> 00:03:36,371
is a high-period orbit.
71
00:03:36,380 --> 00:03:38,598
You may want to confuse this
with a chaotic orbit,
72
00:03:38,606 --> 00:03:40,064
but look how structured it is.
73
00:03:40,083 --> 00:03:42,298
As you can see, it repeats very regularly.
74
00:03:42,321 --> 00:03:45,391
This corresponds to figure 1d.
75
00:03:48,962 --> 00:03:52,197
Finally, the dynamics in figure 2d
are chaotic.
76
00:03:52,200 --> 00:03:54,547
Notice in contrast to figure 2c,
77
00:03:54,569 --> 00:03:57,025
they're a little less regular
than a periodic orbit.
78
00:03:57,041 --> 00:04:00,091
These dynamics correspond to figure 1a.
79
00:04:03,411 --> 00:04:05,436
Finally, let's look at question 5.
80
00:04:05,449 --> 00:04:06,918
As discussed in lecture,
81
00:04:06,957 --> 00:04:10,738
any time there's an intersection
between the line x+1 = x
82
00:04:10,764 --> 00:04:12,890
and the parabola described
by the logistic map,
83
00:04:12,910 --> 00:04:14,097
which is yellow in this plot,
84
00:04:14,135 --> 00:04:15,979
you'll have a fixed point
of the logistic map
85
00:04:16,000 --> 00:04:20,408
As you can see, there are
two such intersections, here and here.
86
00:04:20,418 --> 00:04:24,085
Obviously, the fixed point at (0,0)
is not very interesting.
87
00:04:24,109 --> 00:04:27,341
In question 5, we are asked to solve
for the interesting fixed point
88
00:04:27,353 --> 00:04:30,059
that occurs right after 0.6.
89
00:04:31,368 --> 00:04:32,834
To solve for this fixed point,
90
00:04:32,854 --> 00:04:35,694
we simply need to solve
this system of equations.
91
00:04:35,715 --> 00:04:38,125
To do so, we set them equal to each other,
92
00:04:38,144 --> 00:04:40,576
resulting in the following equation.
93
00:04:41,419 --> 00:04:44,636
Since we are not interested
in the fixed point that occurs at (0,0),
94
00:04:44,663 --> 00:04:48,888
we can eliminate these two x,
resulting in this equation.
95
00:04:49,940 --> 00:04:51,596
Solving this equation,
96
00:04:51,619 --> 00:04:55,200
we then see that the fixed point
occurs at two thirds.
97
00:04:55,796 --> 00:04:57,679
We can now answer question 5a,
98
00:04:57,690 --> 00:05:02,888
as we know the fixed point at r = 3
is at x* = 2/3.