1
00:00:00,090 --> 00:00:03,690
So welcome back to this very short and small additional lecture.

2
00:00:04,110 --> 00:00:10,530
So in previous lecture, we have solved D3 coupled differential equations and we have found a chaotic

3
00:00:10,530 --> 00:00:16,860
solution where we had to use a value for bead that is larger than 25.

4
00:00:17,220 --> 00:00:24,060
So we used 50 and we have seen that we have such a chaotic solution where we have here just a tiny difference

5
00:00:24,060 --> 00:00:25,200
in the starting conditions.

6
00:00:25,200 --> 00:00:31,260
But then after some time, there is really a large difference in the end, coordinates of the two solutions

7
00:00:31,260 --> 00:00:32,460
blue and orange.

8
00:00:33,570 --> 00:00:39,900
So now I want to just show you an example what happens if the value of B is smaller than this critical

9
00:00:39,900 --> 00:00:40,320
value?

10
00:00:41,190 --> 00:00:44,550
So I just go to this point and copy it.

11
00:00:45,000 --> 00:00:46,470
So this is nice thing we can know.

12
00:00:46,470 --> 00:00:54,600
Just change these parameters here without having to redefine the function because we have added them

13
00:00:54,600 --> 00:00:59,880
here and then we have added or the arts optional argument.

14
00:01:01,380 --> 00:01:09,570
So I copied this whole cell and then I copy also the plots, which is this one.

15
00:01:12,490 --> 00:01:22,510
So when I now change the value of B to a value smaller than 24, like 20, for example, I get such

16
00:01:22,510 --> 00:01:23,200
a solution.

17
00:01:24,520 --> 00:01:34,600
And let's see what happens if I add another solution here with a slightly different starting position.

18
00:01:34,600 --> 00:01:35,530
One point one.

19
00:01:36,250 --> 00:01:37,720
Sorry again, the same typo.

20
00:01:38,650 --> 00:01:40,150
1.1 times X0.

21
00:01:40,690 --> 00:01:45,730
And then I plot here both of them.

22
00:01:50,410 --> 00:01:50,840
Hmm.

23
00:01:51,640 --> 00:01:52,960
Sorry, I forgot to run it.

24
00:01:53,350 --> 00:01:53,650
Yeah.

25
00:01:54,130 --> 00:02:01,480
Now you see, we have here on top of each other both of the solutions and maybe just to explain it to

26
00:02:01,480 --> 00:02:02,300
you a bit better.

27
00:02:02,320 --> 00:02:03,940
Let me export the plots.

28
00:02:08,310 --> 00:02:08,940
To

29
00:02:11,460 --> 00:02:13,080
and open it.

30
00:02:16,050 --> 00:02:23,160
OK, so now you see here we have this small difference in the starting positions of X equal to one and

31
00:02:23,160 --> 00:02:23,970
one point one.

32
00:02:24,420 --> 00:02:30,720
And you see they follow exactly or almost exactly the same trajectory and then they end up in the same

33
00:02:30,720 --> 00:02:31,170
point.

34
00:02:32,280 --> 00:02:39,950
This is because by changing the value of B, we have turned the two repulsive points in space to two

35
00:02:39,960 --> 00:02:40,980
attractive points.

36
00:02:41,520 --> 00:02:46,590
And so in this case, the system does not behave chaotic anymore, and it will move towards the same

37
00:02:46,590 --> 00:02:48,540
solution almost.

38
00:02:48,540 --> 00:02:51,000
Or it will always move to the two same solution.

39
00:02:51,000 --> 00:02:56,190
But even it will move on the same trajectory if the starting conditions are just slightly different.

40
00:02:56,850 --> 00:03:02,680
So this is really the difference of chaotic and deterministic behavior.

41
00:03:03,330 --> 00:03:10,410
And so even though this is more of a mathematical model, it teaches us quite a lot about physics as

42
00:03:10,410 --> 00:03:10,680
well.

43
00:03:11,040 --> 00:03:15,870
And I told you in the very beginning that there are some physical applications of this model.

44
00:03:16,290 --> 00:03:24,060
But yeah, I think I said already enough about the background and we see that from such a nice mathematical

45
00:03:24,060 --> 00:03:26,320
model, we can learn quite a lot about physics.

46
00:03:26,670 --> 00:03:30,390
And of course, we can use it as a nice model for our function.

47
00:03:30,390 --> 00:03:36,000
Integrate lots, solve underscore IVP, which had absolutely no problem handling such a system.