Why are periodic systems so unpredictable?

  Рет қаралды 16,193

Concave Cuboid

Concave Cuboid

Күн бұрын

Пікірлер: 45
@blacklistnr1
@blacklistnr1 Жыл бұрын
17:30 quite insane that Dee Dee's "2 steps forward, 1 step back" song from Dexter's Lab is at the heart of a chaos theorem
@filipbanek31514
@filipbanek31514 Жыл бұрын
great video, i’m leaving a comment for the algorithm
@ChaoteLab
@ChaoteLab Жыл бұрын
Im replying for the algorithm. Thx for your repliable comment.
@hammadhassan9125
@hammadhassan9125 Жыл бұрын
top lad!
@asailijhijr
@asailijhijr Жыл бұрын
Engagement for The Engagement God.
@spacelem
@spacelem Жыл бұрын
Ugh, I remember learning "period 3 implies chaos" during my maths degree in the early 2000s, then promptly blanked on it during the exam!
@ottolehikoinen6193
@ottolehikoinen6193 Жыл бұрын
This brings to mind that the ITCZ behavior is inherently chaotic and this might be one of the reasons how El Ninos are developed and to where the worse weather hits sometime afterwards. As the Northern and Southern hemisphere are more land or more ocean they have different style of responses that may delay or hasten the next cycle of ENSO. Thanks.
@Jaylooker
@Jaylooker Жыл бұрын
Sharkovskii’s theorem at 15:24 motivates Bott periodicity being 2-periodic in stable homotopy theory. There are other periodic “v_n” maps of homotopy groups to consider as well.
@maalls
@maalls 4 ай бұрын
My mind is blown 🎉
@Nzargnalphabet
@Nzargnalphabet Жыл бұрын
With the double pendulum, I believe the time the two differ is when they hang together in the air, if you find an example like that in other chaotic systems, you could get very predictable behavior, although would likely be pretty boring
@hvok99
@hvok99 Жыл бұрын
I am dying 😂 "'period 3 implies chaos' is basically mathematical click bait"
@johnchessant3012
@johnchessant3012 Жыл бұрын
Great video!
@smiley_1000
@smiley_1000 Жыл бұрын
I think it would be interesting to consider the intersection I of all the I_k's. Since I is the intersection of a chain of compact sets, it's nonempty. Also we can see that f(I) is a subset of I. Consider x in I. Then for any k, x is in I_(k+1) and therefore f(x) is in I_k. Therefore, f(x) is in I, so we have that f(I) is a subset of I. I would also guess that we even have f(I) = I, but I'm unable to prove it.
@concavecuboid3253
@concavecuboid3253 Жыл бұрын
f(I)=I because f(I_k)=I_{k-1}. f(I) is the intersection of all f(I_k) which is the intersection of all l_{k-1} which is precisely I.
@smiley_1000
@smiley_1000 Жыл бұрын
@@concavecuboid3253 Why is f(I) the intersection of all the f(I_k)?
@concavecuboid3253
@concavecuboid3253 Жыл бұрын
Suppose x in I. Then, for all k, x in I_k, so f(x) is in I_k-1. Therefore, f(x) is in I. Suppose x not in I. Then, there is a k so that x is not in I_k. So f^-1(x) is disjoint from I_k+1. Therefore, f^{-1}(x) is disjoint from I and x is not in f(I). I think that I just consists of a single point with period 1.
@smiley_1000
@smiley_1000 Жыл бұрын
@@concavecuboid3253 I don't see what you're proving here. In the first part, you've proven the following claim: "Given x in I, f(x) is in I". This means that f(I) is a subset of I. In the second part, you've proven the following claim: "Given x not in I, x is not in f(I)". By contrapositive, this is equivalent to the claim "Given x in f(I), x is in I". This means that f(I) is a subset of I. So you've just proven the same claim two times. I'm not sure myself of how to prove that applying f to I will not make it smaller. Also, I don't think that I generally consists of a single point only. For example, if f is the identity function on some subinterval of [b, c], then our construction of the I_k's will stabilize, so we will keep getting the same subinterval, meaning that I will also be that subinterval instead of just a single point.
@smiley_1000
@smiley_1000 Жыл бұрын
@@concavecuboid3253 I think I figured out how to prove that I is a subset of f(I). Take x in I. Then, for all k, x is in I_k, meaning that there is y_k in I_(k+1) with f(y_k) = x. Now, (y_k)_k is a bounded sequence which contains a convergent subsequence by the Bolzano-Weierstrass theorem. Let y be the limit of that convergent subsequence. Then we have by continuity of f that f(y) = x. Additionally, since for any k, the terms of the sequence are contained in I_k from some point on, and I_k is a closed set, y is also contained in I_k. Therefore, y is contained in I. Therefore, for an arbitrary x in I, we have found y in I with f(y) = x. Therefore, I is a subset of f(I). Since we have already proven that f(I) is a subset of I, we can conclude that f(I) = I.
@gametimewitharyan6665
@gametimewitharyan6665 Жыл бұрын
Such an intensive video, good work
@bashirabdel-fattah9499
@bashirabdel-fattah9499 Жыл бұрын
This is a very interesting video, and it's a shame that it doesn't have more views...
@superawesomename5027
@superawesomename5027 Жыл бұрын
Amazing video! Keep up the good work!
@RickyMud
@RickyMud Жыл бұрын
Tasteful animation
@trsarathi
@trsarathi Жыл бұрын
Simple things can create chaos. Chaos can not be simple.
@dmitryvolovich4357
@dmitryvolovich4357 Жыл бұрын
8:15 Why is there a last argument? What if there are infinitely many such arguments, and no last one between them?
@dmitryvolovich4357
@dmitryvolovich4357 Жыл бұрын
And actually, if you look at a continuous function like x*sin(1/x), there are infinitely many points where it is 0 in any interval that contains 0
@concavecuboid3253
@concavecuboid3253 Жыл бұрын
The interval is closed. If there are infinitely many such arguments, then the sequence of points that all evaluate to the same value must have a limit point (and the problem you discuss is only an issue when ) but by continuity, the function evaluated at the limit point must also evaluate to that same value. xsin(1/x) is only continuous on [0, 1] if you define f(0)=0.
@mightymoe333_
@mightymoe333_ Жыл бұрын
Excellent video! I have not encountered Sharkovskii's theorem before. Can you explain why the logistic map for r=1+sqrt(8) does not appear to have higher order periods or chaos? Is the additional assumption of an unstable 3-cycle necessary?
@japanada11
@japanada11 Жыл бұрын
From the Wikipedia article on Sharkovskii's theorem: "Sharkovskii's theorem does not state that there are stable cycles of those periods, just that there are cycles of those periods. For systems such as the logistic map, the bifurcation diagram shows a range of parameter values for which apparently the only cycle has period 3. In fact, there must be cycles of all periods there, but they are not stable and therefore not visible on the computer-generated picture."
@concavecuboid3253
@concavecuboid3253 Жыл бұрын
If you set f(x)=rx(1-x) and graph y=f^5(x) and y=x, you will see that these lines intersect (apart from where f(x)=x). This means that there is a point with period 5. But the period 5 is unstable, so does not appear in the bifurcation diagram for the logistic map. The same thing is true for all other periods, which is why the bifurcation diagram only has 3 points on it.
@juancristi376
@juancristi376 Жыл бұрын
Nice video!
@gabitheancient7664
@gabitheancient7664 Жыл бұрын
have I got tricked into learning real analysis? you mf good video
@concavecuboid3253
@concavecuboid3253 Жыл бұрын
lol glad you liked it
@user-pr6ed3ri2k
@user-pr6ed3ri2k Жыл бұрын
0:42 collatz and tent things
@user-pr6ed3ri2k
@user-pr6ed3ri2k Жыл бұрын
binary shift chaot
@user-pr6ed3ri2k
@user-pr6ed3ri2k Жыл бұрын
2:17 insane functioj
@user-pr6ed3ri2k
@user-pr6ed3ri2k Жыл бұрын
5:23 intermediate val
@user-pr6ed3ri2k
@user-pr6ed3ri2k Жыл бұрын
5:34 baking
@octosaurinvasion
@octosaurinvasion Жыл бұрын
comment
@aweebthatlovesmath4220
@aweebthatlovesmath4220 Жыл бұрын
Replay
@blacklistnr1
@blacklistnr1 Жыл бұрын
0:30 "Continuous system" **shows discrete animation of discrete simulation** 😂
@concavecuboid3253
@concavecuboid3253 Жыл бұрын
increase your framerate to infinite ;)
@pierreabbat6157
@pierreabbat6157 Жыл бұрын
Sharkovsky's initials are АНШ in Russian, but ОМШ in Ukrainian.
@federook78
@federook78 Жыл бұрын
∀ r ∈ script, r = v 😂
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