## Monday, November 30, 2009

### Chaos Theory and Fractals

Disclaimer : This is an interesting piece but may be too technical. I have explained it in simple terms. NOT WORTH SKIPPING :-D

What if I say there is orderliness in chaos? People!! People!! I have not lost my mind; I am very much sane.
That's what Chaos theory is all about.

Chaos theory is the study of dynamic systems that are apparently chaotic; lacking order ; but studying them in depth reveals their orderliness. It imbibes physics and maths to explain that the unpredictable conditions occurring in nature are due to the initial conditions of the system. The predictions can be arrived from simple deterministic mathematical equations.

A very minuscule irrelevant and random occurrence can produce drastic results; by triggering a series of events. They call it the Butterfly Effect. Why so? Because it seems that the flapping of a butterfly's wings in one corner can set off a tsunami in other corner of the world. That is; the flapping of wings are the minuscule changes in the initial condition that can trigger a chain of events leading to a mighty phenomenon. Had the butterfly not flapped its wings, the chain of events might have been different.

So, being deterministic doesn't necessarily mean the events are predictable!!

There are lot of scientific terminologies which i wont use here; I want you folks to come back to my blog :-)

In other words, small and random things can cause complex and drastic events and these seemingly random events can be actually predicted from simple deterministic equations.

One of the most interesting subject in chaos theory is Fractals. They make the study of chaos theory simpler and applying the theory is easier when the subject is 'fractalized'.

Fractals are everywhere around you. Nature abounds fractals.

Fractals are geometrical figures; but you don't find perfect square, circle or rectangle in nature. The shapes are complex, chaotic but observe them in depth; you find definite patterns and order.

They have a unique property - self-similarity. A small split portion from a whole figure is a miniature replica of the whole.

The best example in nature is the Snow Flake and the Fern Leaf. Observe a portion of the whole and you can sense the portion is similar to the whole.

There are lot of examples in nature - The coastlines, the shell on a snail, cauliflower, water flowing, blood capillaries......

These fractals are generated to study the behavior of chaos theory by using recursive mathematical equations and recursive algorithms.

Few more fractals in nature

Fractals are applied in every field known to man - astronomy, medicine, meteorology, architecture, study of soils, seismology, enzymology, computer graphics, biology, robotics, economics, engineering, finance.....the list is endless.

-Images are compressed in computer storage technology on the theory of fractals.
- Architecture and construction - It is said that a tiny nail or a screw in a skyscraper can predict the strength and resiliency of the entire building
- Fluid mechanics - study of aircraft turbulence
- Special effects for movies; fractals achieve realism and the images need very less storage space
- Weather and natural calamities predictions
- Study of landscapes like mountains, terrains, coastlines and rivers
- Market prediction in economy
- Water retention properties of soil in the science of agronomy.
- Study of population growth
- Studying diffusion characteristics

Diffusing fractal

And many many more!!!!

Fractals generated using recursive algorithms

There are detailed texts all over the net. There are fractal generating software and one can write recursive algorithm to generate beautiful fractals. Please try them if interested. Its intoxicating :-)

PS : None of these snaps belong to me. And I seem to have lost my collections of Fractal Generating programs :-(

1. Many years back, they used to have UGC progs on DD. I saw one on fractals then. The subject has made me wonder on the creation of the Universe. No I find a like minded person in you. Yoohoo. Nice pic collection

2. hii

nice yaar! very interesting post! and amazing images :)

3. oh my, i was not even aware about most of the things...fractals - i knew the concept, but never noticed them around me...so many times I have seen them around...this shows how negligent I am, and to an extent a duffer too :(

and a pat on ur back...I completely understood what the chaos theory is.. and trust me, that's saying something..

very interesting read...enjoyed reading this, and will enjoy flaunting my knowledge even more now :P

4. and btw, the last image reminds me of NZ cricket team logo :D

so a special thank u for that :)))

5. nice info...took me a while to understand the fractals images...thanks for this valuable info...

6. Hi Holy Lama,

Did they? You wont believe it if I say I have been fascinated over them since then.

Ended up deriving mathematical equations; applying them and deriving algorithms and generating fractals using computer languages!!

They are just out of this world isn't it? My fascination for them never ends.

Pictures - not mine this time. :-)

7. Hi AS,

Thank you :-)

8. Hi Neha,

Thats how we are isn't it? We are too busy chasing our life that we dont notice such beautiful things that are all around us.

I am so happy that you understood Chaos Theory. I am crazy about it and hence wanted to share this beautiful stuff with people.

Oh yeah!! Please flaunt; people will stare at you in awe. :-)

9. LOL!!

NZ team logo. Yeah...remember that now :-D

10. Hi Gayathri,

Ultimately you understood isn't it? If so, me a happy person.

Coming to your question; I can say I am enchanted by these chaos theory and fractals since the day I discovered their existence. I researched a lot on them, learnt their properties as much my brain could understand, their applications. I then started deriving mathematical equations; deriving recursive algorithms and started generating fractals using Java - computer language rich in graphics utilities.

I have been wanting to share about this fascinating topic for quite a long time; as its everywhere around us but most of us dont know!!! But was not sure if people would have any interest as its too mathematical and scientific. But I still wanted people to know these amazing thing. Thats why this 'hatke' post :-)

And mind you; I have not used any scientific jargon but only explained simple things. There are lot more to it; but I didnt want people to run away from my blog

Hope you enjoyed reading it :-)

11. very interesting ... never thought about it before ..

12. Hi Insignia!

One of the blogs I follow had a post on precisely the same subject, and probably because your as well as that blogger's primary source could be the same, some of the examples you have given are common:

Chaos Theory - Is there a pattern in randomness? « Destination Infinity (click)

So while reading your post, I just couldn't help but subconsciously, compare you post with his. :) Sorry about that!

Anyway, coming to chaos theory!

First disclaimer is I have never read the chaos theory article myself!...

13. ...Now this is going to be a long comment simply because the subject of your post spans a wide areas of human knowledge--most importantly, science, the philosophy of science, philosophy, epistemology and such!

One thing I could gather was, chaos theory attempts to help up predict outcomes that are "unpredictable".

This usage of 'chaotic' or 'unpredictable' is what inspires the feeling of awe in our minds when thinking of the systems being illustrated as subjects of the chaos theory.

I'm glad you'd drawn a distinction between 'deterministic' and 'predictable' right in the beginning itself, which means a few less sentences to type out here! :)

What all possibly makes a system unpredictable/chaotic?

1. The individual causes governing a phenomenon are unknown. Here the greatest challenge lies in being able to say with certainty that these causes would never be known even in the future! So as science progresses and we learn of more cause-and-effect relations, obviously fewer and fewer things will remain in this domain of 'random'. And well, if we reach a 'consensus' that a particular event is so very random that it occurs without any cause whatsoever, then it no longer remains determinstic! And no longer does it remain open to exploration through the chaos theory. :)

2. Number of factors/causes governing an event are overwhelmingly large, and which in turn interact with each other in a complex, but deterministic/predictable manner....

14. ...Yes such events would indeed seem chaotic and unpredictable to the 'human eye'. But again, being able to predict them is only a matter of technological advancement. With faster computing more and more things would become inherently predictable.

3. Lack of sufficient data. We find many things chaotic simply because no one would have observed them for long enough to find a pattern. But we can always keep record more meticulously!

The reason I'm telling all this is simply 'cuz most of what all our superficial examination compels us to think as unpredictable, was never that way. And also that there is a 'degree' of unpredictability, which on the whole keeps on decreasing with scientific understanding and better technology.

Also, I did not understand the relevance of fractals to chaos theory.

The best I could get is this:

large, asymmetrical, chaotic phenomena/objects are composed of small-scale, but identical motifs.

And there is a premis that if we are able to make out what these motifs are, and if we are able to make out how they interact at a micro level, then we can make out, what the 'large-scale' interaction (the ultimate chaotic phenomenon under study) would end up like.

Am I right on above two counts?

Assuming, I have understood the concepts correctly, I proceed further to point out problems with this premis:

1. In my opinion, some of the superstructures, like fern-leaf pattern are that way for a reason. So there is nothing special about how they're arranged....

15. ...You'll find the reason in my response to Destination Infinity's post.

2. Some of the examples are not even perfect fractals. Here this emphasis on 'perfect', because our subjects--the chaotic systems are inherently very sensitive to even minor changes in 'input factors'. So we cannot deal in approximations. Because an approximation is simply a deviation from the 'actual' input value. So in that sense, if you see the figure of lightning is not a perfect fractal. Or it least my eyes couldn't find a perfect fractal in it. I could see too many forks/prongs that erupted from the main stem of the lightning at unique angles (uniqueness of these angles goes against self-sameness).

3. It is wrong to say there are no perfect geometrical figures in nature. Think of an NaCl crystal! It is a perfect cube right from its smallest units. And no wonder, the macrocrystals formed are also perfect cubes. But here I'll digress slightly and ask you one question, if you've thought as to why do we think of asymmetry as 'natural'? In fact I feel, every system if uninfluenced by external factors, would inherently be symmetrical. Whenever there is asymmetry we are left wondering why one 'direction' was 'chosen' over other. We learn about many 'causes' simply 'cuz they would be 'external' to the system and introduce asymmetry.

The implication of the last point is that it goes against the idea of fractals that natural things are asymmetrical. I rather feel, 'nature' tends to keep things symmetrical....

17. ...If you're wondering (not unlikely since my comment has been highly unorganized), what am I trying to say above, then I would summarize it here:

1. If chaos theory is just a sophisticated way of saying, "Yes, we can do it! Yes, what we used to think of as random and cause-less if further examined, does indeed have causes, only that we need to resolve the factors at play better." then I agree with it, but would wonder, if it indeed does fall in the domain of science (and mathematics).

2. But if chaos theory is proposing a new 'method' of finding pattern in randomness by positing that: "we can predict outcomes in all seemingly complex systems, only by fishing for the self-same units, i.e., motifs called fractals, without finding the underlying cause-and-effect relation that results in replication of self-sameness to form larger superstructures", then I feel this postulation is misguiding, because it might help us predict only very few outcomes, and that too only where self-sameness would be an outcome of a REASON (e.g., NaCl crystals, fern pattern). So in that case, this method wouldn't work RELIABLY. :)

And hence I could not understand the relevance of chaos theory!

But I've to concede, I've yet not made a sincere attempt to understand chaos theory. My understanding of it is based largely on two blog posts (yours and DI's)....

18. ...But yes, fractals are an interesting thing. And some of the human-generated fractals represent a very high level of intellectual accomplishment, and not to forget, fill our world with beauty and mind, with wonder! But not sure if they can explain everything about our Universe! ;)

And I sincerely thank you for bringing me closer to the understanding of chaos theory, and also very much appreciate your own efforts to undertand and convey it to your readers. :)

TC.

PS: The peacock feather fractal was the funniest! I understand, the individual golden spade-shaped motifs look like one peacock's plumage. But how do we make this self-sameness grow further? Meaning, how do we ask many peacocks to to stand in a 'dahi-handi' configuration to form a larger peacock? :P

19. Hi Naveen,

Fractals are interesting!!!

20. Gautam!!

Dear, you know how fascinated I am with this thing. I thought I can relive the excitement by sharing it here :-)

I am still waiting for you to be free so that we can discuss more on these. :-)

21. Hi Ketan,

Thanks for all the comments. I am in awe seeing your interest in chaos theory.

I wanted to explain in simplest terms as possible. Hope it worked. Regarding examples; few of them stated here are the easiest way to make someone understand fractals and most basic classes of fractals start with quoting these examples. So its not common to see these examples quoted elsewhere.

I would surely discuss all the points with you OFFLINE as I dont want any more protests :-P

Would just like to clarify one thing. When I said "we dont find perfect shapes in nature" I didnt necessarily mean that all things in nature are asymmetrical.

The above sentence was only to quote most things in nature when observed are not in perfect shapes; one defining properties of fractal is self similarity in these asymmetrical shapes :-)

22. ah yes i have read quite a bit on it.aniakum athaan sonen.theres a pattern in everything.nothings random.if they cant find a pattern they say its random.
no two snow flakes are alike.yet the small peaks in a big mountain and the ultimate big peak have a similar pattern..free

23. Hi Insignia!

I just skimmed through the article on chaos theory on Wikipedia, and of course, probably could understand only 2 to 3 percent of that article. :) One thing I realized is that the word 'chaotic' has a very stringent definition in mathematics. And it obviously does not seek to explain the whole of the Universe!

Systems, apart from being deterministic or chaotic, could also be 'stochastic', too. One more thing is that the theory deals more with statistical observations of system rather than the actual governing mechanisms.

I also realized that the systems that form the subjects of chaos theory are actually limited. Also, such systems have to be multi-dimensional, usually (but not always) where one of the dimensions is time. And for something to be chaotic, the equations relating the various dimensions must be non-linear, otherwise the 'errors' found in one dimension on changing the value in some other dimension would not be exponential, which (greater than 1 order of equation) is the prerequisite for a system to be chaotic!

But again, I could not entirely understand where fractals exactly fit in the picture!!! The Wikipedia article seems to have suddenly brought in fractals, with rather loose connection with the concepts being discussed hitherto!

Anyway, what I did understand from all this is that with my limited knowledge of mathematics, if I've to understand chaos theory, I'll have to dedicate at least two entire weeks of very, very, very intense learning on Wikipedia, and that too I might end up plucking my residual hair out!

Which is not to mean, I won't try. And at that point your post would be my inspiration. So, thanks in advance!

I realize my previous comments are very amateur with very superficial (qualitative) and wrong understanding of the word 'chaotic', so you could largely ignore them!

You could've easily pointed out my understanding of 'chaotic' was wrong. I wouldn't mind. :)

And this comment was largely typed before reading your above response. So all the concepts I have put in it, are for your approval, i.e., if what I've grasped is right or not.

With regard to carrying the discussion offline, I could only say that a small disadvantage of it is that other readers would not get to learn whatever we discuss offline (assuming other readers are desirous of learning that 'whatever').

But thanks, again!

And are you like 'ultra-into' mathematics? ;) That makes me look at you in great awe!

TC.

24. Soin,

You got it!!! Attaboy!!
:-D

25. Hi Ketan,

I replied to all your queries offline.
Chaos theory is difficult to comprehend; as you said; there are lot of parameters to it and also; science, math and philosophy is involved.

There are no concrete proof and research is going on in this field.

I do not have an idea about any definite proof; that yeah "We did avoid a skyscraper from falling as we applied fractals and chaos theory and could well predict the outcome"

But thats where they want to reach :-)

I would say its difficult to understand unless we get into research of chaos theory :-D

Regarding people missing out on new learning from the discussion; I would post the offline reply; if anyone requests. But I guess none would. They dont want to pluck out their already receding hairs; as you are probably doing :-P

I wouldnt say I am 'ultra-into' mathematics yet. I sincerely want to :-)

26. And yes, it is indeed funny to ask the peacocks to stand as it is.....to form a larger fractal :-P

27. Wow! Thanks for this:) though have heard about the chaos theory, never had the courage to dig it..But I must say that your disclaimer helpded as it gave me enough courage to read on ..smart move :)Btw..on the fractals..wish I knew this when I was in college..All I needed to peek was one portion of my neighbours answer sheet and then I could visualize the whole paper;)

28. Chaos theory is one hell of an interesting topic, isn't it? Although chaotic systems appear to be random, they are not. Beneath the random behavior, patterns emerge. Strange, but beautiful!
I saw a movie a couple of months ago called '╥' (pi) which was awesome. Recognizing that the stock market is a non-linear, dynamic, chaotic system, the film's hero uses chaos theory in order to determine the pattern behind random nature of market prices.
Hope you've seen it. If not don't miss it.

As usual, a wonderful and interesting post. :-)
Keep coming up with such posts. :-)

29. ' There are fractal generating software and one can write recursive algorithm to generate beautiful fractals'
There are many other key words like 'deterministic' for example.
Wow! you really are a nerd:P

Nice informative post, Have you seen the butterfly effect and chaos theory? good movies, they are.

30. Rohit,

Hehehehe thanks. I didnt want people swearing at me; hence the disclaimer. Glad it helped.

The name "Chaos theory" itself is too scaring to dig on that subject isn't it?

Hahahaha, you are asking for too much. They would have asked you to derive a new equation/algorithm to apply it elsewhere instead of the pictures directly!!! :-P

31. Hi Karthik,

You bet it is!! :-)

Have heard about the movie 'Pi' havent seen. For sure wont miss it now :-)

Thank you, its encouraging of you :-)

32. Hehehehe.. Am I, Kish?

I have done all that - writing recursive algorithm and generating fractals and all.....May be I am a nerd!!!
:-P

Havent seen those movies. Will sure watch :-)

33. Very informative post. One word. Excellent. We always say in a different slang….There is a method to my madness.

Remember I visited Stonehenge in June. A Julia factor Crop Circle appeared in Stonehenge on July 7, 1996. It just appeared suddenly. It would have taken several days to even design that pictogram.

34. Hi SG,

Thank you. "There is a method to my madness" Woooow, thats a statement.

This is a new piece of information about the Julia set on Stonehenge. I will dig more into it now. Thanks for the trivia

35. the only place fractals work in is a mixer with utter chaos

36. Oh my! Back from a month's break from blogging to read what might be the simplest form of explanation to my favourite theory - The Chaos theory.

Nicely put dear Insignia. Download Apophysis. Open source and freely avaliable, you'll get down to fractal generating in no time. And once you start, it's bloody addictive!

P.S. Good to be back!!

37. Very nice post. Am so glad that you have written this post. And where did you find all these pics! They make the blog look even beautiful. Try reading more about multi-universe and the relation ship between quantum physics to this chaos theory. Very interesting too. Will be back to form soon dear. Happy blogging till then.

38. Hi Shaunak,

Welcome back!!!

Glad you liked it and found the explanation simple. Thats what I wished to :-)

Dont need an Apophysis :-) I used to do it with Java using graphics classes.
That was sometime ago; and I have lost all my fractal generating programs :-(

39. Hi diaboli,

Welcome to B Log :-)
Hahahaha, you bet!!

40. Thanks much Gautam.
Pics are all over internet :-D

Oh yes, there's more to it; I will never cease reading, you know that!!

Waiting for you to get back
Take care

41. Never knew about it earlier..interesting info and beautiful pictures.

42. Fractals are applied in every field known to man

And in unknown as well.

Nothing is absolute random

its just we fall short of capabilities to see that how nature plays its games.

We(human being) are trying to understand Rules of this game. still struggling.

43. there much more things like...as you said when butterfly flaps ...it produces a sound wave ..and similarly all other natural events..and it become a music ..and called "Music or World"

you can listen to it...with keeping your soul in sync with Nature.

:)

Keep Smiling.

44. Hi Antarman,

Thank you, glad you liked them

45. Hi Makk,

Yes, and it turns out to be unbelievable as we lack the capability to comprehend it yet :-)

Thanks for those wonderful comments :)

46. Hey I just finished reading the Dan Brown book (the lost symbol) and thought "phew finally its over no more riddles and puzzles" and then you post this :-)

Welcome to B Log.
You just finished The Lost Symbol? Too late!!! :-)

I am happy that you felt this was something close to Dan Brown. But his are all figment of imaginations with some truth, this is science and math :-)

48. Interesting way to tell a otherwise boring thing.

49. I have read about fractals earlier & even now, continue to be fascinated by it. Science is usually quite orderly amidst all that visible 'outer' chaos and mess right?

Very very interesting post. I was stunned when I read about the architecture trivia (the screws gauging the resillience)!!

50. Hi Pramathesh,

Thank you for the compliment. :-)

51. Hi lostworld,

Yeah, I completely agree with you :)

Oh yes, the application part, there are quite lot of stunning theories, but dont know how far they have really applied it

52. sorry to say..but i knew everythin of wat u said..the pics were all very nice.. maybe i didnt think of peacock wen they say fractals.. so i jus hope u have seen the movie butterfly effect.. right?? wat abt part 2.. although part 2 is mokka and is repetiton of part 1..

53. Loved this post...so simple- so intelligent!!!!

54. Hi Vishnu,

I dont see any reason as to why you are sorry at all. You are sorry because you knew it? thats absurd. Be proud that you knew these things; its a beautiful theory.

Havent see any movie on chaos theory yet.
My knowledge is completely based on what I read; researched and did some fractal generation using Java

55. Hi Shrutzz,

Welcome to B Log. Glad you liked it and got some information :-)

56. There is a guy who used to play with such patter generating programs as a kid. He went into it in so much depth that he saw a whole 'New Kind of Science' in it. I first saw the real hardcover book at library of the place where I did my engg. final year project. Now its online too. (And of course the guy Wolfram has become a big name).

http://www.wolframscience.com/thebook.html

By the way if you use programs and formuae, I don't think its 'chaos'. Maybe it will look like chaos to human intuition, but is not. Actually none of your images look chaotic.

Ketan, NKS is quite capable of offering new insights into biology too. You might like. Also you might notice there that many patters are quite naturally unsymmetric.

I have read it only little. You can read the whole book online. There are also forums on various applications

57. Hi Stupido,

Thanks for the information. Guess you havent understood.

The basis of chaos theory is that if you analyze a seemingly random behavior; it does reveal deterministic patterns.

And the deterministic equations are derived thus and the fractals I have put up are IFS which can be derived using iterations.

58. Well, I did understand. Only thing is, many of the patterns in the post are not even 'seemingly random'. They are 'obvioulsy patterned'. Especially snowflake strcture, seashell, and all the first three outputs of recursive algorithms.

But I understand that 'seemingly random' could have an inherent pattern.

And of course, Kinetic theory of gases, which predicts very definite rules is also based on inherent random movement of molecules.

Again, if you take pic of small part of Mona Lisa's smile and zoom it up, it will just look like a random ugly blotch, so only when you view the 'complete picture' at appropriate scale, do fascinating things show themselves clearly.

Even microscopic view of many tissues of body could appear random. But overall it gives rise to very definite phenomenon called living being. Of course many tissues do appeal to human intuition of 'regularity', like many of your pics, but in that case we could zoom into the cells further and find the 'irregular' shapes inside the cells too!

Chaos theory may be just one way of looking at it. Very specific way of looking at it, or very generic way of looking at it, I can't comment as I don't know in depth about chaos theory nor above or other phenomenon I compared it with.

I guess everything could be like pseudorandom numbers generated by EXOR feedback logic or circuits. To the observer, there may not be any relation between one number and next, but if we keep on continuing to generate the numbers, finally the whole cycle repeats.

59. (any 'apparent' relation between one number and next I meant)

60. Stupidosaur and Insignia,

By 'nature', I did not mean ecosystem or events observed in nature. Just that the most 'natural' way we expect events to occur is in all/two directions symmetrically.

For instance, if cells in a plant would divide, they would do so randomly - and when observed after a long time following MANY divisions, it would look like a big ball (sphere is geometrically the most 'symmetrical' structure after a 'point'). The only reason is unequal EXTERNAL forces act on this 'system' of growing plant, most prominently, gravity (geotropism) and light (phototropism). So when I said, 'naturally', I only meant without these external factors.

What you say of kinetic theory of gases sounds more like 'anti-chaotic' theory! ;) Symmetrical grossly, but random in components.

Actually, the symmetry we get in gross events like behavior of a gas in a confined space is simply 'cuz the number of statistical observations is very large!

So it is assumed that the components of linear momentum of 'SO MANY' particles would cancel out each other. Now this was since linear momentum is a vector, but kinetic energy is a scalar, so it does not get canceled out. I'm sure you know this all. So thermodynamics of IDEAL GAS basically rides on this benefit offered by number of observations being large.

But when studying a real gas, so many factors - mostly arising out of electrostatic interactions, which are deterministic, but difficult to determine - make it (possibly) enter the domain of 'chaos theory'.

My earlier string-comments were total garbage; don't take them seriously. I'd employed a very linguistic/colloquial meaning of 'chaotic'.

I'm not sure, if you've yet read the Wikipedia article on it.

I'd read, and could understand only 2 to 3% of it.

Though, am not a mathematician/statistician/physicist/philosopher/logician, let me tell you of MY understanding of chaos theory. If you read it, and find my interpretation grossly wrong, do correct me. Seriously. :) I'm very bad at mathematics.

Firstly, apart from the said system being DETERMINISTIC, 'chaotic' has a very specific MATHEMATICAL definition, and has very little to do with our usage of 'chaotic' in daily language.

Let me try to explain a bit further, then maybe the definition of 'chaotic' will make better sense....

61. ...Most of the relations in physics [and all verifiable+tangible+natural 'knowledge' is basically physics using mathematics :) ] are derived through experimentation (and collection of data), or through derivation from pre-existing 'laws'.

We make changes in one parameter (independent variable, usually labeled as x) and observe the corresponding changes in the dependent variable (usually labeled y). This is a two-dimensional system, in the sense, two variables are under study.

Then we obtain a scatter diagram - such that we plot individual points for every value of x 'used' and 'y' observed.

If you find that most of the points tend to be in a 'line', you ASSUME that the two measured quantities are connected by a linear equation. This is purely theoretical. This ASSUMPTION of what the relation SHOULD be is known as 'regression of' or 'fitting' the function.

Now after you have derived an equation and are sufficiently confident of its correctness 'cuz of SUFFICIENT consistency in OBSERVATIONS, then you would perform more such experiments. But not quite unexpectedly, you'll see that hardly any point ACTUALLY falls on 'the line' you had drawn on the graph!!

Most of the plotted points fall on either side of the line. This difference from predicted value in observed value (difference in 'y') is known as 'error'.

Error arises 'cuz of two reasons:

1. Your equation was an approximation. There are ALWAYS more number of independent variables involved. We study only a limited number of them for simplicity.

2. Observational error.

Now whether a system is mathematically chaotic or not depends on how this error behaves with respect to one of the independent variable!!!

If there is a LINEAR relationship between errors (and NOT observed value 'y') and the independent variables, it is known as a 'mathematically' deterministic system.

If the errors and independent variable are related exponentially - it is a CHAOTIC system. 'Exponent' in exponential is such a trap! It need not be an integer; could be any real number!

If there is NO such relation, then it is a STOCHASTIC system.

So basically, chaos theory seeks to describe and classify systems as they are OBSERVABLE, taking into account observational errors and the fact that the known systems are mereley 'MODELS' with lot many influencing variables left out from the equations, but all this without actually going into the theoretical reasons for those errors....

62. ...Now again, an important criterion to call a system chaotic is that it should be MULTIDIMENSIONAL. Meaning, by default we should be studying at least three parameters (variables). This however holds true for so many physical systems.

Now, coming to the the reason the system is so sensitive to small changes, i.e., a small change in the value of independent variable makes such an extreme difference in the value of observed (dependent) variable (actually HUGE change in ERROR produced in y).

As you shall see, by definition, chaotic system is one, where error is exponentially related to the value of 'x'. But putting the reason in this fashion is simply beating round the bush! :)

The possible reason is that 'y' depends not just on x, but also on other variables (not constants) - 'a' and 'b'.

Now the biggest assumption we usually make in making models is that independent variables - x, y, a and b would be independent of EACH other. But what if they are NOT independent? I think that's where chaos theory starts playing a role.

Let's take an example where y is directly proportional to x; directly proportional to square root of a; and inversely proportional to cube root of b.

Now, imagine, we never take 'b' into consideration because either it would make the calculations too complex or more importantly because, we don't know yet that it influences y or also because we don't know mathematically how y and b are related!!

Now what makes the system further complex is that x would be related to a and b!!!

Let's say x is directly proportional to a and directly proportional to the square of b.

As you shall see now, our mathematical 'MODEL' will only give us predicted values of y for changes in values of x and a. But there will definitely be errors simply 'cuz we totally overlooked b, and also the fact that that x and a are related, was overlooked!

So to 'understand' this system will take a lot of technological advancement, whereby we would be able to resolve ALL the component independent variables.

However, the error we get in above system would depend not just on values of x, a and b, but also on the CONSTANTS used in various equations relating these parameters....

63. ...But the impatient organisms that we are, many times we're interested in being able to predict outcomes in a system, rather than understanding the underlying mechanisms [astrology, for instance; and you might see, how in days to come, chaos theory would be used to explain away a whole lot of BS like astrology, fully and conveniently forgetting that chaos theory is merely a curious observation in mathematically chaotic systems, and NOT an established 'fact' with mathematical/statistical/physical bases!! ;) ].

So what chaos theory attempts to do is to do away with need for resolving all the influencing parameters. It simply tries to find a relation for one of the independent variables and the ERROR. Plus, it tries to establish a trend for the 'behavior' of this error with respect to the chosen independent variable.

Possibly here is where FRACTALS come into play.

Observing the behavior of errors in a small region of the graph, and then assuming that this behavior would be replicated at a much bigger scale (over a larger RANGE of values of x) if the system is ESTABLISHED as mathematically chaotic. Of course, it's an entirely different matter that we might have to resort to 'regression' of function connecting the independent variable and the error in observed variable, when ironically we had already stated that the system is 'sensitive' to small changes (and hence, does not 'like' approximations)! ;)

Possibly, there is no theoretical reason for this assumption to work, but it has been observed that it does work in systems MATHEMATICALLY defined as chaotic! So chaos theory might help us in PREDICTING outcomes in mathematically chaotic systems.

Phew!! After so much mental and digital (fingers') effort, this is all I could speculate and state. And the worst part - my interpretation could be way off mark!!! :(

So possibly, chaos theory is still not entirely reliable, and till at least mathematical basis for 'fractalized' behavior for chaotic systems is found, it might remain andhere mein teer! ;) Am not undermining its importance in ability to predict complex systems, just stating my understanding of the current situation.

Do correct me, if Stupidosaur's or Insignia's interpretations are different.

TC.

64. Along with Ketan's highly detailed interpretation, which left me speechless and eyes watering from staring at the screen without blinking, Stupidosaur's comment reminded me of a quote I'd read:

"Give a monkey a typewriter and a long enough timeline and he'll start writing Shakespeare."

At a large enough scale, what appears random in a microcosm, becomes part of a unified whole, bringing order to the apparent chaos.

65. And Shaunak not to forget, beauty (and also all that is there to behold) lies in the eyes of the beholder. ;)

Sometimes I wonder, how merely quoting something/someone makes me feel as if I said something profound and worthy of being read and responding to! :D

66. Precisely why I quote stuff a lot. Narcissism you see ;)

67. Thanks Stupido and Ketan.
Ketan, thanks to you especially for saving me so much of mental and typing work.

You are right; the first time you commented about chaos; was in its literal sense in English. Now your understand is fine. :-)

Stupido,

the examples I gave, they do have definite patterns you said, especially the Serpinski Triangle, the snow flakes and all. But it requires in depth study to say they are following a pattern; and these fractals are generated after that in depth study; after deriving a deterministic equation. The basis is

new x = bx(1-x)

Seroinski Triangle, Mandelbrot sets, Julia sets are all generated as IFS. Ketan has clearly explained it. So easy for me now :-)

As he clearly states, its not yet an established science. Even though they quote there are many applications of the chaos theory, no one knows how much is really implemented.

But it does seem silly at first, but interesting to read and dig further.

Ketan,

Thanks much for taking this huge burden of explaining Stupido. :-)

68. Shaunak,

Thats a nice monkey quote :-D

69. You're welcome, Insignia!

What is IFS?

Plus, now I also feel, though chaos theory might involve the concept of fractals, (maybe) not all fractals actually qualify as mathematically chaotic systems. Or do they? I'm curious.

Additionally, I think while trying to explain chaos theory through fractals, people tend to get carried away by the 'visual' appeal of fractals, whereas there is actually a deep mathematical/statistical thought behind all this.

It becomes quite a bit like Einstein trying to console common people about the sufficiency of their intelligence and the relevance of special theory of relativity by giving that crappy example of hot oven and hot oven! [And no, I don't find Kareena hot! How would I know till I actually touch her! ;) ] And no, I've never felt like touching her. So, that should explain better! ;)

Another idea that later on struck me was: what if the actual observed errors would in turn 'deviate' from the equation describing errors' relation with x - by the SAME equation used to predict the 'original' errors?!! Please note that the new deviations, by definition are also errors! This error-defining process might also run into an infinite 'recursion'! - So what I wanted to ask Insignia is: in your knowledge, is this serial recursion of 'errors of errors', also a part of definition of chaotic system?

@ Shaunak: Narcissism of the 'highest' form would entail that you consider others so insignificant that you quote only yourself! ;) TC.

70. You're welcome, Insignia!

What is IFS?

Plus, now I also feel, though chaos theory might involve the concept of fractals, (maybe) not all fractals actually qualify as mathematically chaotic systems. Or do they? I'm curious.

Additionally, I think while trying to explain chaos theory through fractals, people tend to get carried away by the 'visual' appeal of fractals, whereas there is actually a deep mathematical/statistical thought behind all this.

It becomes quite a bit like Einstein trying to console common people about the sufficiency of their intelligence and the relevance of special theory of relativity by giving that crappy example of hot oven and hot oven! [And no, I don't find Kareena hot! How would I know till I actually touch her! ;) ] And no, I've never felt like touching her. So, that should explain better! ;)

Another idea that later on struck me was: what if the actual observed errors would in turn 'deviate' from the equation describing errors' relation with x - by the SAME equation used to predict the 'original' errors?!! Please note that the new deviations, by definition are also errors! This error-defining process might also run into an infinite 'recursion'! - So what I wanted to ask Insignia is: in your knowledge, is this serial recursion of 'errors of errors', also a part of definition of chaotic system?

@ Shaunak: Narcissism of the 'highest' form would entail that you consider others so insignificant that you quote only yourself! ;) TC.

71. *crappy example of hot oven and hot woman!

72. IFS or Iterated function system is a way to construct fractals; the results are self-similar.
Each self similar tiny part of the whole is derived from a function.

Once the function is derived; the fractals are constructed by recursively applying them - recursive algorithm.

some fractals are just derived; that is they are not found in nature. Such fractals are orderly and not necessarily chaotic. Fractals are an easy way to explain "self similarity"; one of the parameters in chaos theory. But perfect visually appealing ordered figures are not the ultimate example.

Its just created to make people understand the concept.

Hahahaha..Kareena and hot.. I find both of them so exclusive :-P
Yeah seriously, its just like the Einstein stuff you said w.r.t fractals.

To answer your question on errors of errors, that again results in a different pattern isn't it?
A slight deviation results in a drastically different behavior is what the theory says.

73. That was an interesting post. :) Reminds me of my lost fractals programs too! I was crazy about them sometime back. oh btw, hopped in from Neha's blog :)

74. Hi evanescentthoughts,

Welcome to B Log. Thanks , glad you liked this piece. Oops, I know how it feels, you lost the programs too..:-(

anyways...we had fun creating them. Keep visiting!!

75. Survival group against God?? LOL. Good luck with that. Truth is, no one knows the exact time this will happen except the man upstairs, however, I firmly believe that there are people placed here by God that post the warning signs and it's up to you to take heed.
[url=http://2012earth.net/future_and_past_of_the_earth.html
]2012 doomsday
[/url] - some truth about 2012

76. Ellie Lloyd JonesMay 11, 2011 at 8:14 PM

Dear Insignia,

I am currently writing a project on fractals for my final year at university. Your picture of a sierpinski gasket caught my attention and i would like to use it as a front page image if possible. However, i need to reference it. I was wondering if you know where you initially sourced it from.

Thanks, Ellie

77. Ellie Lloyd JonesMay 17, 2011 at 6:15 PM

Hello, sorry to be a bore but i have a deadline for my project and would really like to know where you sourced you picture of sierpinski's triangle from.
If you can't remember then please just send me a quick reply to say that.

Thanks
Ellie.

78. Ellie,

I dont remember if I did source it or picked it up from a program I myself generated it long ago while in college during my engineering.

I am so sorry I cant recollect.

79. Ellie,

I dont remember if I did source it or picked it up from a program I myself generated it long ago while in college during my engineering.

I am so sorry I cant recollect.

80. Ellie Lloyd JonesMay 26, 2011 at 6:02 PM

Ok, thank-you. If it is ok with you i will put the web address of this page.
It is a very interesting article. I am finding my project topic so interesting - just a shame i'm not doing it for my own purposes because i'm sure having a deadline takes the fun out of it!

Thanks

Ellie

81. Sure Ellie you may refer this link. :-)

Thanks for writing it.

83. Makk,

thanks a lot. My pleasure.

84. Attractive part of content. I simply stumbled upon your weblog and in accession capital to
assert that I acquire actually loved account
your weblog posts. Anyway I'll be subscribing to your feeds and even I fulfillment you access constantly rapidly. you may like the .pdf piece written by : Scott Tucker on
quantum computer wiki
My webpage ; Scott Tucker

1. Thanks Scott. I was passionate about Chaos theory and that's how I got hooked to Fractals. Will surely read your piece. Thanks for stopping by

85. Everytime I feel better... :)

I'd love to know what you thought :-) Please shoot!