Contents a commonly used definition originally formulated by

 ContentsIntroductionDefinition of chaos Concepts of chaosProof that the doubling map is chaoticAnother feature of chaosConclusion (Title page not complete)           The notion of chaos is focused on thebehavior of deterministic dynamical systems whose behavior can in principle bepredicted. Chaotic systems are predicted for a while and then appear to becomerandom. In chaotic systems, the uncertainty in a forecast increasesexponentially.When was chaos first discovered? The firstexperimenter in chaos was Edward Lorenz.

He was working on the problem where ina sequence, the number was 0.506127, and he only used the first three digits,.506, which led to a completely different evolution of the sequence he had.Instead of the same pattern as before, it diverged from the pattern, ending uptotally different from the original.This effect came to be known as the butterflyeffect.

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The initial conditions have very little difference so small that it canbe compared to butterfly flapping its wings.      Theflapping of a single butterfly’s wing today produces a tiny change in the stateof the atmosphere. Over a period, what the atmosphere actually does divergesfrom what it would have done. So, in a month’s time, a tornado that would havedevastated the Indonesian coast doesn’t happen.

Or maybe one that wasn’t goingto happen, does. (Ian Stewart, Does God Play Dice? The Mathematics of Chaos,pg. 141) Thisphenomenon, common to chaos theory, is also known as sensitive dependence oninitial conditions. Just a slight change in the initial condition candramatically lead to a change in the long-term behavior of a system.In common usage “chaos” means a state ofdisorder. However, in mathematics this term is defined more precisely. Althoughthere is no universally accepted mathematical definition of chaos, a commonlyused definition originally formulated by RobertL. Devaney.

There are many possible definitions of chaos in dynamicalsystems, some stronger and some weaker than ours.  Definition: Let  be a set. is said tobe chaotic on  if  1)    has sensitive dependence on the initialconditions.2)    is topologically transitive.3)   periodic points are dense in  A chaotic map consists of three properties whichare unpredictability, non-decomposability, and an element of regularity.

Achaotic system is unpredictable because of the sensitive dependence on initialconditions. It cannot be broken down into two subsystems (two invariant opensets) which do not interact under f because of topological transitivity whichmeans there has to be some intersection. And, in midst of this random nature,we have an element of regularity, namely the periodic points which are dense. Amap is chaotic only if all three properties stated above exists for a dynamicalsystem, absence of any of these wouldn’t make it chaotic.   Definition:Sensitive Dependence on Initial conditions (SDIC)Let be a set.  has sensitive dependence on initial conditionsif there exist  such that for any  and any neighborhood  of  there exists and  such that Intuitively, a map has sensitive dependenceon initial conditions if there exist points close to , say , whicheventually separates from  by at least under iteration of  Also, all points near  need not behave in this manner, but there mustbe at least one such point in every neighborhood of          Inthe illustrated diagram, if there is a  , let and  be a small disc around , whose radius is  , then there exists a in the neighborhood of  suchthat Nomatter how close  is to , it will eventually separate from x after niterations by .

This is called sensitive dependence oninitial conditions which is widely considered as being the central idea inchaos.Hereis an example of dynamical system which has sensitive dependence on initialconditions:1)    Let Ithas sensitive dependence on initial conditions as it doubles every initialcondition with each iteration, but there is no topological transitivity anddense periodic points.Examplewhich has no sensitive dependence on initial conditions:1)    Let  Here, there is no sensitive dependence oninitial conditions upon iterating it number of times.  Definition:Topological Transitivity     Let be a set,  is said to be topologically transitive if forany pair of open sets there exists  such that                                                Intuitively, a topologically transitive maphas points which eventually move from one arbitrary small neighborhood to anyother which means the dynamical system cannot be decomposed into two disjointopen sets which are invariant under the map.In the diagram, let J be the space and U, Vare non-empty open sets. Iterating U forward, at some point after some numberof iterations the image of U intersects with the other non-empty open set V.             For, instance an irrationalrotation of the circle is topologically transitive, but not sensitive toinitial conditions, since all points remain at the same distance apart afteriterations.                               where But when isrational then it has no topological transitivity also no sensitive dependenceon initial conditions.

In both the cases the dynamical systemsare not chaotic, which means all the three aspects must to be fulfilled inorder     to be chaotic for a dynamicalsystem.  Definition:Dense Periodic PointsTo say something is densei.e. a set X is dense in a set V, means that in every non-empty open set Aperiodic point is a point that comes back to itself after number of iterations, i.e.  Let  be theset, so this dynamical system has dense periodic points if periodic points aredense in , now let where is a non-empty open set, then there exists such that for some       Inthe diagram above, let be the space and be a non-empty open set with radius is a very small disc in the space andirrespective of size there is a periodic point in it and if that’s the case theperiodic points are dense in  Devaney refers to this condition as an ‘elementof regularity’. A dense orbit implies topologicaltransitivity because such an orbit will always visit any open non-empty set.Identitymap is the perfect example of dynamical system which has dense periodic points.

Let It has dense periodic points because everypoint is a fixed point and every fixed point is a periodic point. Again, it isnot chaotic. Doubling Map:Thefollowing map is called doubling map as it doubles each angle:                                             Also, Note:doubling map deletes the first digit in the binary expansion of a number.TheDoubling map is chaotic, because it has sensitive dependence on initialconditions, has topological transitivity and has dense periodic points.Proof:Sensitive dependence on initial conditions Considerany , and it has a binary expansion, let …………Given any n, let ……..…….

 ……….………..So, the distance between these two is  So,  Given any to be large enough such that Hence, doubling map has sensitive dependenceon initial conditions. Dense orbit: A   Another common feature that chaoticmaps often have is that they have these invariant distributions which shows nomatter how it starts, what distribution initial conditions it starts with,after some time there will be no memory of initial conditions and there wouldbe just a flat (uniform) distribution with no information of past. 1)     Uniform distributioninitial condition1.a)Uniform distribution after first iteration  1.

d)distribution after tenth distribution  1.c)distribution after third iterationAsit is very much clear from the distributions above that a doubling map takesthe uniformly distributed initial conditions to uniform distributions aftersome number of iterations. 1)     ?-distribution initialconditions                  2.a)distribution after first iteration 2.

b)distribution after second iteration2.c)distribution after third iteration            2.d)distribution after tenth iteration Surprisingly,now when initial conditions are ?-distribution, doubling map takes it touniform distribution which is quite hard to think about or predict about. So,its quite clear from here that chaotic systems are unpredictable. And its hardto say anything about the initial conditions, and seems like all informationabout past has been lost and cannot be retraced.  Chaos has already had a significanteffect on science, yet there is a lot still left to be explored.

Although,chaos is everywhere around the world, but the best and most relatable exampleof chaos is possibly weather, in real world, which is why it’s really difficultfor meteorologists to predict the weather. It is a very complex theory which has shown that nature is far morecomplex and surprising. Chaos has inseparably become part of modern science andchanged from a little-known theory to a full science of its own.  

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