Exploration of Chaos

The doubling map

Group Members: Neeraj Kumar Student ID: M00606724

Elena Draghici Student ID: M00564116

Humaira Rahman Student ID: M00566184

Contents

Introduction Page 2

Concepts of chaos Page 3

The doubling map Page 6

Another feature of chaotic maps Page 8

Conclusion Page 16

Appendices Page 17

References Page 20

The notion of chaos is focused on the behavior of deterministic dynamical systems whose behavior can in principle be predicted. Chaotic systems are predicted for a while and then appear to become random. In chaotic systems, the uncertainty in a forecast increases exponentially.

When was chaos first discovered? The first experimenter in chaos was Edward Lorenz. He was working on the problem where in a 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 up totally different from the original.

This effect came to be known as the butterfly effect. The initial conditions have very little difference so small that it can be compared to butterfly flapping its wings.

The flapping of a single butterfly’s wing today produces a tiny change in the state of the atmosphere. Over a period, what the atmosphere actually does diverges from what it would have done. So, in a month’s time, a tornado that would have devastated the Indonesian coast doesn’t happen. Or maybe one that wasn’t going to happen, does.1

This phenomenon, common to chaos theory, is also known as sensitive dependence on initial conditions. Just a slight change in the initial condition can dramatically lead to a change in the long-term behavior of a system.

In common usage “chaos” means a state of disorder. However, in mathematics this term is defined more precisely. Although there is no universally accepted mathematical definition of chaos, a commonly used definition originally formulated by Robert L. Devaney. There are many possible definitions of chaos in dynamical systems, some stronger and some weaker than ours. 2

Definition: Let be a set. is said to be chaotic on if

1) has sensitive dependence on the initial conditions.

2) is topologically transitive.

3) periodic points are dense in

A chaotic map consists of three properties which are unpredictability, non-decomposability, and an element of regularity. A chaotic system is unpredictable because of the sensitive dependence on initial conditions. It cannot be broken down into two subsystems (two invariant open sets) which do not interact under f because of topological transitivity which means there has to be some intersection. And, in midst of this random nature, there is an element of regularity, namely the periodic points which are dense. A map is chaotic only if all three properties stated above exists for a dynamical system, absence of any of these wouldn’t make it chaotic.

Concepts of chaos

Definition: Sensitive Dependence on Initial conditions (SDIC)

Let be a set. has sensitive dependence on initial conditions if there exist such that for any and any neighborhood of there exists and such that

Intuitively, a map has sensitive dependence on initial conditions if there exist points close to , say , which eventually separates from by at least under iteration of Also, all points near need not behave in this manner, but there must be at least one such point in every neighborhood of

In the illustrated diagram, if there is a , let and be a small disc around , whose radius is , then there exists a in the neighborhood of such that the distance between and after iterations is

No matter how close is to , it will eventually separate from after iterations by . This is called sensitive dependence on initial conditions which is widely considered as being the central idea in chaos.

· Here is an example of a non-chaotic dynamical system which has sensitive dependence on initial conditions:

It has sensitive dependence on initial conditions as it doubles every initial condition with each iteration.

Let

After applying the doubling map for the first time:

After a second iteration:

.

After a third iteration:

As it can be clearly seen that with each iteration the distance between andgets doubled, which makes it sensitively dependent on initial conditions. But there is no topological transitivity and dense periodic points.

· Example which has no sensitive dependence on initial conditions:

Here, there is no sensitive dependence on initial conditions upon iterating points a number of times. Because with iteration the distance between andgets halved.

Let’s consider the same initial conditions as in the example above, so and, so

Now applying the function for the first time:

With the next iteration:

Definition: Topological Transitivity (TT)

Let be a set then is said to be topologically transitive if for any pair of open sets there exists such that

In the diagram, let be the space and are non-empty, open sets. Iterating forward, at some point after some number of iterations the image of intersects with.

· For instance, an irrational rotation of the circle i.e. (where is irrational) is topologically transitive.

whereis an irrational number in

Consider a forward sequence of and it is known that they will never coincide because no point is periodic or eventually periodic. So, given any

there must be integers and such that .

Thus, all points are less than apart from each other.Therefore, the orbit ofis dense in the circle (Dense orbit topological transitivity). But not sensitive to initial conditions, since all points remain at the same distance apart after iterations.

But when is rational, there is no topological transitivity and also no sensitive dependence on initial conditions. Thus, in either cases the functions are not chaotic.

Definition: Dense Periodic Points (DPP)

A periodic point is a point that comes back to itself after number of iterations, i.e. . For some positive integer . Let be the set, so this dynamical system has dense periodic points if periodic points are dense in . Dense means that given where is a non-empty open set, there exists such that for some

In the diagram above, let be the space and be a non-empty, open set with radius is a very small disc in the space and irrespective of size there is a periodic point in it and if that’s the case the periodic points are dense in Devaney refers to this condition as an ‘element of regularity’.

· The identity map, is the perfect example of a dynamical system which has dense periodic points. It has dense periodic points because every point is a fixed point and every fixed point is a periodic point. Again, it is not chaotic.

The Doubling Map

The following map is called the doubling map as it doubles each angle and reduces modulo 1:

Also,

Any results obtained will be in the interval 0,1). It is a circle map since it is convenient to rescale angles so that , however it is not a rotation.

The focus of the report will be the action of the doubling map on numbers in binary. What it does is shift the decimal point to the right and remove any integers before the decimal point (reducing ). For shortness, only the numbers after the decimal point will be considered since the integers are removed by the

It is not enough for a map to have sensitive dependence or have dense periodic points for it to be chaotic but rather to ensure this all 3 concepts mentioned need to be true for the map chosen.

Proof of sensitive dependence on initial conditions

Consider any , and it has a binary expansion, let …………

Given any n, let ……..…….

……….

………..

So, the distance between these two is

So,

Given any wecan chooseto be large enough such that .

Hence, the doubling map has sensitive dependence on initial conditions.

Proof of dense periodic points

Let be a non-empty open set and let be the center-point with binary expansion.

A number close to which is periodic, let

Distance between andis.

Proof of dense orbit topological transitivity (because such an orbit will always visit any non-empty open set)

Let S be a dense countable set of rational numbers on . Let the elements in the set be By combining the first digit from , first two digits from etc. call this number .

Let’s iterate with the doubling map. Given any number in and any There is some at a distance from (since { is dense in 0, 1). So, at some point the orbit of under the doubling map lands on a point which coincides with in its first digits. An can be chosen as large as we like , so choose so that the orbit of comes within a distance of . Thus, the orbit of comes within a distance of of which means S has a dense orbit and thus implies topological transitivity.

More precisely, let andbe two non-empty open sets and

Now let and

Let for any Then there is a such that,

begins as

Then, |.

Thus, the doubling map is chaotic, as it has sensitive dependence on initial conditions, topological transitivity and dense periodic points.

Another feature of chaotic maps

Another common feature associated with chaotic maps is that they have these invariant distributions which shows no matter how it starts, what distribution of initial conditions it starts with, after some time there will be no memory of the initial conditions and there would just be a flat (uniform) distribution with no information of the past.

To demonstrate this characteristic utilizing R Studio, 10000 random points in the interval 0,1 were sampled from the uniform distribution, the ?-distribution and the normal distribution and mapped under the doubling map. Histograms of the resulting points after several iterations are shown below (refer to appendices for the R code:

Initial data from the uniform distribution

Uniform distribution after first iteration

Uniform distribution after fifth iteration

Uniform distribution after tenth iteration

Uniform distribution after hundredth iteration

It is very much clear from the distributions above that a doubling map takes the uniformly distributed initial conditions to uniform distributions after some number of iterations.

Initial data from the ?-distribution

?-distribution after first iteration

?-distribution after fifth iteration

?-distribution after tenth iteration

?-distribution after hundredth iteration

Initial data from normal distribution

Normal distribution after first iteration

Normal distribution after fifth iteration

Normal distribution after tenth iteration

Normal distribution after hundredth iteration

Surprisingly, now when initial conditions from the ?-distribution and the normal distribution are iterated, the doubling map takes it to the uniform distribution almost straight away after a few iterations, which is quite hard to think about or predict. So, it’s quite clear from here that chaotic systems are unpredictable. And it’s hard to say anything about the initial conditions, and it seems like all information about past has been lost and cannot be retraced.

Conclusion

Chaos has already had a significant effect 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 example of chaos is possibly weather, in real world, which is why it’s really difficult for meteorologists to predict the weather. It is a very complex theory which has shown that nature is far more complex and surprising. Chaos has inseparably become part of modern science and changed from a little-known theory to a full science of its own.

1

2 Devaney, R. (n.d.). An Introduction to Chaotic Dynamical Systems. 2nd ed. ebook Addison-Wesley Publishing Company. Available at: http://zangeneh.iut.ac.ir/sites/zangeneh.iut.ac.ir/files//files_course/robert_l.devaneyan_introductiponto_chaotic_dynamical_system.pdf Accessed 1 Jan. 2018.