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. (Ian Stewart, Does God Play Dice? The Mathematics of Chaos,

pg. 141)

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.

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,

we have 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.

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

No

matter how close is to , it will eventually separate from x after n

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 dynamical system which has sensitive dependence on initial

conditions:

1) Let

It

has sensitive dependence on initial conditions as it doubles every initial

condition with each iteration, but there is no topological transitivity and

dense periodic points.

Example

which has no sensitive dependence on initial conditions:

1) Let

Here, there is no sensitive dependence on

initial conditions upon iterating it number of times.

Definition:

Topological Transitivity

Let

be a set, is said to be topologically transitive if for

any pair of open sets there exists such that

Intuitively, a topologically transitive map

has points which eventually move from one arbitrary small neighborhood to any

other which means the dynamical system cannot be decomposed into two disjoint

open sets which are invariant under the map.

In the diagram, let J be the space and U, V

are non-empty open sets. Iterating U forward, at some point after some number

of iterations the image of U intersects with the other non-empty open set V.

For, instance an irrational

rotation of the circle is topologically transitive, but not sensitive to

initial conditions, since all points remain at the same distance apart after

iterations.

where

But when is

rational then it has no topological transitivity also no sensitive dependence

on initial conditions.

In both the cases the dynamical systems

are not chaotic, which means all the three aspects must to be fulfilled in

order to be chaotic for a dynamical

system.

Definition:

Dense Periodic Points

To say something is dense

i.e. a set X is dense in a set V, means that in every non-empty open set

A

periodic point is a point that comes back to itself after number of iterations, i.e. Let be the

set, so this dynamical system has dense periodic points if periodic points are

dense in , now let where is a non-empty open set, then 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’.

A dense orbit implies topological

transitivity because such an orbit will always visit any open non-empty set.

Identity

map is the perfect example of dynamical system which has dense periodic points.

Let 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.

Doubling Map:

The

following 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.

The

Doubling map is chaotic, because it has sensitive dependence on initial

conditions, has topological transitivity and has dense periodic points.

Proof:

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 to be large enough such that

Hence, doubling map has sensitive dependence

on initial conditions.

Dense orbit:

A

Another common feature that chaotic

maps often have is that they have these invariant distributions which shows no

matter how it starts, what distribution initial conditions it starts with,

after some time there will be no memory of initial conditions and there would

be just a flat (uniform) distribution with no information of past.

1)

Uniform distribution

initial condition

1.a)

Uniform distribution after first iteration

1.d)

distribution after tenth distribution

1.c)

distribution after third iteration

As

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.

1)

?-distribution initial

conditions

2.a)

distribution after first iteration

2.b)

distribution after second iteration

2.c)

distribution after third iteration

2.d)

distribution after tenth iteration

Surprisingly,

now when initial conditions are ?-distribution, doubling map takes it to

uniform distribution which is quite hard to think about or predict about. So,

its quite clear from here that chaotic systems are unpredictable. And its hard

to say anything about the initial conditions, and seems like all information

about past has been lost and cannot be retraced.

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.

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