Contents Another feature of chaos Conclusion (Title page

 

Contents

Introduction

Definition of chaos

Concepts of chaos

Proof that the doubling map is chaotic

Another feature of chaos

Conclusion

 

(Title page not complete)

 

 

 

 

 

 

 

 

 

 

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.