Chaos theory deals with nonlinear and complex systems that are highly sensitive to initial conditions. These systems, while deterministic, are largely unpredictable due to this sensitivity. Lorenz discovered this "butterfly effect" through modeling atmospheric convection. Chaotic systems evolve toward attractors, which can be fixed points, limit cycles, or strange attractors exhibiting fractal geometry. This geometry is seen throughout nature. While chaotic systems cannot be precisely predicted, control methods like Ott-Grebogi-Yorke can influence their behavior. Chaos theory has applications across many domains.

2. WHAT IS CHAOS THEORY?
• Branch of mathematics that deals with systems that appear to be orderly
but, in fact, harbor chaotic behaviors. It also deals with systems that appear
to be chaotic, but, in fact, have underlying order.
• Chaos theory is the study of nonlinear, dynamic systems that are highly
sensitive to initial conditions, an effect which is popularly referred to as the
butterfly effect.
• The deterministic nature of these systems does not make them predictable.
This behavior is known as deterministic chaos, or simply chaos.

3. • Edward Lorenz. “Deterministic Nonperiodic Flow”, 1963.
• Lorenz was a meteorologist who developed a mathematical model used to
model the way the air moves in the atmosphere. He discovered the
principle of Sensitive Dependence on Initial Conditions . “Butterfly Effect”.
• The basic principle is that even in an entirely deterministic system the
slightest change in the initial data can cause abrupt and seemingly random
changes in the outcome.

4. CHAOS THEORY
Nonlinearity
Determinism
Sensitivity to initial
conditions
Order in disorder
Long-term prediction
is mostly impossible

5. CHAOTIC SYSTEMS
Dynamic systems Deterministic systems
Chaotic systems are unstable since they tend
not to resist any outside disturbances but
instead react in significant ways.

6. • Dynamic system: Simplified model
for the time-varying behavior of an
actual system. These systems are
described using differential
equations specifying the rates of
change for each variable.
• Deterministic system: System in
which no randomness is involved in
the development of future states of
the system. This property implies
that two trajectories emerging
from two different close-by initial
conditions separate exponentially
in the course of time.
Chaotic systems are unstable since they tend not to resist any outside
disturbances but instead react in significant ways.

7. • Chaotic systems are common in
nature. They can be found, for
example, in Chemistry, in
Nonlinear Optics (lasers), in
Electronics, in Fluid Dynamics,
etc.
• Many natural phenomena can
also be characterized as being
chaotic. They can be found in
meteorology, solar system,
heart and brain of living
organisms and so on.

9. ATTRACTORS
• In chaos theory, systems
evolve towards states called
attractors. The evolution
towards a specific state is
governed by a set of initial
conditions. An attractor is
generated within the system
itself.
• Attractor: Smallest unit which
cannot itself be decomposed
into two or more attractors
with distinct basins of
attraction.

10. TYPES OF ATTRACTORS
a) Point attractor: There is only one outcome for the system. Death is a point
attractor for living things.
b) Limit cycle or periodic attractor: Instead of moving to a single state as in a
point attractor, the system settles into a cycle.
c) Strange attractor or a chaotic
attractor: double spiral which never
repeats itself. Strange attractors are
shapes with fractional dimension;
they are fractals.
c)
b)
a)

11. FRACTALS
• Fractals are objects that have fractional
dimension. A fractal is a mathematical
object that is self-similar and chaotic.
• Fractals are pictures that result from
iterations of nonlinear equations. Using
the output value for the next input value,
a set of points is produced. Graphing
these points produces images.

12. • Benoit Mandelbrot
• Characteristics: Self-similarity and fractional dimensions.
• Self-similarity means that at every level, the fractal image repeats itself.
Fractals are shapes or behaviors that have similar properties at all levels of
magnification
• Clouds, arteries, veins, nerves, parotid gland ducts, the bronchial tree, etc
• Fractal geometry is the geometry that describes the chaotic systems we find
in nature. Fractals are a language, a way to describe this geometry.

14. THE BUTTERFLY EFFECT
"Sensitive dependence on initial conditions.“
• Butterfly effect is a way of describing
how, unless all factors can be accounted
for, large systems remain impossible to
predict with total accuracy because there
are too many unknown variables to track.
• Ex: an avalanche. It can be provoked with
a small input (a loud noise, some burst of
wind), it's mostly unpredictable, and the
resulting energy is huge.

15. ASPECTSOFCHAOS
PREDICTABILITY Computations and
mathematical
equations
CONTROL
Ott-Grebogi-Yorke
Method
Pyragas Method

16. WAYS TO CONTROL CHAOS
The applications of controlling chaos are enormous, ranging from the control
of turbulent flows, to the parallel signal transmission and computation to the
control of cardiac fibrillation, and so forth.
Alter organizational
parameters so that
the range of
fluctuations is limited
Apply small
perturbations to the
chaotic system to try
and cause it to
organize
Change the
relationship between
the organization and
the environment

18. APPLICATIONS OF CHAOS THEORY
Stock
market
Population
dynamics
Biology
Predicting
heart
attacks
Real time
applications
Music and
Arts
Climbing
Random
Number
Generation

20. CHAOS THEORY IN NEGOTIATIONS
Richard Halpern, 2008. Impact of Chaos Theory and Heisenberg Uncertainty
Principle on case negotiations in law
Never rely on someone else's measurement to formulate
a key component of strategy. A small mistake can cause
huge repercussions, better do it yourself.
Keep trying something new, unexpected; sweep the
defence of its feet. Make the system chaotic.
If the process is going the way you wanted, simplify it
as much as possible. Predictability would increase and
chance of blunders is minimized.
If the tide is running against you, add new elements:
complicate. Nothing to lose, and with a little help from
Chaos, everything to gain. You might turn a hopeless
case into a winner.

21. CONCLUSIONS
• Everything in the universe is under control of Chaos or product of Chaos.
• Irregularity leads to complex systems.
• Chaotic systems are very sensitive to the initial conditions, This makes the
system fairly unpredictable. They never repeat but they always have some
order. That is the reason why chaos theory has been seen as potentially
“one of the three greatest triumphs of the 21st century.” In 1991, James
Marti speculated that ‘Chaos might be the new world order.’
• It gives us a new concept of measurements and scales. It offers a fresh way
to proceed with observational data.