Theory of chaos and butterfly effect
When
hearing about chaos, the first thing that comes to mind is Confusion, and
serious disorder , in mathematics we talk about the butterfly effect . the
theory of chaos belongs to the branch of studies of the dynamic system , “a
dynamic system is something that evolves over time” he is a system in which function describes the time dependence of a point in a geometrical space.
whether in
chemistry for example particles or the pondule in physics, more to domain
advantages that make it part of .
below will
discuss the case of Pendulum, to initiate and understand the concept of chaos
and its initial conditions.
Pendulum
A pendulum
is a body suspended from a fixed support so that it swings freely back and
forth under the influence of gravity. When a pendulum is displaced sideways
from its resting, equilibrium position, it is subject to a restoring force due
to gravity that will accelerate it back toward the equilibrium position. When
released, the restoring force acting on the pendulum's mass causes it to
oscillate about the equilibrium position, swinging back and forth. The
mathematics of pendulums are in general quite complicated. Simplifying
assumptions can be made, which in the case of a simple pendulum allow the
equations of motion to be solved analytically for small-angle oscillations.
we will
always have the same results, because, a cause causes the same effects, it is
clear at this stage, that a similar cause causes a similar effect, so we do not
have to fear consequences because they are limited, we are talking here about the
scientific principle.
Double pendulum equation
the
evolution of the trajectory of the double pendulum is not regular, it is
erratic , she is unpredictable , unlike the simple pendulum, its position is
dependent on the initial conditions (the exact value of the angles).And it is
this sensitivity to initial conditions that makes a double pendulum, chaotic
and therefore unpredictable system.so to calculate the evolution, it will be
necessary to calculate its( point of departure), precisely, because the
slightest variation will completely change the result (sensitive to the point
of departure),This
phenomenon of small variation at the beginning, having important consequences
in term, it is what one calls today, the effect butterfly,
To sum up ,let's
finish with this universal notion from Wikipedia,
A double
pendulum is a pendulum with another pendulum attached to its end, and is a
simple physical system that exhibits rich dynamic behavior with a strong
sensitivity to initial conditions. The motion of a double pendulum is
governed by a set of coupled ordinary differential equations and is chaotic.
As the
weather is a thing that evolves over time, it is a dynamic system that can not
be predicted and also complex at first glance contrary to eclipse for example, The
term, closely associated with the work of Edward Norton Lorenz, is derived from the
metaphorical example of the details of a tornado (the exact time of formation,
the exact path taken) being influenced by minor perturbations such as the
flapping of the wings of a distant butterfly several weeks earlier. Lorenz
discovered the effect when he observed that runs of his weather model with
initial condition data that was rounded in a seemingly inconsequential manner
would fail to reproduce the results of runs with the unrounded initial
condition data. A very small change in initial conditions had created a significantly
different outcome.
The idea
that small causes may have large effects in general and in weather specifically
was earlier recognized by French mathematician and engineer Henri Poincaré and
American mathematician and philosopher Norbert Wiener. Edward Lorenz's work
placed the concept of instability of the Earth's atmosphere onto a quantitative
base and linked the concept of instability to the properties of large classes
of dynamic systems which are undergoing nonlinear dynamics and deterministic
chaos.
-must be
sensitive to initial conditions,
-it must be
topologically transitive,
-it must
have dense periodic orbits.
The
computer used to do this simulation in 1960 was a LGP-30, 16 KB ram /0.00012
GHZ ,
” The
LGP-30, standing for Librascope General Purpose and then Librascope General
Precision, was an early off-the-shelf computer. It was manufactured by the
Librascope company of Glendale, California (a division of General Precision
Inc.), and sold and serviced by the Royal Precision Electronic Computer
Company, a joint venture with the Royal McBee division of the Royal Typewriter
Company. The LGP-30 was first manufactured in 1956, at a retail price of
$47,000, equivalent to $433,125 in 2018.
The LGP-30
was commonly referred to as a desk computer. Its height, width, and depth,
excluding the typewriter shelf, was 33 by 44 by 26 inches (84 by 112 by 66 cm).
It weighed about 800 pounds (360 kg), and was mounted on sturdy casters which
facilitated moving the unit”
A year
later, the famous problem launched by lorenz appeared ,you will find the pdf
version below :
Since the
smallest initial difference, can lead to a different evolution, so to have a
prediction just in advance (about 1 month) we should take into consideration
the smallest details, even the smallest, even the flutter of a butterfly as
shown in our example
In chaos
theory, the butterfly effect is the sensitive dependence on initial conditions
in which a small change in one state of a deterministic nonlinear system can
result in large differences in a later state.
The term,
closely associated with the work of Edward Lorenz, is derived from the
metaphorical example of the details of a tornado (the exact time of formation,
the exact path taken) being influenced by minor perturbations such as the
flapping of the wings of a distant butterfly several weeks earlier. Lorenz
discovered the effect when he observed that runs of his weather model with
initial condition data that was rounded in a seemingly inconsequential manner
would fail to reproduce the results of runs with the unrounded initial
condition data. A very small change in initial conditions had created a
significantly different outcome.
The idea
that small causes may have large effects in general and in weather specifically
was earlier recognized by French mathematician and engineer Henri Poincaré and
American mathematician and philosopher Norbert Wiener. Edward Lorenz's work
placed the concept of instability of the Earth's atmosphere onto a quantitative
base and linked the concept of instability to the properties of large classes
of dynamic systems which are undergoing nonlinear dynamics and deterministic
chaos.
Logistic map
If we
change the variable r, we always have an attractor, this is called a periodic
orbit, a periodic oscillation, the point where the diagram changes behavior is
called a bifurcation, the bifurcation diagram has a so-called fractal
structure, that is, it repeats itself each time we zoom in. with a different
trajectory, all accumulate in the same figure this figure acts as a butterfly's
wing.
Edward Norton Lorenz attractor
“In the
mathematical field of dynamical systems, an attractor is a set of numerical
values toward which a system tends to evolve, for a wide variety of starting
conditions of the system. System values that get close enough to the attractor
values remain close even if slightly disturbed.
In
finite-dimensional systems, the evolving variable may be represented
algebraically as an n-dimensional vector. The attractor is a region in
n-dimensional space. In physical systems, the n dimensions may be, for example,
two or three positional coordinates for each of one or more physical entities;
in economic systems, they may be separate variables such as the inflation rate
and the unemployment rate.
If the
evolving variable is two- or three-dimensional, the attractor of the dynamic
process can be represented geometrically in two or three dimensions, (as for
example in the three-dimensional case depicted to the right). An attractor can
be a point, a finite set of points, a curve, a manifold, or even a complicated
set with a fractal structure known as a strange attractor (see strange
attractor below). If the variable is a scalar, the attractor is a subset of the
real number line. Describing the attractors of chaotic dynamical systems has
been one of the achievements of chaos theory.
A
trajectory of the dynamical system in the attractor does not have to satisfy
any special constraints except for remaining on the attractor, forward in time.
The trajectory may be periodic or chaotic. If a set of points is periodic or
chaotic, but the flow in the neighborhood is away from the set, the set is not
an attractor, but instead is called a repeller.
On the
other hand, we know a system which is behavior resembles that of lorenz, it is
the magnitic field due to the movement of the liquid metal in the external
nucleus, in physics which is a little analogous to the atmosphere, all this was
officially approved mathematically in 2002 two years later in 2004 someone
could observe the structure.
What we
should remember ! if he had not beaten
the wings the tornado would have happened anyway but simply at a time, this is
already what lorenz had anticipated in his famous conference.
The chaotic
systems are unpredictable, but if we look at things statistically in the long
run, we find that they are relatively predictable, and incensible to the
conditions of departure, chaotic developments are everywhere in astronomy ...
etc, that's what Jules Henri Poincaré had already noted at the end of the 19th
century, so it is he the first to project into the chaos and the butterfly
effect, as soon as there are three bodies at most in gravitational interaction,
we have a chaotic system, it is the solar system's cs, the orbit of the planet
deveins totally unpredictable and that we should know their initial positions
at some meters In order to be able to do this, the uncertainty is particularly
large for the inner planets: mercury, earth, venus, mars.
As soon as
we have equations of nonlinear evolution, we multiply one varible by another,
we can have a chaotic system, for example in the mechanics of fluids it is he
who causes turbulence.It is found in electrical circuits, chaos heart rate,
chemical rections ... etc.
The idea of
the chaotic system, no matter how powerful our computers are, there is a
fundamental obstacle to not being able to predict this kind of devolution, the
theory of chaos and one of the great scientific advancements of the 20th
century. sciecle, which promises a journey to the infinite
these
two articles will give you the collosolausal impact of this phenomenon on the
actutuality and in the world in which we live.
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