-->
Text to Search... About Author Email address... Submit Name Email Adress Message About Me page ##1## of ##2## Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec

404

Sorry, this page is not avalable
Home

Latest Articles

Theory of chaos and butterfly effect

0

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.



Aucun commentaire:

Publier un commentaire