Chaos Theory

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<ul><li><p>WHAT IS CHAOS THEORY? </p><p> 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. </p><p> 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. </p><p> The deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply chaos. </p></li><li><p> Edward Lorenz. Deterministic Nonperiodic Flow, 1963. </p><p> 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. </p><p> 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. </p></li><li><p>CHAOS THEORY </p><p>Nonlinearity </p><p>Determinism </p><p>Sensitivity to initial conditions </p><p>Order in disorder </p><p>Long-term prediction is mostly impossible </p></li><li><p>CHAOTIC SYSTEMS </p><p>Dynamic systems Deterministic systems </p><p>Chaotic systems are unstable since they tend not to resist any outside disturbances but instead react in significant ways. </p></li><li><p> 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. </p><p> 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. </p><p>Chaotic systems are unstable since they tend not to resist any outside disturbances but instead react in significant ways. </p></li><li><p> Chaotic systems are common in nature. They can be found, for example, in Chemistry, in Nonlinear Optics (lasers), in Electronics, in Fluid Dynamics, etc. </p><p> 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. </p></li><li><p>ATTRACTORS </p><p> 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. </p><p> Attractor: Smallest unit which cannot itself be decomposed into two or more attractors with distinct basins of attraction. </p></li><li><p>TYPES OF ATTRACTORS </p><p>a) Point attractor: There is only one outcome for the system. Death is a point attractor for living things. </p><p>b) Limit cycle or periodic attractor: Instead of moving to a single state as in a point attractor, the system settles into a cycle. </p><p> c) Strange attractor or a chaotic attractor: double spiral which never repeats itself. Strange attractors are shapes with fractional dimension; they are fractals. </p><p>c) </p><p>b) a) </p></li><li><p>FRACTALS Fractals are objects that have fractional </p><p>dimension. A fractal is a mathematical object that is self-similar and chaotic. </p><p> 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. </p></li><li><p> Benoit Mandelbrot </p><p> Characteristics: Self-similarity and fractional dimensions. </p><p> 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 </p><p> Clouds, arteries, veins, nerves, parotid gland ducts, the bronchial tree, etc </p><p> Fractal geometry is the geometry that describes the chaotic systems we find in nature. Fractals are a language, a way to describe this geometry. </p></li><li><p> THE BUTTERFLY EFFECT </p><p> "Sensitive dependence on initial conditions. </p><p> 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. </p><p> 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. </p></li><li><p>ASP</p><p>ECTS</p><p> OF </p><p>CH</p><p>AO</p><p>S PREDICTABILITY Computations and </p><p>mathematical equations </p><p>CONTROL </p><p> Ott-Grebogi-Yorke Method </p><p> Pyragas Method </p></li><li><p> WAYS TO CONTROL CHAOS </p><p>The applications of controlling chaos are enormous, ranging from the control of turbulent flows, to the parallel signal transmission and computation to the </p><p>control of cardiac fibrillation, and so forth. </p><p>Alter organizational parameters so that </p><p>the range of fluctuations is limited </p><p>Apply small perturbations to the chaotic system to try </p><p>and cause it to organize </p><p>Change the relationship between the organization and </p><p>the environment </p></li><li><p>APPLICATIONS OF CHAOS THEORY </p><p>Stock market </p><p>Population dynamics </p><p>Biology </p><p>Predicting heart </p><p>attacks Real time </p><p>applications </p><p>Music and Arts </p><p>Climbing </p><p>Random Number </p><p>Generation </p></li><li><p>CHAOS THEORY IN NEGOTIATIONS </p><p>Richard Halpern, 2008. Impact of Chaos Theory and Heisenberg Uncertainty Principle on case negotiations in law </p><p> Never rely on someone else's measurement to formulate a key component of strategy. A small mistake can cause </p><p>huge repercussions, better do it yourself. </p><p>Keep trying something new, unexpected; sweep the defence of its feet. Make the system chaotic. </p><p> If the process is going the way you wanted, simplify it as much as possible. Predictability would increase and </p><p>chance of blunders is minimized. </p><p> 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 </p><p>case into a winner. </p></li><li><p>CONCLUSIONS </p><p> Everything in the universe is under control of Chaos or product of Chaos. </p><p> Irregularity leads to complex systems. </p><p> 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. </p><p> It gives us a new concept of measurements and scales. It offers a fresh way to proceed with observational data. </p></li></ul>