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Transcript of Chaos Theory
This theory is mainly about and unpredictability. Usually science and math are linear and predictable. However the chaos theory defines things in a nonlinear universe, so things are never definite or predictable. Dynamical systems describes a system that has an abstract set of states that correspond to the system at any given point. They also include a dynamical rule that specifies the time when a variable takes place. A dynamical system can be predictable if there are different state consequences and if you know the theoretical probability to each state. Bibliography/Sources
http://serendip.brynmawr.edu/playground/sierpinski.html Dynamical systems Linear vs. Nonlinear The chaos theory describes an unpredictability in dynamical systems, especially when the system is sensitive to initial conditions. "If a butterfly flapped its wings in Brazil, a tornado could occur in Texas." This means a tiny difference during the initial conditions in a nonlinear system will create a huge difference. The changes don't just happen at once, but rather build up on one another and increase exponentially. Later the consequences will be too far off from the expected pattern or estimate for anyone to have guessed the possible outcomes in the first place. Linear means that things are straight, clear, predictable. Most of the math and science we know are linear, everything is straightforward and there is always an answer.
In the nonlinear world, systems aren't predictable. They don't follow a definite pattern, so results can change easily with variables. “As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.”
-Albert Einstein What is chaos theory? The butterfly effect Butterfly effect In examples that include nature, there are so many possible consequences. Each consequence will have so many more different paths, the possibilities at the beginning are almost endless. Other factors like time make it even more unpredictable. Principles/important words There are several important principles to the Chaos theory.
The butterfly effect
Order vs. Chaos History - 1800s How did the chaos theory originate?
1860s - James Clark Maxwell
1880s - Henri Poincaré
1898 - Jacques Hadamard History - 1900s 1930s - Over the years, more scientists researched on this new theory.
1961 - Edward Lorenz
1970 - Robert May
1971 - David Ruelle and Floris Takens
1975 - Benoit Mandelbrot History - Butterfly effect Edward Lorenz was running computations for forecasting weather. He reran rounded numbers. The results he got were way different from the original results. Such a small difference could create a huge difference, and over time, the results became unpredictable. Conclusion Even after all the research I did, there are still so many things about the Chaos theory that even the smartest scientists can't figure out, and way more for any of us to try and understand. Unpredictability Unpredictability is simple. Since we do not know the details of initial conditions in a complex system, we can't predict the outcome. A small error in the initial condition will cause a big difference in the later stages. Order vs. Chaos Most people will think that chaos is the opposite of order, so order is good whereas chaos is bad. Wrong. Chaos is not disorder, but the transition between order and disorder. Order has a bit of chaos in it and so does disorder. Stable systems can go unstable without chaos and unstable systems can return to stability with it. Mixing Mixing means that with "turbulence", two adjacent points in a system will end up far away in time. Turbulence means anything that disturbs the atmosphere and therefore means variables. Feedback It is more likely for a system to become chaotic if there is feedback. Feedback means that something happens during a stage due to the earlier outcomes. The stages can be modified once feedback is present, and can completely change the outcome of the system. Fractals A fractal is a never ending pattern that repeats the same pattern in different sizes. It is an infinite loop repeating from feedback. Fractals are sort of the image of Chaos, because the mathematical equations in both are about the same. It is also considered a chaotic system. Fractals contd An interesting function on fractals is:
Fractals are a complex system just like many chaotic systems. The Lorenz attractor The Lorenz attractor in the Chaos theory is very important. These three equations are the equations for the attractor.
x, y and z = system state
t = time
o, T and B = system parameters.
These equations are related to "atmospheric conversions". (^Solution, T = 28, o = 10, B = 8/3) Chaos game Real life Chaos vs Determinism The chaos game is basically about fractals. You start with a polygon and start the first sequence/iteration inside it. Then you repeat the process on different scales to create a fractal. A famous example is the Sierpinski's triangle. When people believe in the chaos theory, they believe that everything in the future is completely unpredictable, that there is no way of knowing the future. When people believe in determinism, they don't necessarily believe that the future is predictable, but that every outcome leads up to another. People either believe in one or another. The chaos theory applies to the real world in many ways, mainly because it is a theory that applies to the whole world of math and science. It is almost its own form of science. nature (butterfly effect - weather, animal habits, natural disasters)
human body (body seizures)
chaotic systems in industry (mechanical, biological, electrical) Control? Is it possible to control nonlinear systems? First, we need to understand what makes nonlinear systems chaotic in the first place. We have tried with almost linear parts in a system. Therefore it would be easier to work with stages which are not sensitive to initial conditions and work from there. But it's hard to predict if something chaotic will happen. Instability Instability is when a complex system is unstable. When a little difference occurs, a huge impact is made on the overall system, and it will keep growing. The more complex a system is, the more sensitive it is to initial conditions and therefore more unstable.