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Unformatted text preview: Chaotic Dynamics and Bifurcations Tyler Campos May 2, 2011 Abstract: I decided to take interest in this study due to my new found curiousity for this genre of mathematics. I found myself fascinated with what the mathematical could explain, From the decimal system to the mathematics behind the need for probablity. In this report I explain real world examples, basic uses, explain orbits and their components, properties of chaotic systems, display visual examples of orbits and analysis of attracting and repeling points, basics behind bifurcations, visual examples of transitions to chaos, and symbolic dynamics. In each topic learned, I will explain theorems and give visual examples where needed. 1 1 Real World Applications and History Much of the mathematics that we use today can be studied using dynamics. Dynamical systems is a concept that takes is history from solutions to differential equations. Differential equations have been studied extensively since the 1600’s and can be seen in the earliest of Newton’s models. Newton derived an iteration method called “Newton’s Method” for finding solutions to F ( x ) = 0. N ( x ) = x F ( x ) F x ) ——–insert two approximation graphs for a zero value (9)———– This method is designed to take the previous input and move it successively closer to a finite solution of F ( x ) = 0. A real world example is compounding interest. It is one of the simplest examples of an dynamical system that is measured in iterates as time progresses. We can look at the following, Example 1.1 By using A 1 = A + rA , a 15 percent annual interest rate, and an initial deposit of 1000 dollars, we find the amount of money in the account after 3 years. We use A 1 = 1000 + 0 . 15(1000) to find the total after the first year followed by continuously plugging the new values into the equation. A 1 = 1000 + 0 . 15(1000) = 1150 A 2 = 1150 + 0 . 15(1150) = 1322 . 5 A 3 = 1322 . 5 + 0 . 15(1322 . 5) = 1520 . 875 Essentially, we have an equation A n = 1 . 15 A n 1 As we take the current value and input it into the equation to find the next value, we see it is just multiplying it by 1.15. Through repetition we can conclude that: A 1 = F ( A ) A 2 = F ( F ( A )) = F 2 ( A ) A 3 = F ( F ( F ( A ))) = F 3 ( A ) This pattern shown is considered a “period.” 2 2 Orbits and Periods Orbits are essentially sequences attained by repeating a specific function. The behavior of these sequences is the study of dynamical systems. For this we can look back at our compound interest equation. Each value after an iteration is a point o the orbit. We start with A and obtain A n from A n 1 by A n = F ( A n 1 ). For example, A = 1000 A 1 = 1150 A 2 = 1322 . 5 A 3 = 1520 . 875 In this case A n → ∞ as n → ∞ ....
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 Fall '06
 TWKnott
 Bifurcation theory, Iterated function, Logistic map

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