Nonlinear-Systems

# Nonlinear-Systems - Nonlinear Systems and chaos a brief...

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Nonlinear Systems (. . . and chaos) a brief introduction Tom Carter Computer Science CSU Stanislaus http://csustan.csustan.edu/˜ tom/Lecture-Notes/Nonlinear-Systems/Nonlinear-Systems.pdf November 7, 2011 1

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Our general topics: What are nonlinear systems? 5 A Linear Example (string) 7 Systems of Differential Equations . . . . . . . . . . . . 17 Simple Harmonic Oscillator . . . . . . . . . . . . . . . 21 A Linear Approximation . . . . . . . . . . . . . . . . . . 26 Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . 34 Discrete Linear Systems 37 A Discrete Nonlinear System 41 The Logistics Equation (derivation) . . . . . . . . . . 42 The Logistics Equation (analysis) . . . . . . . . . . . . 47 Cobweb Diagrams . . . . . . . . . . . . . . . . . . . . . 57 Bifurcation Diagrams . . . . . . . . . . . . . . . . . . . 94 Universality . . . . . . . . . . . . . . . . . . . . . . . . . 113 2
The Xarkovski¨i Theorem 119 A Hint of Topics (likely :-) to Come: 122 The Cantor Set . . . . . . . . . . . . . . . . . . . . . . 123 Definition of chaos . . . . . . . . . . . . . . . . . . . . 127 A Chaotic Map . . . . . . . . . . . . . . . . . . . . . . 137 Some other chaotic maps . . . . . . . . . . . . . . . . 141 Julia Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 147 The Mandelbrot Set . . . . . . . . . . . . . . . . . . . . 168 Fixed points, limit sets, stable sets, attractors . . . . . 172 Basins of attraction . . . . . . . . . . . . . . . . . . . . 173 Statistical measures . . . . . . . . . . . . . . . . . . . . 174 Fractal dimensions, related measures . . . . . . . . . . 181 3

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Strange attractors . . . . . . . . . . . . . . . . . . . . 191 About fractals . . . . . . . . . . . . . . . . . . . . . . . 191 Continuous systems and flows . . . . . . . . . . . . . . 191 Some history (e.g., Poincaré) . . . . . . . . . . . . . . 192 Some other history (e.g., catastrophe theory) . . . . . 193 Poincaré sections . . . . . . . . . . . . . . . . . . . . . 193 Entropy and information theory . . . . . . . . . . . . . 193 Diagnostics and control of chaotic systems . . . . . . 194 Other discrete systems – e.g.: . . . . . . . . . . . . . . 194 Cellular automata . . . . . . . . . . . . . . . . . . . . 194 Iterated function systems . . . . . . . . . . . . . . . 195 Complex adaptive systems . . . . . . . . . . . . . . . . 195 and complex systems more generally . . . . . . . . . 195 Various other topics . . . . . . . . . . . . . . . . . . . . 196 Appendix 1 197 Appendix 2 200 References 201 4
What are nonlinear systems? Let’s start by adding another word to this, and ask the question, "What are nonlinear dynamical systems?" (and we’ll go back to front . . . ) A reasonable thing to say is that a system is a collection of entities that we can treat (for some purpose, in some context) as a unity of interacting parts or elements. At various times we will treat various collections of entities as systems, or subsystems. We may at times ignore certain elements that might otherwise be included. There will also often be times when we will engage in abstraction, and refer to a “system” when we are actually discussing, manipulating, or analyzing an abstraction from the real (physical) system we are interested in. 5

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A dynamical system is one which changes over time. It is generally not unreasonable to assume that there is some form of energy flow involved in such a system. A nonlinear dynamical system is, as the name implies, a system whose best description (behavior) is not linear . There are various contexts and forms of description for the concept linear . We’ll look at various examples as we go along. 6
A Linear Example (string) (a little string theory :-) It will be worth our while to have some very specific examples available, so let’s start with this one – consider a string stretched tightly between two fixed endpoints: We can pull the center of the string up: 7

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At the moment we release it, there will be forces acting on the string. Let’s focus our attention on the center point of the string (where we had taken hold of it to pull it up).
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