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xivcontents12.8 Signals of Chaos: Lyapunov Coefficients ⊙ 29812.8.1 Shannon Entropy ⊙ 29912.9 Unit I Quiz 30012.10 Unit II. Pendulums Become Chaotic (Continuous) 30212.11 Chaotic Pendulum ODE 30212.11.1Free Pendulum Oscillations 30312.11.2Solution as Elliptic Integrals 30412.11.3Implementation and Test: Free Pendulum 30512.12 Visualization: Phase Space Orbits 30512.12.1Chaos in Phase Space 30712.12.2Assessment in Phase Space 31112.13 Exploration: Bifurcations of Chaotic Pendulums 31312.14 Alternative Problem: The Double Pendulum 31512.15 Assessment: Fourier/Wavelet Analysis of Chaos 31712.16 Exploration: Another Type of Phase Space Plot 31712.17 Further Explorations 31812.18 Unit III. Coupled Predator–Prey Models ⊙ 31912.19 Lotka–Volterra Model 32012.19.1LVM with Prey Limit 32112.19.2LVM with Predation Efficiency 32212.19.3LVM Implementation and Assessment 32312.19.4Two Predators, One Prey (Exploration) 32413 Fractals & Statistical Growth 32613.1 Fractional Dimension (Math) 32613.2 The Sierpiński Gasket (Problem 1) 32713.2.1 Sierpiński Implementation 32813.2.2 Assessing Fractal Dimension 32813.3 Beautiful Plants (Problem 2) 32913.3.1 Self-affine Connection (Theory) 33013.3.2 Barnsley’s Fern Implementation 33113.3.3 Self-affinity in Trees Implementation 33213.4 Ballistic Deposition (Problem 3) 33213.4.1 Random Deposition Algorithm 33213.5 Length of the British Coastline (Problem 4) 33413.5.1 Coastlines as Fractals (Model) 33413.5.2 Box Counting Algorithm 33513.5.3 Coastline Implementation and Exercise 336−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.34811_fm — 2008/2/13 — Page xiv