Chaos and Fractals: An Elementary IntroductionThis book provides the reader with an elementary introduction to chaos and fractals, suitable for students with a background in elementary algebra, without assuming prior coursework in calculus or physics. It introduces the key phenomena of chaos - aperiodicity, sensitive dependence on initial conditions, bifurcations - via simple iterated functions. Fractals are introduced as self-similar geometric objects and analyzed with the self-similarity and box-counting dimensions. After a brief discussion of power laws, subsequent chapters explore Julia Sets and the Mandelbrot Set. The last part of the book examines two-dimensional dynamical systems, strange attractors, cellular automata, and chaotic differential equations. The book is richly illustrated and includes over 200 end-of-chapter exercises. A flexible format and a clear and succinct writing style make it a good choice for introductory courses in chaos and fractals. |
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algebra approximate attracting fixed point average winnings bifurcation diagram box-counting dimension calculator Cantor set chaos and fractals chaos game chapter complex numbers Consider the function cycle dependence on initial determine deterministic differential equation distribution dynamical systems Euler’s method example Exercises exponential figure final-state diagram function f(x graph graphical iteration growth rate Hénon map histogram illustrated in Fig infinite number initial condition x0 initial conditions input iterated functions itinerary Julia set Koch curve line segment logistic equation long-term behavior look Lorenz equations Mandelbrot set mathematical move one-dimensional orbit output parameter period period-doubling Petersburg game phase line phase space power law prediction rabbit population random Koch curve result rule SDIC self-similar sensitive dependence sequence series plot shape shown in Fig Sierpinski triangle smaller solutions step strange attractor Suppose symbol temperature tend toward infinity tion trajectories two-dimensional variables zero zooming