Encounters with chaos and fractals PREFACE The far reaching interest in chaos and fractals are outgrowths of the computer age. On the one hand the notion of chaos is related to dynamics or behavior of physical systems. On the other hand fractals are related to geometry and appear as delightful but in nitely complex shapes on the line in the plane or in space. Encounters with Chaos and Fractals is designed to give an introduction both to chaotic dynamics and to fractal geometry. During the past fty years the topics of chaotic dynamics and fractal geometry have become increasingly popular. Applications have extended to disciplines as diverse an electric circuits weather prediction orbits of satellites chemical reactions analysis of cloud formations and complicated coast lines and the spread of disease. A fundamental reason for this popularity is the power of the computer with its ability to produce complex calculations and to create fascinating graphics. The computer has allowed scientists and mathematicians to solve problems in chaotic dynamics that hitherto seemed intractable and to analyze scienti c data that in earlier times appeared to be either random or awed. Fractals on the other hand are basically geometric but depend on many of the same mathematical properties that chaotic dynamics do. Mathematics lies at the foundation of chaotic dynamics and fractals. The very concepts that describe chaotic behavior and fractal graphs are mathematical in nature whether they be analytic geometric algebraic or probabilistic. Some of these concepts are elementary others are sophisticated. There are many books that discuss chaos and fractals in an expository manner as there are treatises on chaos theory and fractal geometry written at the graduate level The far reaching interest in chaos and fractals are outgrowths of the computer age. On the one hand the notion of chaos is related to dynamics or behavior of physical systems. On the other hand fractals are related to geometry and appear as delightful but in nitely complex shapes on the line in the plane or in space. Encounters with Chaos and Fractals is designed to give an introduction both to chaotic dynamics and to fractal geometry. During the past fty years the topics of chaotic dynamics and fractal geometry have become increasingly popular. Applications have extended to disciplines as diverse an electric circuits weather prediction orbits of satellites chemical reactions analysis of cloud formations and complicated coast lines and the spread of disease. A fundamental reason for this popularity is the power of the computer with its ability to produce complex calculations and to create fascinating graphics. The computer has allowed scientists and mathematicians to solve problems in chaotic dynamics that hitherto seemed intractable and to analyze scienti c data that in earlier times appeared to be either random or awed. Fractals on the other hand are basically geometric but depend on many of the same mathematical properties that chaotic dynamics do. Mathematics lies at the foundation of chaotic dynamics and fractals. The very concepts that describe chaotic behavior and fractal graphs are mathematical in nature whether they be analytic geometric algebraic or probabilistic. Some of these concepts are elementary others are sophisticated. There are many books that discuss chaos and fractals in an expository manner as there are treatises on chaos theory and fractal geometry written at the graduate level