## Chaos: A Mathematical IntroductionWhen new ideas like chaos first move into the mathematical limelight, the early textbooks tend to be very difficult. The concepts are new and it takes time to find ways to present them in a form digestible to the average student. This process may take a generation, but eventually, what originally seemed far too advanced for all but the most mathematically sophisticated becomes accessible to a much wider readership. This book takes some major steps along that path of generational change. It presents ideas about chaos in discrete time dynamics in a form where they should be accessible to anyone who has taken a first course in undergraduate calculus. More remarkably, it manages to do so without discarding a commitment to mathematical substance and rigour. The book evolved from a very popular one-semester middle level undergraduate course over a period of several years and has therefore been well class-tested. |

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### Contents

MAKING PREDICTIONS | 1 |

11 MATHEMATICAL MODELS | 2 |

12 CONTINUOUS GROWTH MODELS | 7 |

13 DISCRETE GROWTH MODELS | 11 |

14 NUMERICAL SOLUTIONS | 17 |

15 DYNAMICAL SYSTEMS | 20 |

MAPPINGS AND ORBITS | 27 |

21 MAPPINGS | 28 |

81 DIVERGING ITERATES | 142 |

82 DEFINITION OF SENSITIVITY | 144 |

83 USING THE DEFINITION | 149 |

84 SENSITIVITY AND CHAOS | 155 |

INGREDIENTS OF CHAOS | 157 |

91 SENSITIVITY EVERYWHERE | 158 |

92 DENSE ORBITS AND TRANSITIVITY | 161 |

93 DENSE PERIODIC POINTS | 166 |

22 TIME SERIES | 33 |

23 ORBITS | 38 |

PERIODIC ORBITS | 45 |

31 FIXED POINTS AND PERIODIC POINTS | 46 |

32 FINDING FIXED POINTS | 50 |

33 EVENTUALLY PERIODIC ORBITS | 57 |

ASYMPTOTIC ORBITS I LINEAR AND AFFINE MAPPINGS | 61 |

41 COBWEB DIAGRAMS | 62 |

42 LINEAR MAPPINGS | 66 |

43 AFFINE MAPPINGS | 70 |

ASYMPTOTIC ORBITS II DIFFERENTIABLE MAPPINGS | 75 |

51 DIFFERENTIABLE MAPPINGS | 76 |

52 MAIN THEOREM AND PROOF | 83 |

53 PERIODIC POINTS | 86 |

FAMILIES OF MAPPINGS AND BIFURCATIONS | 91 |

61 THREE FAMILIES OF MAPPINGS | 92 |

62 FAMILIES OF FIXED POINTS | 98 |

63 FAMILIES OF PERIOD 2 POINTS | 106 |

64 THE BIFURCATION DIAGRAM | 113 |

GRAPHICAL COMPOSITION WIGGLY ITERATES AND ZEROS | 117 |

71 GRAPHING THE SECOND ITERATE | 118 |

72 ONEHUMP MAPPINGS | 125 |

73 DENSE SETS OF ZEROES | 132 |

74 ZEROES UNDER BACKWARD ITERATION | 135 |

SENSITIVE DEPENDENCE | 141 |

94 WHAT IS CHAOS? | 168 |

95 CONVERSE LEMMAS | 170 |

SCHWARZIAN DERIVATIVE AND WOGGLES | 179 |

101 WIGGLES AND WOGGLES | 180 |

102 THE SCHWARZIAN DERIVATIVE | 186 |

103 TESTING FOR CHAOS | 191 |

104 PROVING THEOREM 1031 | 193 |

CHANGING COORDINATES | 203 |

111 CHANGE OF VARIABLE | 204 |

112 COORDINATES | 207 |

113 MAPPINGS ON THE LINE | 211 |

CONJUGACY | 217 |

121 DEFINITION AND EXAMPLES | 218 |

122 APPROXIMATING A CONJUGACY | 225 |

123 EXISTENCE OF A CONJUGACY | 238 |

124 CONJUGACY AND DYNAMICS | 247 |

WIGGLY ITERATES CANTOR SETS AND CHAOS | 255 |

131 ITERATING ONEHUMP MAPPINGS | 256 |

132 CONSTRUCTING AN INVARIANT SET | 265 |

133 WIGGLY ITERATES AND CANTOR SETS | 272 |

134 DENSE IN A CANTOR SET | 278 |

135 CHAOS ON CANTOR SETS | 282 |

136 TESTS FOR CHAOS AND CONJUGACY | 284 |

287 | |

### Common terms and phrases

affine mapping bifurcation diagram Cantor set chaos chaotic behaviour cobweb diagram codomain conjugacy h conjugate to g converges coordinate system definition dense set Devaney difference equation differential equation domain Dynamical Systems eventually periodic Example Let Exercise f and g f has wiggly fixed point formula Fractals function gives graph of f Hartmut Jürgens Hence homeomorphism hump of f initial value Intermediate Value Theorem interval 0,1 interval I containing invariant set iterates of f Let f logistic mapping logistic mapping Q4 mapping f mapping h mappings with wiggly Matching Zeroes mathematical mathematical induction negative Schwarzian derivative nth iterate open interval orbit portrait period-2 periodic orbit periodic points plot PMJPSY point of f population prime period Proof prove Section sensitive dependence sensitivity constant sequence of iterates shown in Figure sketch the graph solution strictly increasing tangent tent mapping wiggly iterates woggle zeroes of f