000			03654nmm a22002415i 4500
005			20230705150625.0
008			121227s1996 xxu| s |||| 0|eng d
020			_a9781441987327 _9978-1-4419-8732-7
082			_a515 _223
100			_aHolmgren, Richard A. _919806
245			_aA First Course in Discrete Dynamical Systems _h[electronic resource] / _cby Richard A. Holmgren.
250			_a2nd ed. 1996.
260			_aNew York, NY : _bSpringer New York : _bImprint: Springer, _c1996.
300			_aXV, 223 p. 4 illus. _bonline resource.
505			_a1.1. Phase Portraits -- Exercise Set 1 -- A Quick Look at Functions -- Exercise Set 2 -- The Topology of the Real Numbers -- Exercise Set 3 -- Periodic Points and Stable Sets -- 4.1 Graphical Analysis -- Sarkovskii's Theorem -- Exercise Set 5 -- Differentiability and Its Implications -- Exercise Set 6 -- Parametrized Families of Functions and Bifurcations -- Exercise Set 7 -- The Logistic Function Part I: Cantor Sets and Chaos -- 8.1. A First Look at the Logistic Function when r > 4 -- 8.2. Cantor Sets -- 8.3. Chaos and the Dynamics of the Logistic Function -- 8.4. A Few Additional Comments on Cantor Sets -- The Logistic Function Part II: Topological Conjugacy -- Exercise Set 9 -- The Logistic Function Part III: A Period-Doubling Cascade -- Exercise Set 10 -- The Logistic Function Part IV: Symbolic Dynamics -- 11.1. Symbolic Dynamics and Metric Spaces -- 11.2. Symbolic Dynamics and the Logistic Function -- Newton's Method -- 12.1 Newton's Method for Quadratic Functions -- 12.2 Newton's Method for Cubic Functions -- 12.3 Intervals and Rates of Convergence -- Numerical Solutions of Differential Equations -- Exercise Set 13 -- The Dynamics of Complex Functions -- 14.1. The Complex Numbers -- 14.2. Complex Functions -- 14.3. The Dynamics of Complex Functions -- 14.4. The Riemann Sphere -- 14.5. Newton's Method in the Complex Plane -- The Quadratic Family and the Mandelbrot Set -- 15.1 Generating Julia and Mandelbrot Sets on a Computer -- A.l. Iterating Functions -- Finding the Value of a Point Under Iteration -- Tables of Iterates -- Controlling the Precision of the Computations -- Graphing Iterated Functions -- A.2. Graphical Analysis -- A.3. Bifurcation Diagrams -- A.4. Julia Sets -- A.5 The Mandelbrot Set -- A.6 Stable Sets of Newton's Method -- References -- Dynamical Systems -- General Interest Books on Dynamics -- Topics in Mathematics -- Computer Programs and Algorithms.
520			_aDiscrete dynamical systems are essentially iterated functions. Given the ease with which computers can do iteration, it is now possible for anyone with access to a personal computer to generate beautiful images whose roots lie in discrete dynamical systems. Images of Mandelbrot and Julia sets abound in publications both mathematical and not. The mathematics behind the pictures are beautiful in their own right and are the subject of this text. The level of the presentation is suitable for advanced undergraduates with a year of calculus behind them. Students in the author's courses using this material have come from numerous disciplines; many have been majors in other disciplines who are taking mathematics courses out of general interest. Concepts from calculus are reviewed as necessary. Mathematica programs that illustrate the dynamics and that will aid the student in doing the exercises are included in an appendix.
650			_aMathematical analysis. _919807
650			_aManifolds (Mathematics). _919808
650			_aAnalysis. _919809
650			_aManifolds and Cell Complexes. _919810
856			_uhttps://doi.org/10.1007/978-1-4419-8732-7
942			_cEBK
999			_c13571 _d13571