A first course in discrete dynamical systems
Discrete dynamical systems are essentially iterated functions. The book under review is suitable for a one-semester course on discrete dynamical systems. It contains an extensive quantity of interesting exercises. Let us specify its substance by chapters: the concept of the function; topology of the real numbers; periodical points and stable sets at the iteration of real functions, graphical analysis; Sharkovskij's theorem; dynamical information contained in the derivative of the function, attracting and repelling points; parametrized families of functions and bifurcations (the definition of the bifurcations of parametrized families of functions, bifurcation diagrams, saddle-node, pitchfork, transcritical and period doubling bifurcations); symbolic dynamics and chaos; the logistic function \(h_ r(x) = rx(1 - x)\), \(r > 0\); Newton's method from the point of view of discrete dynamics, the iterations for quadratic and cubic functions; numerical solution of differential equations, Euler's method; the dynamics of complex functions, dynamics of the quadratic maps, Newton's method in the complex plane; Mandelbrot and Julia sets. The Appendix is devoted to computer algorithms (iterating functions, graphical analysis, bifurcation diagrams, Julia and Mandelbrot sets, stable sets of Newton's method).
scientific article; zbMATH DE number 575713