Discrete Dynamical Systems

with an Introduction to Discrete Optimization Problems
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254 pages
Language:
 English
This book covers topics like stability, hyperbolicity, bifurcation theory and chaos, which are essential in order to understand the fascinating behavior of nonlinear discrete dynamical systems.
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About the author

Dr Arild Wikan is an associate professor at Harstad University College, Harstad, Norway. He also holds a position at the University of Tromsø. Norway. His research interests are mainly in discrete dynamical systems, in particular problems within population dynamics. Wikan has published several scient...

Description
Content

This book covers important topics like stability, hyperbolicity, bifurcation theory and chaos, topics which are essential in order to understand the fascinating behavior of nonlinear discrete dynamical systems. The theory is illuminated by several examples and exercises, many of them taken from population dynamical studies. Solution methods of linear systems as well as solution methods of discrete optimization (control) problems are also included. In an Appendix it is explained how to estimate parameters in nonlinear discrete models.

  1. Part 1 One-dimensional maps
    1. Preliminaries and definitions
    2. One-parameter family of maps
    3. Fixed points and periodic points of the quadratic map
    4. Stability
    5. Bifurcations
    6. The flip bifurcation sequence
    7. Period 3 implies chaos. Sarkovskii’s theorem
    8. The Schwarzian derivative
    9. Symbolic dynamics I
    10. Symbolic dynamics II
    11. Chaos
    12. Superstable orbits and a summary of the dynamics of the quadratic map
  2. Part II n-dimensional maps
    1. Higher order difference equations
    2. Systems of linear difference equations. Linear maps from Rn to Rn
    3. The Leslie matrix
    4. Fixed points and stability of nonlinear systems
    5. The Hopf bifurcation
    6. Symbolic dynamics III (The Horseshoe map)
    7. The center manifold theorem
    8. Beyond the Hopf bifurcation, possible routes to chaos
    9. Difference-Delay equations
  3. Part III Discrete Time Optimization Problems
    1. The fundamental equation of discrete dynamic programming
    2. The maximum principle (Discrete version)
    3. Infinite horizon problems
    4. Discrete stochastic optimization problems
  4. Appendix (Parameter Estimation)
  5. References