Bifurcation diagrams of discrete dynamical systems

Abstract

The goal of this thesis is the study and analysis of bifurcation diagrams of discrete dynamical systems. Bifurcation theory focuses on the changes in properties of dynamical systems under varying system parameters, and it is possible to use the techniques from this field of study in various engineering areas. Focus will be placed on bifurcation diagrams of generic one and two dimensional spaces, namely the interval, the circle and the plane. Needed theory will be provided and supported with relevant examples and theorems. Investigated systems (Logistic map, Tent map, Hénon map, Avishai-Berend map) are introduced and explained. Simulation results performed in Java are presented, along with relevant algorithm scripts published on GitHub. Several bifurcation types are also mentioned -- transcritical, saddle node, pitchfork.