Bifurcation Formulas and Algorithms of Constructing Central Manifolds of Discrete Dynamical Systems

Abstract

Ones of the main questions in theory of local bifurcations and its applications are questions about direction of bifurcations (sub- or supercriticality) and on stability of the solutions arising in neighborhood of a nonhyperbolic equilibrium point or cycle dynamic system. We consider problems of local bifurcations in dynamical systems with discrete time. New features are proposed to orientation of bifurcations and properties stability of bifurcation solutions for problems on basic scenarios of bifurcations. We also propose new algorithms for constructing central manifolds of the corresponding problems, allowing to obtain new bifurcation formulas, in particular, formulas to calculate Lyapunov quantities. Proposed algorithms and formulas are based on the common operator method the study of problems on local bifurcations and allow under the new conditions effective qualitative analysis of bifurcations in terms of the initial equations.