Numerical Detection and Analysis of Strong Resonance Bifurcations with a Reflection Symmetry and Some Applications in Economics and Neural Networks

Abstract

This paper is concerned with the strong resonance bifurcations with a reflection symmetry i.e. [Formula: see text]-symmetry in maps. We compute the normal form of [Formula: see text] resonance and [Formula: see text] resonance bifurcations with [Formula: see text]-symmetry. We use standard normal form techniques in order to obtain the reduced map. Then, we will obtain explicit formulae for normal form coefficients of bifurcations with [Formula: see text]-symmetry. By using critical coefficients, we avoid the computation of the center manifold and the transformation of the linear part of the map into Jordan form. So this method can be used in the study of bifurcations with [Formula: see text]-symmetry in general problems. To illustrate our results, we will analyze local bifurcations of the strong resonance bifurcations with [Formula: see text]-symmetry numerically and then we will present some applications from economics and neural networks.