AN ANALYTICAL APPROACH TO BIFURCATIONS IN SOME TWO-DIMENSIONAL CUBIC MAPPINGS

Abstract

This paper investigates the dynamics of two-dimensional mixed-type mappings that combine quadratic and cubic nonlinearities. Extending classical results on purely quadratic or cubic systems, we study the geometric and topological properties of the corresponding Mandelbrot and Julia sets. An analytical framework is developed to examine the stability of fixed points, bifurcations, and the structure of the parameter space. Particular emphasis is placed on the interplay between the algebraic form of the mapping and the shape of the filled Julia set. Our results include criteria for the existence of attractors, the identification of bifurcation curves, and the classification of parameter regions based on dynamical behavior.