The stability of complex dynamics for two families of coquaternionic quadratic polynomials

Abstract

In this work, we begin by demonstrating that attractors, both periodic and aperiodic, of the one-parameter family of complex quadratic maps x2+ c, where c is a complex number, maintain their stability when we transition from the complex plane C to the coquaternions Hcoq as the map’s phase space. Next, we investigate the same question for a different family of quadratic maps, x2+ bx, and find that this is not the case. In fact, the situation for this family of maps turns out to be quite complicated. Our results show that there are complex attractors that undergo changes in their stability, while others maintain it. However, the most intriguing result is that certain regions of the parameter space, known as bulbs, which correspond to the existence of attracting cycles of some fixed period n, exhibit a mixture of stability behavior when we consider coquaternionic quadratics.