Iteration of quadratic maps on coquaternions

This paper is concerned with the study of the iteration of the quadratic coquaternionic map fc(q) = q2 + c, where c is a fixed coquaternionic parameter. The fixed points and periodic points of period two are determined, revealing the existence of a type of sets of these points which do not occur in the classical complex case: sets of nonisolated points. This brings the need to consider a different concept of stability. The analysis of the stability, in this new sense, of the sets of fixed points and periodic points is performed and a discussion of certain type of bifurcations which occur, in the case of a real parameter c, is also presented.Research at CMAT was financed by Portuguese funds through FCT - Funda¸c˜ao para a Ciˆencia e a Tecnologia, within the Project UID/MAT/00013/2013. Research at NIPE has been carried out within the funding with COMPETE reference number POCI-01-0145-FEDER-006683, with the FCT/MEC’s (Funda¸c˜ao para a Ciˆencia e a Tecnologia, I.P.) financial support through national funding and by the ERDF through the Operational Programme on “Competitiveness and Internationalization – COMPETE 2020” under the PT2020 Partnership Agreement.info:eu-repo/semantics/publishedVersio

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

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.FCT -Fundação para a Ciência e a Tecnologia(UIDB/00013/2020

Remarks on the zeros of quadratic coquaternionic polynomials

In this paper we focus on the study of monic quadratic polynomials whose coefficients are coquaternions and present several new results concerning the number and nature of its zeros. Examples specially constructed to illustrate the diversity of cases that can occur are also presented.FCT - Fundação para a Ciência e a Tecnologia(UIDB/00013/2020, UIDP/00013/2020 and UIDB/03182/2020

Mathematica tools for coquaternions

Coquaternions form a four dimensional real algebra generalizing complex numbers and were introduced by James Cockle at about the same time that Hamilton discovered the famous algebra of quaternions. Although not as popular as quaternions, in recent years one can observe an emerging interest among mathematicians and physicists on the study of these numbers. In this work we revisit a Mathematica package for implementing the algebra of coquaternions - Coquaternions - and discuss a set of Mathematica functions - CoqPolynomi al - to deal with coquaternionic polynomials. These two sets of functions provide the basic tools necessary for manipulating coquaternions and unilateral coquaternionic polynomials, reflecting, in its present form, the recent interests of the authors in the area.Supported by FCT - Fundacao para a Ciencia e a Tecnologia, within the Projects UIDB/00013/2020, UIDP/00013/2020 and UIDB/03182/2020

Fixed points for cubic coquaternionic maps

This paper deals with the dynamics of a special two-parameter family of coquaternionic cubic maps. By making use of recent results for the zeros of one-sided coquaternionic polynomials, we analytically determine the fixed points of these maps. Some numerical examples illustrating the theory are also presented. The results obtained show an unexpected richness for the dynamics of cubic coquaternionic maps when compared to the already studied dynamics of quadratic maps.FCT - Fundação para a Ciência e a Tecnologia(UIDB/00013/2020, UIDP/00013/2020, UIDB/03182/2020

Symbolic computations over the algebra of coquaternions

Coquaternions, introduced by Sir James Cockle in 1849, form a four dimensional real algebra generalizing complex numbers. In recent years one can observe an emerging interest among mathematicians and physicists on the study of these numbers. In this work we present a Mathematica package for implementing the algebra of coquaternions. This package provides the basic mathematical tools necessary for manipulating coquaternions and coquaternionic polynomials, reflecting, in its present form, the recent interests of the authors in the area.This paper was financed by national funds of the FCT { Portuguese Foundation for Science and Technology { within the projects UID/ECO/03182/2019 and UID/MAT/00013/2013