Iteration of Quadratic Maps on Coquaternions

Abstract

This paper is concerned with the study of the iteration of the quadratic coquaternionic map [Formula: see text], 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.

On the Roots of Coquaternions

“…Some geometric applications of coquaternions can be found in [16,18,20] and the relation between coquaternions and complexified mechanics is discussed in [3]. The use of coquaternions in dynamical systems was recently considered in [11,12].…”

Fixed Points for Cubic Coquaternionic Maps

“…In a previous study [4] the authors discussed the dynamics of the family of coquaternionic quadratic maps of the form x 2 + c and, more recently, they considered quadratic maps of the form x 2 + bx [7]. The present paper can be seen as continuation of the studies initiated in [4] and [7] and is a first step to deal with the -naturally much more interesting, but also much more demanding -problem of the dynamics of cubic coquaternionic maps.…”

“…In this section, we present a summary of the results on coquaternions and coquaternionic polynomials which are essential to the understanding of the rest of the paper. For more details, we refer the reader to [4,5,6,7].…”

Dynamics of the coquaternionic maps x2 + bx

“…The authors considered in a previous study [13] the family of coquaternionic quadratic maps of the simple form x 2 + c. The fixed points and periodic points of period two of this family of maps were determined and an interesting feature of the coquaternionic dynamics was observed: the appearance of sets of non-isolated such points. For this type of sets, the usual concept of stability has to be appropriately adapted.…”