Dynamics of the coquaternionic maps x2 + bx

Abstract

This paper deals with the dynamics of the one-parameter family of coquaternionic quadratic maps x 2 + bx. By making use of recent results for the zeros of one-sided coquaternionic polynomials, the fixed points are analytically determined. The stability of these fixed points is also addressed, where, in some cases, due to the appearance of sets of non-isolated points, a suitably adapted notion of stability is used. The results obtained show clearly that this family is not dynamically equivalent to the simpler family x 2 + c previously studied by the authors. Some numerical examples of other dynamics beyond fixed points are also presented.

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].…”

“…Since x(x − b) is a right factor of P (x), its zeros are always zeros of P (x); this means that the fixed points of the quadratic g b (x) = x 2 − (b + 1)x -which is a kind of family studied in detail by the authors in [7] -are always fixed points of f a,b . This will enable us to highlight the richness of the dynamics of cubic coquaternionic maps when compared to the dynamics of quadratic coquaternionic maps.…”

Product of Hessians and Discriminant of Critical Points of Level Function Attached to Sphere Arrangement

“…In [14], there is an interesting argument on zero points of coquaternionic polynomials using characteristic polynomials, which enable to linearize the problem by Euclidean algorithm. Moreover, our argument goes along the similar line in a more complicated situation.…”

“…Under (H 0 ), (H 1 ) and ρ 12 = ρ 14 Throughout this section, in addition to the imposed conditions (H 0 ) and (H 1 ), we suppose further…”