Basins of attraction for a quadratic coquaternionic map

Abstract

In this paper we consider the extension, to the algebra of coquaternions, of a complex quadratic map with a real super-attractive 8-cycle. We establish that, in addition to the real cycle, this new map has sets of non-isolated periodic points of period 8, forming four attractive 8cycles. Here, the cycles are to be interpreted as cycles of sets and an appropriate notion of attractivity is used. Some characteristics of the basins of attraction of the five attracting 8-cycles are discussed and plots revealing the intertwined nature of these basins are shown.

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].…”

Remarks on the Zeros of Quadratic Coquaternionic Polynomials

“…These results motivated the authors of this paper to start studying the dynamics of the quadratic map in the coquaternionic setting [5,6,7] leading to preliminary results which can be considered as even richer than the ones obtained in the well known complex case. A complete understanding of the dynamics requires a deep knowledge of the quadratic equation: number of zeros; coexistence of zeros of different nature, etc.…”

Dynamics of the coquaternionic maps x2 + bx

“…For this parameter value, the lines L = q 0 + αi + (q 0 + 1 2 )j + αk : α ∈ R and L ′ = q ′ 0 + αi + (q 0 + 1 2 )j + αk : α ∈ R , where q 0 = 1 40 (−53 − √ 129) and q ′ 0 = 1 40 (−53 + √ 129), are attractive sets of periodic points of period two such that f b (L ) = L ′ , i.e. the sets L and L ′ form what we call a 2-set cycle [14].…”