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Dynamics of the coquaternionic maps x2 + bx

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Abstract

This paper deals with the dynamics of the one-parameter family of coquaternionic quadratic maps \(x^2+\mathsf {b} x\). 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 \(x^2+\mathsf {c}\) previously studied by the authors. Some numerical examples of other dynamics beyond fixed points are also presented.

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Code availability

A Mathematica add-on application implementing the algebra of coquaternions, written by the authors, is available at the site http://w3.math.uminho.pt/ Coquaternions. Some Matlab programs used in our computations can be obtained upon request to the corresponding author.

Notes

  1. This is no longer true, however, for the case of periodic points of period two.

  2. This polynomial is more commonly referred to as the characteristic polynomial of the coquaternion \(\mathsf {q}\). Since this polynomial is an invariant of the class, we find it more convenient to adopt our denomination.

  3. Since the product of two polynomials in \({\mathbb {H}}_\mathrm{{coq}}\) is defined in the usual manner, we can use the “Euclidean Division Algorithm" to perform the division of two polynomials, provided that the leading coefficient of the divisor is non-singular, which is obviously the case here.

  4. Note that, in the case we are considering, we have four distinct eigenvalues and hence such a basis always exists.

  5. In this case, we do not have four linearly independent eigenvectors, but we can consider a basis of \(\mathbb {R}^4\) formed by generalized eigenvectors.

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Funding

Research at CMAT was partially financed by Portuguese funds through FCT - Fundação para a Ciência e a Tecnologia, within the Projects UIDB/00013/2020 and UIDP/00013/2020. Research at NIPE has been financed by National Funds of the FCT - Fundação para a Ciência e a Tecnologia, within the Project UIDB/03182/2020. The authors have no relevant financial or non-financial interests to disclose.

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Authors and Affiliations

  1. Center of Mathematics, University of Minho, Campus de Gualtar, 4710-057, Braga, Portugal

    Maria Irene Falcão & Fernando Miranda

  2. Department of Mathematics, University of Minho, Campus de Gualtar, 4710-057, Braga, Portugal

    Ricardo Severino

  3. NIPE and Department of Mathematics, University of Minho, Campus de Gualtar, 4710-057, Braga, Portugal

    Maria Joana Soares

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  1. Maria Irene Falcão
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  2. Fernando Miranda
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Correspondence to Maria Joana Soares.

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Falcão, M.I., Miranda, F., Severino, R. et al. Dynamics of the coquaternionic maps x2 + bx. Rend. Circ. Mat. Palermo, II. Ser 72, 959–975 (2023). https://doi.org/10.1007/s12215-021-00715-6

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