Basins of attraction for a quadratic coquaternionic map

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 8-cycles. 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.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

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

Sistemas dinâmicos discretos em álgebras

Neste trabalho é feito o estudo de sistemas dinâmicos discretos em álgebras de matrizes. Este tema é explorado recorrendo a várias ferramentas da álgebra linear, com o objectivo de tirar partido da estrutura algébrica do espaço. É estudada a aplicação quadrática matricial, tomando uma matriz como parâmetro, aliando as propriedades algébricas à teoria das aplicações quadráticas escalares já existente, no caso real e complexo. São exploradas diversas características da dinâmica, tais como, a existência de ciclos comutativos e não-comutativos, a sua estabilidade, entre outras. São estudadas possíveis generalizações para o caso matricial das noções de conjunto de Mandelbrot e de conjunto de Julia. Os resultados atingidos são aplicados ao estudo da dinâmica da aplicação quadrática em diferentes álgebras hipercomplexas. É explorada a iteração quadrática no conjunto das matrizes estocásticas simétricas; as conclusões ilustram o comportamento do sistema dinâmico discreto definido no espaço das cadeias de Markov reversíveis; ABSTRACT: In this work we study discrete dynamical systems in matrix algebras. This subject is explored using different tools of linear algebra, in order to take advantage of the algebraic structure of the space. It is studied the iteration of a quadratic family in the algebra of real matrices, with a parameter matrix, combining the properties of the algebraic theory with the theory of the quadratic map in the real and complex cases. Several characteristics of the dynamics are explored, such as, the existence of commutative and non-commutative cycles, its stability, among others. Possible generalizations of the Mandelbrot set and Julia set are considered and studied. The results obtained are applied to the study of the quadratic dynamic in different hypercomplex algebras. Quadratic iteration is explored in the set of symmetric stochastic matrices; the findings illustrate the behavior of the discrete dynamical system on the space of reversible Markov chains

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition

Bifurcation analysis of the Topp model

In this paper, we study the 3-dimensional Topp model for the dynamicsof diabetes. We show that for suitable parameter values an equilibrium of this modelbifurcates through a Hopf-saddle-node bifurcation. Numerical analysis suggests thatnear this point Shilnikov homoclinic orbits exist. In addition, chaotic attractors arisethrough period doubling cascades of limit cycles.Keywords Dynamics of diabetes · Topp model · Reduced planar quartic Toppsystem · Singular point · Limit cycle · Hopf-saddle-node bifurcation · Perioddoubling bifurcation · Shilnikov homoclinic orbit · Chao

Chiral Random Matrix Theory: Generalizations and Applications

Kieburg M. Chiral Random Matrix Theory: Generalizations and Applications. Bielefeld: Fakultät für Physik; 2015