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Master's Thesis Defense

Asma Zangana, SLU. Thursday, May 6th, 2021, from 1:30 p.m. to 3:30 p.m. via Zoom. Write to This email address is being protected from spambots. You need JavaScript enabled to view it.  for zoom information.

Title:

Computation of Minimal and Formal Periodic Points for Dynamical Systems in Projective Space.

Abstract:

Computing minimal and formal periodic points of a given period in one-dimensional dynamics is a straightforward polynomial division. However, computing minimal and formal periodic points of a given period of a system of multivariable polynomial equations in higher-dimensional dynamics has not been done. Hence, the focus of this thesis is to compute minimal and formal periodic points for dynamical systems in projective space using saturation and deformation.

Computing minimal periodic points of a given period using saturation of ideals proves to be the effective method to compute all the minimal periodic points of a given period. However, the multiplicity information is lost under saturation due to the geometric nature of saturation. Computing formal periodic points for a given period using saturation of ideals alone proves to not produce all the formal periodic points for a given period due to the geometric nature of saturation. Thus, using deformation first by adding a parameter to the coordinates of the dynamical system and then applying saturation of ideals is the effective method to produce all the formal periodic points of a given period with the correct multiplicity. All computations have been done using Sage, and the Appendix provides the complete Sage code that has been used for each example in this thesis.