MSc Defence: Analysis of Discrete-Time Planar Systems Using Julia Tool (Hewage S.I. Wijerathna)

CANDIDATE: Hewage S.I. Wijerathna

ABSTRACT: We introduce a tool, ‘DiscretePhasePortrait.jl’, written in Julia, that generates augmented phase portraits for the analysis of two-dimensional discrete mappings, to generalize the results of the paper by Sabrina H. Streipert and Gail S.K. Wolkowicz, “An augmented phase plane approach for discrete planar maps: Introducing next-iterate operators”, Mathematical Biosciences 355 (2023) 108924. The purpose of our research is to augment their phase portrait with additional information to allow for the determination of the global dynamics of planar discrete systems with complex behaviors, where their approach yielded inconclusive results. The first generalization considers isoclines for any given set of directions rather than simply those associated with the coordinate directions. We show that choosing directions aligned with eigenvectors of the Jacobian at a fixed point aids in making conclusions about global dynamics. The second generalization is to add a curve indicating where the determinant of the Jacobian is zero and the image of that curve. This helps identify the range of the mapping, effectively reducing the space one needs to consider when determining global dynamics. This tool also allows the user to generate phase portraits for multiple iterations of the map, thus, a phase portrait for a twice-iterated map can be used to eliminate the complicating oscillatory behaviour of orbits near fixed points that have negative eigenvalues for their Jacobian. We describe the use of each augmentation throughout the manuscript, applying it to several different systems, showing how it can be used to establish the global stability fixed points of interest. We also use the tool and some additional analysis to provide new solutions to four previously open problems about the local stability of some bifurcating fixed points for certain maps. We further discuss the potential limitations of the augmented phase portrait that arise when determining the global stability through illustrative examples.