Ordinary differential equations over the quaternions.

• Grant number: 25/27872-5
• Support Opportunities: Scholarships in Brazil - Master
• Start date: March 01, 2026
• End date: February 29, 2028
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Paulo Ricardo da Silva
• Grantee: Paulo Hissao Simoce Araujo
• Host Institution: Instituto de Biociências, Letras e Ciências Exatas (IBILCE). Universidade Estadual Paulista (UNESP). Campus de São José do Rio Preto. São José do Rio Preto , SP, Brazil
• Associated research grant: 24/15612-6 - Ergodic and qualitative theories of dynamical systems III, AP.TEM
• Associated scholarship(s): 26/11984-1 - Geometric Singular Perturbation Theory and Piecewise Smooth Systems, BE.EP.MS

Abstract

The set of quaternions consists of all numbers that can be written as the sum of a real part and three imaginary parts, each multiplied by specific imaginary units traditionally denoted by ``i'', ``j'', and ``k''. In the first part of this work, we study the dynamics of real four-dimensional polynomial differential equations whose terms are constructed from products involving the unknown quaternion, its conjugate, and a fixed quaternionic parameter, each raised to non-negative integer powers. The algebraic structure of the quaternions plays an essential role in theproofs and in the formulation of the equations. In the second part, we analyze piecewise-smooth systems in four dimensions, composed of two vector fields belonging to the aforementioned class. The study of piecewise-smooth dynamical systems is an area of significant relevance, with applications in physics, engineering, economics, and biology. Among the foundational texts in the field is the classical work of Filippov, which provides the basis for understanding dynamics with discontinuities.Subsequent developments have established a deep connection between piecewise-smooth systems and singular perturbation theory, especially through geometric desingularizationmethods such as blow-up. These advances culminated in a solid geometric framework for the regularization of discontinuous foliations. (AU)