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Degree, quaternions and periodic solutions.

Jean Mawhin1

  • 1Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, Louvain-la-Neuve, Belgium.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|January 4, 2021
PubMed
Summary

This study calculates the Brouwer degree for homogeneous polynomials on quaternions. These findings aid in determining the existence and number of periodic solutions for quaternionic differential equations.

Keywords:
Brouwer degreecoincidence degreeperiodic solutionsquaternionic equations

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Area of Science:

  • Mathematics
  • Differential Equations
  • Topology

Background:

  • Homogeneous polynomials on quaternions are foundational in advanced mathematical analysis.
  • Understanding periodic solutions in differential equations is crucial for modeling dynamic systems.
  • Topological degree theory provides powerful tools for analyzing solutions to equations.

Purpose of the Study:

  • To compute the Brouwer degree for specific classes of homogeneous polynomials defined on quaternions.
  • To apply these computed degrees, alongside coincidence degree theory, to analyze quaternionic differential equations.
  • To establish the existence and multiplicity of periodic solutions for these equations.

Main Methods:

  • Computation of the Brouwer degree for homogeneous polynomials over quaternions.
  • Application of a continuation theorem from coincidence degree theory.
  • Analysis of systems of quaternionic valued ordinary differential equations.

Main Results:

  • The Brouwer degree for selected homogeneous polynomials on quaternions was successfully computed.
  • The study demonstrated the utility of these results in conjunction with coincidence degree theory.
  • Conditions for the existence and multiplicity of periodic solutions were established.

Conclusions:

  • The Brouwer degree computation provides a novel approach to solving quaternionic differential equations.
  • The findings contribute to the broader understanding of topological degree and fixed point theories in differential equations.
  • This work offers a framework for analyzing the periodicity of solutions in complex mathematical systems.