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Simultaneity of centres in ℤ -equivariant systems
Jaume Giné1, Jaume Llibre2, Claudia Valls3
1Departament de Matemàtica, Inspires Research Centre, Universitat de Lleida, Avda. Jaume II, 69; 25001 Lleida, Catalonia, Spain.
Proceedings. Mathematical, Physical, and Engineering Sciences
|June 12, 2018
Summary
This study investigates the simultaneous existence of centers for two families of planar Z2-equivariant systems. We establish necessary and sufficient conditions for Z2-equivariant cubic and quintic systems to have simultaneous centers.
Area of Science:
- Differential Equations
- Dynamical Systems Theory
- Symmetry in Mathematics
Background:
- Planar Z2-equivariant systems are a significant class of differential equations exhibiting symmetry.
- The study of centers in dynamical systems is crucial for understanding qualitative behavior and bifurcations.
- Previous research has explored conditions for centers in individual families of equivariant systems.
Purpose of the Study:
- To investigate the simultaneous existence of centers for two distinct families of planar Z2-equivariant systems.
- To provide a comprehensive review of Z2-equivariant systems.
- To derive the necessary and sufficient conditions for the simultaneous existence of centers in Z2-equivariant cubic and quintic systems.
Main Methods:
- Review of existing literature on Z2-equivariant systems.
- Analytical derivation of conditions for the existence of centers.
- Application of algebraic and geometric methods in the theory of differential equations.
Main Results:
- Established necessary and sufficient conditions for the simultaneous existence of centers in Z2-equivariant cubic systems.
- Derived the necessary and sufficient conditions for the simultaneous existence of centers in Z2-equivariant quintic systems.
- Presented a unified framework for analyzing simultaneous centers in these systems.
Conclusions:
- The conditions derived provide a complete characterization for the simultaneous existence of centers in the studied families of Z2-equivariant systems.
- This work contributes to a deeper understanding of the qualitative behavior of symmetric dynamical systems.
- The findings can be extended to analyze other types of equivariant systems and higher-order polynomial systems.
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