The connection between the dynamical properties of 3D systems and the image of the energy-Casimir mapping
Mingxing Xu, Shaoyun Shi, Kaiyin Huang
Abstract
We investigate the connection between the dynamical properties of a class of 3D systems and the geometric characteristics of the image of the energy-Casimir mapping. By examining the energy-Casimir mapping for such systems, we can explore the stability of the equilibrium states, the distribution of the periodic solutions, and the existence of homoclinic or heteroclinic orbits. We apply our findings to investigate the dynamic behavior of two specific equations, and provide a topological classification of the fibers of the energy-Casimir mapping for the two systems.
Topics & Concepts
Casimir effectHomoclinic orbitConnection (principal bundle)Dynamical systems theoryPhysicsEnergy (signal processing)Classical mechanicsClass (philosophy)Stability (learning theory)Mathematical analysisTopology (electrical circuits)MathematicsComputer scienceGeometryNonlinear systemQuantum mechanicsBifurcationCombinatoricsMachine learningArtificial intelligenceAdvanced Differential Geometry ResearchAdvanced Differential Equations and Dynamical SystemsBlack Holes and Theoretical Physics