Dynamics and integrability of polynomial vector fields on the n-dimensional sphere

2024/12/03 by Supriyo Jana, Soumen Sarkar, Jana, Supriyo +1
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #advanced mathematical theories #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2412.02190

Abstract

In this paper, we characterize arbitrary polynomial vector fields on Sⁿ. We establish a necessary and sufficient condition for a degree one vector field on the odd-dimensional sphere S^2n-1 to be Hamiltonian. Additionally, we classify polynomial vector fields on Sⁿ up to degree two that possess an invariant great (n-1)-sphere. We present a class of completely integrable vector fields on Sⁿ. We found a sharp bound for the number of invariant meridian hyperplanes for a polynomial vector field on S². Furthermore, we compute the sharp bound for the number of invariant parallel hyperplanes for any polynomial vector field on Sⁿ. Finally, we study homogeneous polynomial vector fields on Sⁿ, providing a characterization of their invariant (n-1)-spheres.

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