## “School of Mathematics”

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Paper   IPM / M / 17076
 School of Mathematics Title: Leaf-normal form classification for n-tuple Hopf singularities Author(s): Majid Gazor (Joint with A. Shoghi) Status: Published Journal: Commun. Math. Phys. Vol.: https://doi.org/10.1007/s00220-022-04470-2 Year: 2022 Supported by: IPM
Abstract:
We are concerned with complete normal form characterization and classification of non-resonant $n$-tuple Hopf singular differential systems with radial and rotational nonlinearities. Our analysis is facilitated by using several reduction techniques. These include an invariant cell-decomposition of the state space, a family of smooth flow-invariant foliations, leaf-reduction of differential systems and leaf-normal forms. Each leaf of the foliations is a minimal flow-invariant realization of the state space for all radial and rotational differential systems. Complete simplest normal form characterization for singular flows are provided using a family of leaf-reductions and infinite level (simplest) formal leaf-normal forms. In this direction, we introduce Lie algebra structures on invariant leaf manifolds for the local leaf-normal classifications. Since leaf-manifolds foliate the state space, leaf normal forms for all flow-invariant leaves are required for a complete normal form characterization of the $2n$-dimensional system. Thus, we further discuss the geometry and spectral impact of leaf variations on the infinite level leaf-normal form coefficients, leaf-finite determinacy and leaf-universal unfoldings. There are infinitely many leaf-systems for such a $2n$-dimensional system. However, we show that a $2n$-dimensional system can admit at most a finite number of topologically non-equivalent leaf-normal form systems. These are the ones that classify the 2n-dimensional singular family.