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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
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.

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