Leaf-Normal Form Classification for n-Tuple Hopf Singularities.

This is the first instance in the extensive literature of more than three decades for complete normal form classification of a singular family on a 2n-dimensional center manifold for arbitrary n. 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. [ABSTRACT FROM AUTHOR]