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Paper IPM / M / 17076  


Abstract:  
We are concerned with complete normal form characterization and classification of nonresonant \(n\)tuple Hopf singular differential systems with radial and rotational nonlinearities. Our analysis is facilitated by using several reduction techniques. These include an invariant celldecomposition of the state space, a family of smooth flowinvariant foliations, leafreduction of differential systems and leafnormal forms. Each leaf of the foliations is a minimal flowinvariant 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 leafreductions and infinite level (simplest) formal leafnormal forms. In this direction, we introduce Lie algebra structures on invariant leaf manifolds for the local leafnormal classifications. Since leafmanifolds foliate the state space, leaf normal forms for all flowinvariant 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 leafnormal form coefficients, leaffinite determinacy and leafuniversal unfoldings. There are infinitely many leafsystems for such a \(2n\)dimensional system. However, we show that a \(2n\)dimensional system can admit at most a finite number of topologically nonequivalent leafnormal form systems. These are the ones that classify the 2ndimensional singular family.
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