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We are concerned with bifurcation analysis and control of nonlinear Eulerianflows with non-resonant n-tuple Hopf singularity. The analysis is involved withCW complex bifurcations of flow-invariant Clifford hypertori, where we refer tothese toral manifolds by toral CW complexes. We observe from primary totertiary flow-invariant toral CW complex bifurcations for one-parametricsystems associated with two most generic cases. In a particular case, atertiary toral CW complex bifurcates from and resides outside a secondary toralCW complex. When the parameter varies, the secondary internal toral CW complexcollapses with the origin. However, the tertiary external toral CW complexcontinues to live even after the secondary internal toral manifold disappears.Our analysis starts with a flow-invariant primary cell-decomposition of thestate space. Each open cell admits a secondary cell-decomposition via a smoothflow-invariant foliation. Each leaf of the foliations is a minimalflow-invariant realization of the state space configuration for all Eulerianflows with n-tuple Hopf singularities. Permissible leaf-vector field preservingtransformations are introduced via a Lie algebra structure for nonlinear vectorfields on the leaf-manifold. Complete parametric leaf-normal formclassification is provided for singular leaf-flows. Leaf-bifurcation analysisof leaf-normal forms are performed for three most leaf-generic cases associatedwith one to three unfolding bifurcation-parameters. Leaf-bifurcation varietiesare derived. Leaf-bifurcations provides a venue for cell-bifurcation control ofinvariant toral CW complexes. The results are implemented and verified usingMaple for practical bifurcation control of such parametric nonlinearoscillators.

Toral CW complexes and bifurcation control in Eulerian flows with multiple Hopf singularities