  We are concerned with bifurcation analysis and control of nonlinear Eulerian
flows with non-resonant n-tuple Hopf singularity. The analysis is involved with
CW complex bifurcations of flow-invariant Clifford hypertori, where we refer to
these toral manifolds by toral CW complexes. We observe from primary to
tertiary flow-invariant toral CW complex bifurcations for one-parametric
systems associated with two most generic cases. In a particular case, a
tertiary toral CW complex bifurcates from and resides outside a secondary toral
CW complex. When the parameter varies, the secondary internal toral CW complex
collapses with the origin. However, the tertiary external toral CW complex
continues to live even after the secondary internal toral manifold disappears.
Our analysis starts with a flow-invariant primary cell-decomposition of the
state space. Each open cell admits a secondary cell-decomposition via a smooth
flow-invariant foliation. Each leaf of the foliations is a minimal
flow-invariant realization of the state space configuration for all Eulerian
flows with n-tuple Hopf singularities. Permissible leaf-vector field preserving
transformations are introduced via a Lie algebra structure for nonlinear vector
fields on the leaf-manifold. Complete parametric leaf-normal form
classification is provided for singular leaf-flows. Leaf-bifurcation analysis
of leaf-normal forms are performed for three most leaf-generic cases associated
with one to three unfolding bifurcation-parameters. Leaf-bifurcation varieties
are derived. Leaf-bifurcations provides a venue for cell-bifurcation control of
invariant toral CW complexes. The results are implemented and verified using
Maple for practical bifurcation control of such parametric nonlinear
oscillators.
