A primary spectral submanifold (SSM) is the unique smoothest nonlinear
continuation of a nonresonant spectral subspace E of a dynamical system
linearized at a fixed point. Passing from the full nonlinear dynamics to the
flow on an attracting primary SSM provides a mathematically precise reduction
of the full system dynamics to a very low-dimensional, smooth model in
polynomial form. A limitation of this model reduction approach has been,
however, that the spectral subspace yielding the SSM must be spanned by
eigenvectors of the same stability type. A further limitation has been that in
some problems, the nonlinear behavior of interest may be far away from the
smoothest nonlinear continuation of the invariant subspace E. Here we remove
both of these limitations by constructing a significantly extended class of
SSMs that also contains invariant manifolds with mixed internal stability types
and of lower smoothness class arising from fractional powers in their
parametrization. We show on examples how fractional and mixed-mode SSMs extend
the power of data-driven SSM reduction to transitions in shear flows, dynamic
buckling of beams and periodically forced nonlinear oscillatory systems. More
generally, our results reveal the general function library that should be used
beyond integer-powered polynomials in fitting nonlinear reduced-order models to
data.Comment: To appear in Chao