  This letter describes a method for estimating regions of attraction and
bounds on permissible perturbation amplitudes in nonlinear fluids systems. The
proposed approach exploits quadratic constraints between the inputs and outputs
of the nonlinearity on elliptical sets. This approach reduces conservatism and
improves estimates for regions of attraction and bounds on permissible
perturbation amplitudes over related methods that employ quadratic constraints
on spherical sets. We present and investigate two algorithms for performing the
analysis: an iterative method that refines the analysis by solving a sequence
of semi-definite programs, and another based on solving a generalized
eigenvalue problem with lower computational complexity, but at the cost of some
precision in the final solution. The proposed algorithms are demonstrated on
low-order mechanistic models of transitional flows. We further compare accuracy
and computational complexity with analysis based on sum-of-squares optimization
and direct-adjoint looping methods.
