  Assessment of degree of boundedness and stability of multidimensional
nonlinear systems with time-dependent and especially nonperiodic coefficients
is an important applied problem which has no adequate resolution yet. Most of
the known techniques mostly provide computationally intensive and conservative
stability criteria in this area which frequently fail to gage the degrees of
stability and especially boundedness of solutions to the corresponding systems.
Recently, we outline a new approach to this task resting on analysis of
solutions to a scalar auxiliary equation bounding from above time-histories of
the norms of solutions to the original systems. This paper develops a new
technique casting the auxiliary equation in a simplified form which, in turn,
amplifies its application domain and reduces the computational hamper of our
prior approach. Consequently, we develop novel boundedness and stability
criteria and estimated the trapping and stability regions for some
multidimensional nonlinear systems with time - dependent coefficients. This let
us to assess in target simulations the degree of boundedness and stability of
multidimensional nonlinear and non-autonomous systems which were intractable to
our prior methodology
