  This paper develops a new approach to the estimation of the degree of
boundedness or stability of multidimensional nonlinear systems with
time-dependent nonperiodic coefficients-an essential task in various
engineering and natural science applications. Known approaches to assessing the
stability of such systems rest on the utility of Lyapunov functions and
Lyapunov first approximation methodologies, typically providing conservative
and computationally elaborate criteria for multidimensional systems of this
category. Adequate criteria of boundedness of solutions to nonhomogeneous
systems of this kind are rare in the contemporary literature. Lately, we
develop a new approach to these problems which rests on bounding the evolution
of the norms of solutions to initial systems by matching solutions of a scalar
auxiliary equation we introduced in [1], [2] and [3]. Still, the technique
advanced in [3] rests on the assumption that the average of the linear
components of the underlying system is defined by a stable matrix of general
position. The current paper substantially amplifies the application domain of
this approach. It is merely assumed that the time-dependent linear block of the
underlying system can be split into slow and fast varying components by
application of any smoothing technique. This dichotomy of temporal scales is
determined by the optimal criterion reducing the conservatism of our estimates.
In turn, we transform the linear subsystem with slow-varying matrix in a
diagonally dominant form by successive applications of the Lyapunov transforms.
This prompts the development of novel scalar auxiliary equations embracing the
estimation of the norms of solutions to our initial systems. Next, we formulate
boundedness or stability criteria and estimate the relevant regions of the
underlying systems using analytical and abridged numerical reasoning.
