This paper develops a new approach to the estimation of the degree of boundedness/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 that typically provide 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 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 Pinsky and Koblik (Math Probl Eng 2020), Pinsky (J Nonlinear Dyn 103:517–539, 2021, Stability and boundedness of solutions to some multidimensional nonautonomous nonlinear systems, 2022). Still, the technique advanced in Pinsky (Math. Problems in Eng, 2022:5098677, https://doi.org/10.1155/2022/5098677) 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 a 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/stability criteria and estimate the relevant regions of the underlying systems using analytical and abridged numerical reasoning. Lastly, we authenticate the developed methodology in inclusive simulations.

本文开发了一种新方法来估计具有时间相关非周期系数的多维非线性系统的有界/稳定性程度 - 这是各种工程和自然科学应用中的一项重要任务。评估此类系统稳定性的已知方法依赖于 Lyapunov 函数和 Lyapunov 第一近似方法的效用，这些方法通常为此类多维系统提供保守的和计算复杂的标准。此类非齐次系统解的有界性的适当标准在当代文献中很少见。最近，我们开发了一种解决这些问题的新方法，该方法依赖于通过匹配我们在 Pinsky 和 ​​Koblik（Math Probl Eng 2020）、Pinsky（J Nonlinear Dyn 103： 517–539, 2021, 一些多维非自治非线性系统的解的稳定性和有界性, 2022)。尽管如此，Pinsky 提出的技术（Math. Problems in Eng, 2022:5098677, https://doi.org/10.1155/2022/5098677）仍基于以下假设：基础系统的线性分量的平均值定义为一般位置的稳定矩阵。目前的论文大大扩大了这种方法的应用领域。仅假设底层系统的时间相关线性块可以通过应用任何平滑技术分为慢速和快速变化的组件。这种时间尺度的二分法是由减少我们估计的保守性的最佳标准决定的。反过来，我们通过连续应用 Lyapunov 变换以对角线占优形式对具有缓慢变化矩阵的线性子系统进行变换。这促使了新的标量辅助方程的发展，包括对我们初始系统解的范数的估计。接下来，我们制定有界性/稳定性标准，并使用分析和简化的数值推理来估计基础系统的相关区域。最后，我们在包容性模拟中验证了开发的方法。