arXiv subjects

Mark A. Pinsky

Publications and source records attributed to Mark A. Pinsky.

12 recordsLinked to original sources

arXiv2026

Estimating the Evolution of Solution Norms in Vector Delay Nonlinear Systems: Stability and Boundedness

Existing methods rarely capture the temporal evolution of solution norms in vector nonlinear DDEs with variable delays and coefficients, often leading to overly conservative boundedness and stability criteria. We develop a framework that constructs scalar counterparts of vector DDEs whose solutions upper-bound the evolution of the original solution norms when the corresponding history functions are matched. This reduction enables boundedness and stability assessment of vector DDEs through the dynamics of their scalar counterparts, using straightforward simulations or simplified analytical reasoning. New boundedness and stability criteria and the estimates of the radii of balls containing history functions that yield bounded or stable solutions for the original vector systems were derived and validated through representative simulations.

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arXiv2026

Bilateral Solution Bounds and Successive Estimation of Boundedness and Stability Regions for Vector Delay Nonlinear Time-Varying Systems

Stability and boundedness analysis for vector nonlinear systems with variable delays and coefficients remains challenging due to the conservatism of existing methods. Moreover, estimates of the transient behavior of solution norms remain insufficiently developed. This paper presents an approach to estimate the temporal evolution of solution norms and applies it to the analysis of boundedness and stability of vector nonlinear systems with variable delays and coefficients. The method is based on a novel scheme for successive approximations of the original solutions, complemented by the estimates of the corresponding residual norms. This leads to the construction of a scalar nonlinear delay equation whose solutions provide upper bounds for the evolution of residual norms. As a result, bilateral bounds on the original solution norms are obtained, yielding effective boundedness and stability criteria and enabling estimation of the associated regions. Simulations demonstrate that the proposed approximations rapidly approach the reference boundaries of the regions of interest as the iteration count increases. Moreover, the bilateral bounds progressively approach each other and the norm of the reference solution when the initial function remains within the considered regions.

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arXiv2024

A New Approach to Reducing Vector Delay Nonlinear Systems: Boundedness and Stability Analysis

This paper introduces a new method for assessing the boundedness and stability of certain vector nonlinear systems with delays and variable coefficients. The approach is based on developing scalar counterparts to the given vector systems. We prove that the solutions to these scalar nonlinear equations, which also include delays and variable coefficients, provide upper bounds for the norms of solutions to the original vector equations if the history functions for both equations are properly matched. This enables the evaluation of the boundedness and stability characteristics of a vector system by analyzing the abridged dynamics of its scalar counterparts. This assessment can be carried out through straightforward simulations or by applying simplified analytical methods. As a result, we introduce new criteria for boundedness and stability and estimate the radii of the balls that contain history functions stemming bounded or stable solutions for the original vector systems. Finally, we validate our inferences through representative simulations that also assess the accuracy of our approach.

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arXiv2024

Bilateral Bounds for Norms of Solutions and Boundedness/Stability and Instability of Some Nonlinear Systems with Delays and Variable Coefficients

This paper presents a novel methodology for evaluating the boundedness, stability, and instability of some vector nonlinear systems with multiple time-varying delays and variable coefficients. The proposed technique develops two scalar counterparts for the initial vector system. The solutions to these scalar nonlinear equations, which also incorporate delays and variable coefficients, provide upper and lower bounds for the norms of solutions to the original vector equations with corresponding history functions. This enables evaluation of the dynamics of a vector system through the analysis of its scalar counterparts. This analysis can be accomplished using simplified analytical reasoning or straightforward simulations, which remain effective even for systems with a large number of coupled equations. Consequently, we introduced some novel boundedness, stability and instability criteria and estimated the radiuses of the balls containing the history functions which stem bounded/stabile solutions to the vector systems. Lastly, we validate our results in representative simulations that also assess their accuracy.

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arXiv2024

Application of a Novel Model Reduction Technique to the Assessment of Boundedness/Stability of Some Delay Time-Varying Vector Nonlinear Systems

Assessing the boundedness and stability of vector nonlinear systems with variable delays and coefficients remains a challenging problem with broad applications in science and engineering. Existing methods tend to produce overly conservative criteria that offer limited practical value and often fail to explicitly characterize the temporal evolution of solution norms. This paper presents a novel framework for evaluating the evolution of solution norms in such systems. This approach constructs scalar counterparts for the original vector equations. We prove that the solutions to these scalar nonlinear equations, which also include delays and variable coefficients, provide upper bounds for the norms of the original solutions, if the history functions for both equations are properly matched. This reduction enables the evaluation of the boundedness and stability of vector systems through the analysis of the dynamics of their scalar counterparts, which can be performed via straightforward simulations or simplified analytical reasoning. Consequently, we introduce new criteria for boundedness and stability and estimate the radii of the balls containing history functions that yield bounded or stable solutions for the original vector systems. Finally, we validated our inferences through representative simulations that also assessed the accuracy of the proposed approach.

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arXiv2022

Optimal Dichotomy of Temporal Scales and Boundedness and Stability of Time-Varying Multidimensional Nonlinear Systems

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.

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arXiv2022

Stability and Boundedness of Solutions to Some Non-autonomous Multidimensional Nonlinear Systems

Assessment of the degree of boundedness/stability of multidimensional nonlinear systems with time-dependent and nonperiodic coefficients is an important problem in various applied areas which has no adequate resolution yet. Most of the known techniques provide computationally intensive and conservative stability criteria in this field and frequently fail to estimate the regions of boundedness/stability of solutions to the corresponding systems. Recently, we outlined a new approach to this problem which is based on the analysis of solutions to a scalar auxiliary equation bounded from the above time histories of the norms of solutions to the original system. This paper develops a novel technique casting the auxiliary equation in a modified form which extends the application domain and reduces the computational hamper of our prior approach. Consequently, we developed more general boundedness/stability criteria and estimated trapping/stability regions for some multidimensional nonlinear systems with nonperiodic time-dependent coefficients that are common in various application domains. This let us to assess in target simulations the extent of boundedness/stability of some multidimensional, nonlinear and time-varying systems which were intractable with our prior technique.

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arXiv2020

Successive Estimations of Bilateral Bounds and Trapping/Stability Regions of Solution to Some Nonlinear Non-autonomous Systems

Estimation of the degree of stability and the bounds of solutions to non-autonomous nonlinear systems present major concerns in numerous applied problems. Yet, current techniques are frequently yield overconservative conditions which are unable to effectively gage these characteristics in time-varying nonlinear systems. This paper develops a novel methodology providing successive approximations to solutions that are stemmed from the trapping and stability regions of these systems and estimate the errors of such approximations. In turn, this leads to successive approximations of both the bilateral bounds of solutions and the boundaries of trapping and stability regions of the underlying systems. Along these lines we formulate enhanced stability and boundedness criteria and contrast our inferences with inclusive simulations.

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arXiv2020

Solution Bounds, Stability and Estimation of Trapping/Stability Regions of Some Nonlinear Time-Varying Systems

Estimation of solution norms and stability for time-dependent nonlinear systems is ubiquitous in numerous engineering, natural science and control problems. Yet, practically valuable results are rare in this area. This paper develops a novel approach, which bounds the solution norms, derives the corresponding stability criteria, and estimates the trapping/stability regions for some nonautonomous and nonlinear systems, which arise in various application domains. Our inferences rest on deriving a scalar differential inequality for the norms of solutions to the initial systems. Utility of the Lipschitz inequality linearizes the associated auxiliary differential equation and yields both the upper bounds for the norms of solutions and the relevant stability criteria. To refine these inferences, we introduce a nonlinear extension of the Lipschitz inequality, which improves the developed bounds and allows estimation of the stability basins and trapping regions for the corresponding systems. Finally, we confirm the theoretical results in representative simulations.

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arXiv2018

Solution Bounds, Stability and Attractors Estimates of Some Nonlinear Time-Varying Systems

Estimation of solution norms and stability for time-dependent nonlinear systems is ubiquitous in numerous applied and control problems. Yet, practically valuable results are rare in this area. This paper develops a novel approach, which bounds the solution norms, derives the corresponding stability criteria, and estimates the trapping/stability regions for a broad class of the corresponding systems. Our inferences rest on deriving a scalar differential inequality for the norms of solutions to the initial systems. Utility of the Lipschitz inequality linearizes the associated auxiliary differential equation and yields both the upper bounds for the norms of solutions and the relevant stability criteria. To refine these inferences, we introduce a nonlinear extension of the Lipschitz inequality, which improves the developed bounds and estimates the stability basins and trapping regions for the corresponding systems. Finally, we conform the theoretical results in representative simulations.

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arXiv2011

Growth Properties of Fourier Transforms

In a recent paper by the authors, growth properties of the Fourier transform on Euclidean space and the Helgason Fourier transform on rank one symmetric spaces of non-compact type were proved and expressed in terms of of a modulus of continuity based on spherical means. The methodology employed first proved the result on Euclidean space and then, via a comparison estimate for spherical functions on rank one symmetric spaces to those on Euclidean space, we obtained the results on symmetric spaces. In this note, an analytically simple, yet overlooked refinement of our estimates for spherical Bessel functions is presented which provides significant improvement in the growth property estimates.

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arXiv1995

Speed of convergence of two-dimensional Fourier integrals

Recently we found necessary and sufficient conditions for the convergence at a preassigned point of the spherical partial sums of the Fourier integral in a class of piecewise smooth functions in Euclidean space. These yield elementary examples of divergent Fourier integrals in three dimensions and higher. Meanwhile, several years ago Gottlieb and Orsag observed that in two dimensions we may expect slower convergence at certain points, specifically for Fourier-Bessel series of radial functions. In this paper we investigate the rate of convergence of the spherical partial sums of the Fourier integral for a class of piecewise smooth functions. The basic result is an asymptotic expansion which allows us to read off the rate of convergence at a pre-assigned point.

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