I've decided to expand significantly the analysis of the example I added recently to my paper "The Lyapunov Concept of Stability from the Standpoint of Poincare Approach: General Procedure of Utilization of Lyapunov Functions for Non-Linear Non-Autonomous Parametric Differential Inclusions". Why? This example is remarkable to my eye because not only for being quite simple (3D system of non-autonomous ODEs can be visually illustrated by graphs), but also for helping clarify geometric and topological nature not only Lyapunov functions and Lyapunov Second method for stability but also Lyapunov exponent. Moreover, the example considerably assists with understanding what factors play the main role in geometric-topological structure of phase spaces far off from singular points or without them at all. I hope the detailed analysis of it will be useful not only to PhD students working in Applied Mathematics, Physics and Engineering but also everyone who has to deal with Mathematical Modeling and Simulation.
Your "question" stimulated my memory, as I did my thesis on the asymptotic stability of homogeneous open physicochemical systems 39 years ago. Are you aware of Zubov's construction for Lyapunov functions? I used this concept to generate my first paper, J Chem Phys 1973, 58, 4025-4026, and when I used it to provide analytic solutions for the global asyptotic stability of a nolinear parabolic system in IEEE Trans At Control 1978, 23, 1114. In a 1974 paper I tied these concepts to Poincare's limit cycles at a Faraday Symposium, Limit cycles in the plane. An equivalence class of homogeneous systems, Faraday Symp. Chem. Soc., 1974,9, 254-262.
Yes, I am. There are a lot of papers published in sixties and seventies that followed the Zubov's approach especially for control theory. I'd be grateful if you send me the copies of your papers.
Hi John,
I've placed full-text requests on your Researchgate account for your above-mentioned papers and several more. Please upload them. I'm interested in chemical math models.
OK, I am teaching noon to 9pm today and am very busy. Plase be patient.
Wei Zhang, actually this is not a question. My purpose has been to invite the ResearchGate community to discuss the paper, which is in its working version. I'd like the researchers to ask the questions they are interested in, to share their results on the subject, if they have, with the aim of collaboration, to indicate the mistakes if they have found, to make their remarks and to express the thoughts, etc. One of my areas of research is Stability Theory. Its results can be applied to the practically unlimited numbers of the spheres of science and technology. In other words in any problem that uses a mathematical model. Thus, any question, remark, comment, example and counterexample, etc is very welcomed. That's true that it is necessary to have some specific skills and knowledge to understand the paper. To facilitate the process of understanding I've added the example that illustrates visually and geometrically the main ideas. The problem of Lyapunov functions, I've studied in the paper, is quite old and was originated by very Aleksandr Lyapunov in 1892. Many great theoretical results have been obtained since then by mathematicians and engineers including the Zubov's works as Prof. John Texter has mentioned. The main difficulty in Lyapunov functions is to construct them. If you've constructed it for your mathematical model then there are Lyapunov theorems that help you determine whether your system is stable or unstable. But the prime obstacle in constructing Lyapunov functions lies in the absence of the clear understanding of the nature of these functions. You cannot construct what, the nature of which you don't understand or don't know. My work gives the answer just to this question. I show topological and geometrical nature of Lyapunov functions. To solve this problem I've used the methods of topology. And I am here to help you understand the solution.
How can I use the lagrange decomposition methods to uncapacitate location problem?
A little bit later as I’ll be able to find some time I am going to add the additional remarks explaining the principle “why”-s and “what”-s that may come to the minds of readers of my paper, namely.
1. Why has the concept of foliations been used? This is a cornerstone question giving the answer to the next vital question.
2. What is the crucial difference between my results and the ones of all previous researchers starting from A. M. Lyapunov, N. G. Chetaev, V. I. Zubov, N. N. Krasovskii, E. A. Barbashin, J. P. LaSalle, S. Lefschetz and etc?
3. Why have the tools of fiber bundles (mostly used in differential and algebraic topology) and sheaves (mostly used in differential and algebraic geometry) been used?
4. What is the relation between the classification of covering mappings and the properties of Lyapunov functions and why do we need the concept of covering mappings in general?
5. How can engineers and researchers benefit from the results of this theoretical paper working on real practical problems?
6. What are the further directions of development of Stability Theory in my personal opinion?
I guess that I have to explain all these points because in fact the paper crowns the thirty years of my intensive research work in Nonlinear Dynamical Systems and Control.
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