On the Status of the Stability Theory of Discontinuous Dynamical Systems

Author(s):  
Anthony N. Michel ◽  
Ling Hou
Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1228
Author(s):  
Manseob Lee ◽  
Jumi Oh

Expansiveness is very closely related to the stability theory of the dynamical systems. It is natural to consider various types of expansiveness such as countably-expansive, measure expansive, N-expansive, and so on. In this article, we introduce the new concept of countably expansiveness for continuous dynamical systems on a compact connected smooth manifold M by using the dense set D of M, which is different from the weak expansive flows. We establish some examples having the countably expansive property, and we prove that if a vector field X of M is C 1 stably countably expansive then it is quasi-Anosov.


2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Codruţa Stoica

The aim of this paper is to highlight current developments and new trends in the stability theory. Due to the outstanding role played in the study of stable, instable, and, respectively, central manifolds, the properties of exponential dichotomy and trichotomy for evolution equations represent two domains of the stability theory with an impressive development. Hence, we intend to construct a framework for an asymptotic approach of these properties for discrete dynamical systems using the associated skew-evolution semiflows. To this aim, we give definitions and characterizations for the properties of exponential stability and instability, and we extend these techniques to obtain a unified study of the properties of exponential dichotomy and trichotomy. The results are underlined by several examples.


2003 ◽  
Vol 2003 (2) ◽  
pp. 75-85 ◽  
Author(s):  
Ziad Zahreddine

Using some properties of the matrix measure, we obtain a general condition for the stability of a convex hull of matrices that will be applied to study the stability of interval dynamical systems. Some classical results from stability theory are reproduced and extended. We present a relationship between the matrix measure and the real parts of the eigenvalues that make it possible to obtain stability criteria.


Author(s):  
Wassim M. Haddad ◽  
Sergey G. Nersesov

This chapter describes a fundamental stability theory for nonlinear dynamical systems using vector Lyapunov functions. It first introduces the notation and definitions before developing stability theorems via vector Lyapunov functions for continuous-time and discrete-time nonlinear dynamical systems. It then extends the theory of vector Lyapunov functions by constructing a generalized comparison system whose vector field can be a function of the comparison system states as well as the nonlinear dynamical system states. It also presents a generalized convergence result which, in the case of a scalar comparison system, specializes to the classical Krasovskii–LaSalle theorem. In the analysis of large-scale nonlinear interconnected dynamical systems, several Lyapunov functions arise naturally from the stability properties of each individual subsystem.


2008 ◽  
Vol 08 (04) ◽  
pp. 625-641 ◽  
Author(s):  
ZHENXIN LIU ◽  
SHUGUAN JI ◽  
MENGLONG SU

In the stability theory of dynamical systems, Lyapunov functions play a fundamental role. In this paper, we study the attractor–repeller pair decomposition and Morse decomposition for compact metric space in the random setting. In contrast to [7,17], by introducing slightly stronger definitions of random attractor and repeller, we characterize attractor–repeller pair decompositions and Morse decompositions for random dynamical systems through the existence of Lyapunov functions. These characterizations, we think, deserve to be known widely.


2012 ◽  
Vol 22 (08) ◽  
pp. 1250183 ◽  
Author(s):  
JIANENG TANG ◽  
CAIRONG ZOU ◽  
SHAOPING WANG ◽  
LI ZHAO ◽  
PINGXIANG LIU

In this paper, the synchronization problem of Chen systems with time-varying delays is discussed based on the stability theory of time-delay systems. Through the analysis of the error dynamical systems, the time-delay correlative synchronization controller is designed to achieve chaos synchronization. And finally, numerical simulations are provided to verify the effectiveness and feasibility of the developed method.


2020 ◽  
Vol 93 (4) ◽  
pp. 215-226
Author(s):  
V. V. Zubkov ◽  
◽  
P.G. Sidorov ◽  

The article presents the results of a pilot sociological study of migration perceptions of the population, the reasons for their formation, as well as the factors and conditions under which the willingness to live and work in the Khabarovsk Territory is realized. The analysis of the results of the survey, which according to the research methodology was conducted in two target groups ("residents of the region" and "student youth"), indicates the stability and reproduction of migration intentions as a determined willingness and desire to leave the place of permanent residence in the Khabarovsk territory. The sociological approach to the study of migration perceptions of the target groups under study consists in determining the target attitudes, guidelines and expectations from moving, due to the status-role set and personal attitudes of respondents.


Author(s):  
Nam Parshad Bhatia ◽  
George Philip Szegö

Author(s):  
Przemysław Banasik ◽  
Katarzyna Metelska-Szaniawska ◽  
Małgorzata Godlewska ◽  
Sylwia Morawska

AbstractThe goal of this paper is to identify factors which affect judges’ productivity and career choice motives with the view of increasing judicial efficiency. Specifically, the investigation focuses on such aspects as judges’ remuneration, promotion, threat of judgment revocation, service/mission, periodic assessment, the threat of a complaint about protracted proceedings or of disciplinary proceedings, the threat of destabilization of the employment relationship, status/prestige of the profession, power/authority, social recognition, leisure, as well as administrative supervision and self-monitoring. To this end, a survey was conducted among judges of three of the largest Polish regional courts and subordinate district courts. The descriptive and statistical analyses show that judges’ care for the number of cases resolved, proxying for their productivity, is significantly correlated with self-monitoring of their adjudication activity. The stability of employment, the status/prestige of the profession and a relatively high remuneration are the most important factors in terms of judges’ career choices. In their care for the number of cases resolved remuneration is, albeit, no longer a relevant factor. Judges monitor their productivity due to reasons other than remuneration, possibly the sense of service/mission and the threat of various adverse consequences, the evidence for which is, however, also rather weak.


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