Stability Theory Of Dynamical Systems Research Articles

Finite/Prescribed-Time Cluster Synchronization of Complex Dynamical Networks With Multiproportional Delays and Asynchronous Switching

This article focuses on the finite-time and prescribed-time cluster synchronization problems of switched complex networks with proportional delays. Considering that the matched controller often switches behind the switched systems, asynchronous switching law and switching parameters have been introduced to the complex dynamical networks. By applying the stability theory of dynamical systems and the average dwell time approach, some sufficient criteria in form of linear matrix inequalities have been developed to guarantee such networks finite/prescribed-time cluster synchronization under the pinning control scheme. Finally, one numerical example is presented to show the effectiveness of the theoretical results.

Robust Finite-Time H∞ Control for the Uncertain Singular System

This paper addresses the robust finite-time $H_\infty $ control problem for the uncertain singular system by using the stability theory of dynamical systems. Firstly, a lemma is provided to show that the singular system is finite-time stability by using the state space decomposition approach. Similar with the proof method of this Lemma, the singular system is divided into a differential system and an algebra one. Then, some conditions are derived to ensure the singular system being finite-time $H_\infty $ stability based on the obtained lemma, and the state feedback control law is designed. These conditions are provided in the form of the linear matrix inequalities and can be easily solved. Finally, a numerical example is given to illustrate the effectiveness of the obtained results.

Nonexistence of invariant manifolds in fractional-order dynamical systems

Invariant manifolds are important sets arising in the stability theory of dynamical systems. In this article, we take a brief review of invariant sets. We provide some results regarding the existence of invariant lines and parabolas in planar polynomial systems. We provide the conditions for the invariance of linear subspaces in fractional order systems. Further, we provide an important result showing the nonexistence of invariant manifolds (other than linear subspaces) in fractional order systems.

Uniform Detectability of Linear Time Varying Systems With Exponential Dichotomy

Exponential dichotomies play a central role in stability theory for dynamical systems. They allow to split the state space into two subspaces, where all trajectories in one subspace decay whereas all trajectories in the other subspace grow, uniformly and exponentially. This letter studies uniform detectability and observability notions for linear time varying systems, which admit an exponential dichotomy. The main contributions are necessary and sufficient detectability conditions for this class of systems.

Мэтр качественной теории и теории устойчивости динамических систем (к 100­-летию со дня рождения Александра Андреевича Шестакова)

The article presents the biography of Alexander A. Shestakov — ScD in Physics and Mathematics, Professor, a major scientist who had a wide range of scientific interests. His researches are devoted to the key problems of quality theory of differential equations, stability theory of dynamic systems, and also various applied aspects of these theories to problems of dynamics of transport systems. Professor Shestakov and his students and followers have obtained significant results in various fields of fundamental and applied mathematics, as well as mechanics. The article provides a brief overview of the scientific and pedagogical heritage of the scientist who is an organizer and founder of the famous scientific school on quality theory of dynamic processes and stability theory of differential equation solutions. He and his numerous students obtained fundamental results related to the development of the theory of stability in the sense of A.M. Lyapunov and the theory of stability in the sense of N.E. Zhukovsky, with the development of methods for the analysis of orbital stability, limitation and convergence of solutions of differential equations, with the development of limit equations and method of localization of limit sets, with theoretical and applied problems of stability of mathematical models of dynamics of railway transport. Its manuals, textbooks and monographs are the standard of fundamentality of the issues presented in them and have not lost their relevance to our days.

Global fixed-time synchronization of chaotic systems with different dimensions

This paper studies the global fixed-time synchronization (GFTS) of chaotic systems with different dimensions. First, with the help of fixed-time stability theory of dynamic systems, a novel control protocol is put forward, which can achieve globally synchronization of two different dimensional chaotic systems (DDCS) in fixed time. Second, the GFTS of DDCS with uncertain parameters is also considered. The appropriate adaptive laws are designed to address the unknown parameters of the systems. Then, by the adaptive control, a new controller is presented to ensure the realization of DDCS within a given fixed time. Third, the GFTS is considered to networked DDCS, and the controller is also proposed accordingly. Different from conventional finite-time synchronization, the upper bound of settling time is independent of initial conditions of systems in GFTS. Finally, the effectiveness of the obtained results is demonstrated by corresponding numerical simulations.

Global Stability of Fractional Order Coupled Systems with Impulses via a Graphic Approach

Based on the graph theory and stability theory of dynamical system, this paper studies the stability of the trivial solution of a coupled fractional-order system. Some sufficient conditions are obtained to guarantee the global stability of the trivial solution. Finally, a comparison between fractional-order system and integer-order system ends the paper.

Qualitative behaviors of the continuous-time chaotic dynamical systems describing the interaction of waves in plasma

In this paper, a new Lorenz-like chaotic system describing the interaction of three resonantly coupled waves in plasma is studied. Explicit ultimate boundedness and global attraction domain are derived according to stability theory of dynamical systems. The innovation of the paper is that this paper not only proves this chaotic system is globally bounded for the parameters of this system but also gives a family of mathematical expressions of global exponential attractive sets for this system with respect to the parameters of this system. Furthermore, the exponential rate of the trajectories is also obtained. Finally, numerical localization of attractor is presented.

On global boundedness of the Chen system

This paper deals with the open problem of the global boundedness of the Chen system based on Lyapunov stability theory, which was proposed by Qin and Chen (2007). The innovation of the paper is that this paper not only proves the Chen system is global bounded for a certain range of the parameters according to stability theory of dynamical systems but also gives a family of mathematical expressions of global exponential attractive sets for the Chen system with respect to the parameters of this system. Furthermore, the exponential rate of the trajectories is also obtained.

Exponential synchronization via pinning adaptive control for complex networks of networks with time delays

This paper is concerned with the pinning adaptive synchronization control problem for a class of complex networks of networks. The complex networks of networks under consideration are composed of both leaders' network and followers' networks (also called “subnetworks”), where the leaders' network and subnetworks are regarded as the nodes of the networks of networks, each subnetwork can receive the information from leaders' network, but not the reverse. In order to achieve the synchronization for the networks of networks, pinning control strategy is adopted, and adaptive controllers are designed for the controlled nodes. By utilizing the stability theory of dynamical systems and some analysis techniques such as the Barbalat lemma, several sufficient criteria are obtained to ensure global exponential synchronization for the controlled complex networks of networks. Finally, a numerical simulation example is given to verify the theoretical results.

Qualitative analysis of a new Lorenz‐type chaotic system and its simulation

In this paper, the ultimate bound for a new chaotic system is derived based on stability theory of dynamical systems. The meaningful contribution of this article is that the results presented in this paper contain the existing results as special cases. Finally, numerical simulations are given to verify the effectiveness and correctness of the obtained results. Copyright © 2016 John Wiley & Sons, Ltd.

A New Q–S Synchronization Results for Discrete Chaotic Systems

This paper investigates the problem of Q–S synchronization for different dimensional chaotic dynamical systems in discrete-time. Based on two control laws and stability theory of dynamical systems in discrete-time, new synchronization schemes are derived. Numerical examples demonstrate the effectiveness and feasibility of the proposed control techniques.

Dynamical behavior of a generalized Lorenz system model and its simulation

In this article, the dynamical behavior of a generalized Lorenz system is derived based on stability theory of dynamical systems. The meaningful contribution of this article is that the domain of attraction of the new chaotic system is studied in detailed. Finally, numerical simulations are given to verify the effectiveness and correctness of the obtained results. © 2015 Wiley Periodicals, Inc. Complexity 21: 99–105, 2016

On a model of the dynamical systems describing convective fluid motion in rotating cavity

In this paper a Lorenz-like system, describing the process of rotating fluid convection, is considered. Explicit estimations of the bounds are derived based on stability theory of dynamical systems. Numerical localization of attractor is presented.

Quasi-Optimal Regulators for Nonholonomic Systems Driven by Rough Paths

Nonholonomic systems such as port-Hamiltonian systems are well known to be difficult to control. To reduce the difficulty, in this paper, we derive quasi-optimal regulators for nonholonomic systems such as a typical chained system with some restriction to the form of control inputs. To achieve the solution, we employ a notion of stability in roughness, which is Lyapunov stability theory for dynamical systems driven by rough paths. The rough paths are capable of transforming some nonholonomic systems into holonomic systems with “hidden control inputs”. Thus, the control problems are simplified in exchange for the degradation of some control performances.

Detection of salient regions in crowded scenes

The increasing number of cameras and a handful of human operators to monitor the video inputs from hundreds of cameras leave the system ill equipped to fulfil the task of detecting anomalies. Thus, there is a dire need to automatically detect the regions that require immediate attention for more effective and proactive surveillance. A framework that utilises the temporal variations in the flow field of a crowd scene to automatically detect salient regions is proposed, while eliminating the need to have prior knowledge of the scene or training. The flow fields are deemed to be a dynamic system and adopt the stability theory of dynamic systems, to determine the motion dynamics within a given area. In this context, the salient regions refer to the areas with high motion dynamics, where the points in a particular region are unstable. The experimental results on public, crowd scenes have shown the effectiveness of the proposed method in detecting salient regions which correspond to an unstable flow, occlusions, bottlenecks, and entries and exits.

Chaos synchronization between the coupled systems on network with unknown parameters

This work investigates the adaptive network synchronization of different coupled chaotic (or hyper-chaotic) systems with unknown parameters. The definition of network synchronization is presented between the coupled systems where there are 2N chaotic systems, the former N systems are coupled each other and so are the later N systems. Then the sufficient conditions for achieving this synchronization are derived based on the theory of the stability theory of dynamical systems with 2N coupled chaotic systems. By the adaptive control technique, the control laws and the corresponding parameters update laws are proposed such that this network synchronization of non-identical chaotic (or hyper-chaotic) systems is to be obtained. These results obtained are in good agreement with the existing one in open literature and it is shown that the technique introduced here can be further applied to various network synchronizations between coupled dynamical systems. Finally, all our analytic results are confirmed by extensive numerical simulations of the model.

Lyapunov stability for semigroup actions

In this paper we introduce a theory of Lyapunov stability of sets for semigroup actions on Tychonoff spaces. We also present the main properties and the main results relating these new concepts. We generalize several concepts and results of Lyapunov stable sets from Bhatia and Hajek (Local Semi-Dynamical Systems. Lecture Notes in Mathematics, vol. 90. Springer, Berlin, 1969), Bhatia and Szego (Dynamical Systems: Stability Theory and Applications. Lecture Notes in Mathematics, vol. 35. Springer, Berlin, 1967; and Stability Theory of Dynamical Systems. Springer, Berlin, 1970).

Symmetric/asymmetric bifurcation behaviours of a bogie system

Based on the bifurcation and stability theory of dynamical systems, the symmetric/asymmetric bifurcation behaviours and chaotic motions of a railway bogie system under a complex nonlinear wheel–rail contact relation are investigated in detail by the ‘resultant bifurcation diagram’ method with slowly increasing and decreasing speed. It is found that the stationary equilibrium solution and the periodic motions coexist due to the sub-critical Hopf bifurcation in the railway bogie system. It is also found that multiple solutions coexist in many speed ranges. The coexistence of multiple solutions may result in a jump and hysteresis of the oscillating amplitude for different kinds of disturbances. It should be avoided in the normal operation. Furthermore, it is found that symmetry-breaking of the system through a pitchfork bifurcation leads to asymmetric chaotic motions in the railway bogie system. The speed ranges of asymmetric chaotic motions are, however, small.

ATTRACTOR–REPELLER PAIR, MORSE DECOMPOSITION AND LYAPUNOV FUNCTION FOR RANDOM DYNAMICAL SYSTEMS

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.