Stability Of Dynamical Systems Research Articles

On the stability of discrete-time homogeneous polynomial dynamical systems

On the stability of discrete-time homogeneous polynomial dynamical systems

Exponential Stability of Dynamical Systems on Time Scales with Application to Multi-Agent Systems

The exponential stability criteria of systems with time delays on time scales are established, which unifies and generalizes the continuous and discrete cases. The time derivatives of Lyapunov functions (functionals) along solutions are allowed to be indefinite, namely, to take both negative and positive value, which reduces conservatism of the criteria. Moreover, the stability criteria are applicable to both linear and nonlinear systems on time scales, which expands the scope of application of the criteria. Furthermore, the improved stability theorem is applied to solve a leader-following consensus problem of multi-agents on time scales. Sufficient conditions are derived for the leader-following consensus of multi-agent systems under directed interaction topology. A numerical example is given to illustrate the feasibility and effectiveness of the theoretical results.

On the Stability of the Mixing Flow Dynamical System Perturbed with a Logistic Term

This paper continues some recent work in a powerful mathematical domain, with applications in connected scientific fields, namely the stability of dynamical systems. In fluid mechanics, stabilizing a dynamical system is a challenging task and it can be done by various ways. Stabilizing a dynamical system could be often easier if we approach controllable systems, because in this form, there can be imposed some bounds on its behavior, by studying the improvement of the operators that describe the system. In this paper, the mixing flows dynamical systems are taken into account, more exactly the kinematics of mixing flows. The stability analysis of the mixing flow is taken into account, in the case of perturbation with a logistic term. The results can be extended to some other versions of the model.

Investigation of controllability and stability of fractional dynamical systems with delay in control

The primary objective of this research is to investigate the controllability and Hyers–Ulam stability of fractional dynamical systems represented by ψ-Caputo fractional derivative with delay in control. To establish the necessary and sufficient conditions for assessing the controllability of linear fractional systems, which are notably distinguished by the presence of Mittag-Leffler functions, we employ the positivity of the Grammian matrix. Also, we present sufficient conditions for the controllability requirements for nonlinear fractional systems utilizing the fixed-point technique. The Hyers-Ulam stability technique is used to determine the sufficient condition for the stability of fractional nonlinear systems with delay in control. Numerical instances are provided to enhance comprehension of the theoretical findings.

100 years of extremum seeking: A survey

Extremum seeking (ES) is a powerful approach to the optimization and stabilization of unknown dynamic systems and is an active field of research in control theory. This paper surveys recent ES algorithm developments. ES has evolved from its origins as a simple but brilliant engineering invention in the 1922 paper of Leblanc to the breakthrough work of Krstić and Wang in 2000 in which the first general stability analysis was given, to the most recent and numerous analytical results which prove convergence for a wide range of general dynamic systems. This survey provides a thorough overview of the dynamic systems for which ES stabilization and optimization results have been provided, the various analytical methods for proving convergence, and an overview of many real world applications for which ES has been utilized in industry and at scientific laboratories.

Revisiting complex dynamic systems theory: Empowering language teachers and teaching

AbstractThis article revisits complex dynamic systems theory (CDST) and provides a new synthesis that, the authors argue, may empower language teachers. The particular focus is on the interplay between change and stability in complex dynamic systems, including language teaching and learning. The article identifies four key dynamical properties of complex dynamic systems, which reflect change, stability, and the interplay between these two. The dynamical properties are historicity, adaptiveness, nonlinearity, and emergence. The authors associate these dynamic properties with four recognisable types of activity: normative, contingent, creative, and purposeful teaching and learning. The central argument is that language teaching and learning are interconnected, and language teaching that promotes these four types of teaching and learning activity may also promote opportunities for language learning. The authors conclude with a discussion of how teachers may use this CDST perspective to reflect on three aspects of their professional context—teaching, language, and agency—and how such reflection, followed by action, may promote opportunities for language learning in their contexts.

Frequency criterion for stability «as a whole» of modulation-type pulse voltage converters according to Lyapunov

Background. The use of methods for researching and designing stable nonlinear dynamic systems by constructing the frequency characteristics of the loop gain of the negative feedback of a pulsed voltage converter through time characteristics obtained by numerical methods for solving nonlinear differential equations that describe electromagnetic processes in the converter takes a lot of time and has low accuracy in calculating optimal modes, when you need to calculate hundreds of options.
 Aim. Therefore, it seems necessary to further develop the theory of stability of modulation-type pulse voltage converters using frequency stability criteria, Lyapunov functions, amplitude and phase frequency characteristics of the converter loop gain.
 Methods. Stability in the «small» is solved using linearized models, which are constructed using the so-called first Lyapunov method - the «first approximation» method. The second method for studying the stability of nonlinear dynamic systems is the harmonic linearization method, when the nonlinear element is replaced by a harmonically linearized link with a transmission coefficient only for the first harmonic.
 Results. The Lyapunov function for a pulse voltage converter with nonlinear relay-type elements was found. Conclusion. The results obtained in the work make it possible to simplify the design of Lyapunov-stable pulse voltage converters through the use of the method of amplitude-frequency characteristics used for stationary linear systems.

Optimum Coordination of Multi‐Band Power System Stabilizers and STATCOM Using the Modified Particle Swarm Algorithm

In this paper, to improve the small signal stability of the power system, the optimal coordination of multi‐band power system stabilizers (MB‐PSSs) and static synchronous compensator (STATCOM) has been done. The study is conducted on a standard four machines and two areas power system. To create coordination, a modified version of the particle swarm algorithm as well as the integral time absolute error (ITAE) criterion are proposed. To evaluate the proposed stability method in different operating conditions and in the form of two scenarios, a three‐phase fault with a duration of 200 ms has been applied in the communication line between the two areas. After the simulation, the values of overshoot, undershoot, settling time, and ITAE criteria for the proposed stability method have been calculated and their values have been compared with the results of power system stabilizer (PSS) and MB‐PSS stabilization methods. The results indicate that the proposed combined method of MB‐PSS and STATCOM performed better than the other two methods in improving the dynamic stability of the power system and has the lowest value of the criteria. The values of ITAE for the PSS, MB‐PSS stabilization methods and the proposed MB‐PSS and STATCOM combined method in the first scenario are equal to 0.002354, 0.0.001496, and 0.001242 p.u. and in the second scenario it is equal to 0.0.007418, 0.004015, and 0.003435 p.u., respectively.

A new approach to solving isomorphism problems of classical dynamical systems using algebraic structures

Abstract There exists a fixed rule in classical dynamical systems that describes a point in geometric space over time. In this paper, based on the algebraic structure perspective, the dynamical system is defined as a category characterized by ordered state projections, and the dynamical system is inscribed using the algebraic structure, covering the phase space, continuous self-maps containing a single parametric variable, and the dynamical system itself. Meanwhile, two types of self-isomorphisms of algebraic maps are explored. One is the self-isomorphism of ideal inclusion maps on an algebra Mn(Fq) consisting of full matrices of order n over a finite field Fq . The second is the self-isomorphism of ideal relational graphs on a finite field Fq . It is proved that any self-isomorphism problem of graph Mn (Fq ) when n >3 can be used with both criteria on it. Finally, a classical model of a dynamical system obtained from f(x) = cos x iterations is studied and its global convergence is discussed. Numerical experimental results show that the discrete dynamical system generated by function f(x) = cos x iteration has a unique ω limit point of 0.735, indicating that the stability and predictability of classical dynamical systems can be achieved using algebraic structures, as well as revealing the complexity, instability, and chaos of the system.

МОДЕЛЬ ИМПУЛЬСНОГО УПРАВЛЕНИЯ УСТОЙЧИВОСТЬЮ СИСТЕМЫ ИНФОРМАЦИОННОЙ БЕЗОПАСНОСТИ

The article considers in detail issues related to organizing management of the stability in information security systems. A comparative analysis of literature sources was carried out to describe current methods, tools and algorithms for managing the stability of operation and security of information systems based on various management models. It also considers the main approaches to formalizing the control of the structural dynamics of information security systems based on the mathematical theory of reachability sets, as well as issues of ensuring comprehensive information security. The authors propose approaches to the construction of models for strategic and tactical management of the loss of stability of an information security system (ISS) under conditions of its natural wear and tear, as well as in the presence of external disturbing influences of a random impulse. There are given dynamic models that implement the principles of highly effective control of the stability in an information security system when it is upset, as well as in the presence of external disturbing influences of a random impulse nature. The basic principles of strategic management of the stability in the information security system are formed based on the criteria-based approach. Formation of tactical control of its stability was carried out based on regulation by deviation. The concept of pulse control using a regulator is proposed to maintain the dynamic stability of security information systems.

Characterization of stability of dynamic particle ensemble systems using topological data analysis.

Holes are ubiquitous structures in phase space, and their time evolution could indicate an instability in the dynamics of the system. However, the properties of these holes are difficult to study directly due to their theoretical complexity and lack of computational tools. This study proposes the use of persistent homology (PH), a technique from topological data analysis, as a computational tool for analyzing the properties of these phase-space holes, or more formally the H1 homology class according to PH. Initially, by using a toy data set, it is shown that the time evolution and the growth rate of a H1 class in phase space could be obtained by PH. For further validation, PH is applied to particle ensemble systems, such as the Hamiltonian flow and the two-stream instability (TSI). Both the stable case, where no H1 forms, and the unstable case, where H1 forms, were analyzed. It was shown that PH can distinguish between the stable and unstable cases purely from the phase-space time evolution plots. In unstable TSI, the PH also distinguished the transition of the H1 class from linear to non-linear growth. The growth rate, thus, obtained is in excellent agreement with the growth rate of the particle energy in the TSI system.

Investigating Global Stability and Bifurcation in an Ecological Dynamical System

We consider a continuous‐time model describing the interaction between phytoplankton and zooplankton using a Holling type‐II response. We then transform this continuous‐time model into a discrete‐time counterpart using a fractional‐order discretization method. The paper explores the local stability of this obtained system concerning all equilibrium points and establishes the global asymptotic stability of its positive fixed point. The study also demonstrates that, under specific mathematical conditions, the system undergoes a Neimark–Sacker bifurcation around its positive equilibrium point. To effectively manage this bifurcation, two modified hybrid control techniques are introduced. The paper concludes by presenting illustrative numerical examples that validate the theoretical findings and assess the effectiveness and feasibility of the newly proposed control strategies. In addition, a comparative analysis is conducted between the modified hybrid techniques and an existing hybrid approach.

Conical Transition Graphs for Analysis of Asymptotic Stability in Hybrid Dynamical Systems

A method is proposed for analyzing asymptotic stability in the conical approximation of a hybrid system. Specifically, this paper introduces the conical transition graph (CTG) to simplify the analysis of asymptotic stability in conical approximations by converting solutions to a hybrid system into walks through a discrete graph. By exploiting the fact that pre-asymptotic stability in a conical approximation implies pre-asymptotic stability in the original system, a CTG-based approach can establish asymptotic stability in hybrid systems that have nonlinear flow maps and jump maps without needing to construct a Lyapunov function.

On the existence, stability and chaos analysis of a novel 4D atmospheric dynamical system in the context of the Caputo fractional derivatives

<p>In this study, changes in westerly waves and their connections to increased global warming under the influence of greenhouse gases were investigated via a Caputo fractional four-dimensional atmospheric system. The idea of the existence of chaotic behavior in the westerly wind's motion was depicted. It has been noted that westerlies are becoming stronger due to rising air temperatures. An analysis of the existence, uniqueness, boundedness, stability of equilibrium points, and conservative behavior of the solutions was conducted. To prove the existence of chaos in the modified model, the Lyapunov exponents, Poincaré map, and bifurcation were computed. A sliding mode controller to control the chaos in this novel fractional-order system was designed, and conditions for the global stability of the controlled system with and without external disturbances and uncertainties were derived. The finite-time interval for the system to reach the sliding surface was computed. The developed controller's performance was evaluated with respect to both commensurate and non-commensurate fractional derivatives. In each scenario, the impact of fractional orders was investigated. Numerical simulations were used to support theoretical statements about how the controller affects the system.</p>

ИНТЕГРАЦИЯ МЕТОДА ФУНКЦИЙ ЛЯПУНОВА ДЛЯ АНАЛИЗА УСТОЙЧИВОСТИ ЭЛЕКТРООБОРУДОВАНИЯ АПК

The functioning of agro-industrial enterprises is considered from electrical equipment operation, quality and stability of automated controlled systems. Energy consumption management is an important task of agro-industrial enterprises, directly related to the stability of electric energy conversion devices. The stability analysis model implemented by the authors makes it possible to improve the quality of control of electrical equipment. (Research purpose) The research purpose is developing a method for ensuring the stability of electrical equipment of agricultural enterprises using Lyapunov functions. (Materials and methods) Proposed the integration of the Lyapunov function method into the agro-industrial complex management systems for stability analysis and forecasting of the state of electrical equipment. The stator currents of the drive motor and the speed characteristics of the rotor rotation were analyzed. We studied the electric motor as a second-order link, evaluated the equilibrium position of the system under consideration, and discussed the stability range of the system described by a homogeneous differential equation. The method of Lyapunov functions was used to assess the stability of dynamic systems. This allows stability analysis and determining critical points of failure of electrical equipment. (Results and discussion) It was determined that the stability conditions found out using Lyapunov functions depend both on the selected function V(x) and on the equations of the system in state variables. Practical examples of the integration of the Lyapunov function method into automated agricultural production systems, which demonstrate an increase in the stability of electrical equipment, were given. (Conclusions) The possibility and method of using Lyapunov functions for estimating the parameters of an electrical control system have been shown, which makes it possible to successfully solve the problem under consideration, and to propose this approach for solving similar applied problems related to estimating the parameters of ordinary differential equations. It was revealed that the proposed method has shown its efficiency and can be used to automate control facilities in agricultural production.

Influence of incubation delays on COVID-19 transmission in diabetic and non-diabetic populations – an endemic prevalence case

Abstract The study of dynamics of diabetic population infected by COVID-19 is of pressing concern as people with diabetes are considered to be at higher risk of severe illness from COVID-19. A three-compartment mathematical model to describe the interactions of diabetic population and non-diabetic population both infected by COVID-19 with a susceptible population is considered. Time delays in incubation periods of COVID-19 in diabetic and non-diabetic populations are introduced. Besides the basic properties of such a dynamical system, both local and global stability of endemic equilibrium, are studied. The lengths of time delays are estimated for which the stability of the system is preserved locally, while sufficient conditions on system parameters are obtained for global stability. Numerical examples are provided to establish the theory, and simulations are provided to visualize the examples. It is noted that an increase in length of time delay in either of infected populations leads to oscillations in susceptible population but has no impact on infected populations.

A new study for global dynamics and numerical simulation of a discrete-time computer virus propagation model

Truong Ha Hai, Pham Hoai Thu, Hoang Manh Tuan Abstract— This work is devoted to conducting a new study for global dynamics and numerical simulation of a discrete-time computer virus propagation model, which was constructed in our recent work. By utilizing well-known results on asymptotic stability of discrete-time dynamical systems, we establish the global asymptotic stability of a unique viral equilibrium point, whereas only its local asymptotic stability was previously analyzed. After that, we investigate convergence and provide an error bound for the discrete-time model. Next, the step doubling strategy is applied to control errors. The result is that the accuracy of approximations generated by the discrete-time model is enhanced. The obtained results not only improve the ones constructed in the benchmark work, but also can be useful to study reliable numerical methods for mathematical models of malware. Finally, we present two numerical experiments that support and illustrate the theoretical findings of this study.

Nonlinear Vibration Analysis of Bistable Morphing Composite Laminated Shell Based on High-Dimensional Dynamical Model

The bistable morphing composite laminated shell presents promising potential for aerospace morphing structures design, as it is capable of shape-changing under small energy inputs and maintaining two stable states without external loading. The bistable nonlinearity is distinct from the typical geometrical nonlinearity, as it arises from thermal residual stresses that cannot be equalized. During the derivation of a three-degree-of-freedom nonlinear dynamic model, Sanders’ nonlinear strain is incorporated into the strain–displacement relationship, while thermal residual stresses are taken into account in the constitutive relationship. The nonlinear dynamic model effectively captures the snap-through phenomena with orthogonal asymmetry in bistable morphing composite laminated shells. In order to reveal the resonance response and chaotic characteristics due to the interaction of vibrations between degrees of freedom, high-dimensional nonlinear dynamical model is derived by multiple scales method based on the 1:1:1 internal resonance relationship. The resonance response curves reflect the parametric influence law of the dynamical system stability. The nonlinear dynamic behaviors are described by the bifurcation diagrams and the maximum Lyapunov exponent. The significant motions under particular excitation conditions are visualized by phase portraits, time histories and Poincaré maps.

Quantifying the underlying landscape, entropy production and biological path of the cell fate decision between apoptosis and pyroptosis

Pyroptosis, a recently identified type of cell death, and apoptosis are fundamental and complex biological processes that underlie a multitude of diseases. However, the crosstalk between pyroptosis and apoptosis, as well as their molecular regulatory mechanisms governing decision-making, remain to be fully elucidated. In particular, comprehending how internal driving forces, including vital components of life such as proteins and reactions, collaborate with the environment to dictate cell fate remains an ongoing challenge. To address these issues, a cell death decision module model of the crosstalk between pyroptosis and apoptosis was developed. Stability analysis revealed the presence of three steady-state attractors within the death decision system: the apoptosis state attractor, the pyroptosis state attractor, and the concurrence state attractor. Landscape theory was employed to study the stochastic dynamic and global stability of the death decision system, allowing us to quantitatively describe the uncertainty of cell death decisions by measuring the production of Shannon entropy. In addition, we identified the dominant kinetic paths among different death mode attractors. We found that the dominant kinetic paths between different cell death modes usually do not pass through the minimum potential point. Through quantifying the underlying driving force of the system's dynamics, we determined that this phenomenon is attributed to the dependence of the driving force in non-equilibrium systems on both the gradient force and the curl force. In summary, this study offers a natural and physical groundwork for understanding the crosstalk network between pyroptosis and apoptosis, providing valuable insights and therapeutic strategies for the regulation of cell death modes in mammals.

A Study on Some Properties for Weak Stability of Non-Autonomous Discrete Dynamical Systems

The interest for non-autonomous discrete dynamical systems has been increasing in recent years, because they are adequate to tell of real activities. For examples, when the mapping is disturbed in each iteration because of external factors, or model of some phenomena in physics, biology and economy, in specific, the population of human and weather forecasting, plus to solve problems generated in mathematics. In mathematics, stability theory addresses the stability of solutions of trajectories of dynamical systems under small perturbations of initial conditions. Besides that, the topological dynamics is the main method used in this paper, since we studied the non-autonomous discrete dynamical systems on a topological space. Next, we present a conception of weak stability (stableness) of non-autonomous discrete dynamical systems (NADDS). We evaluate a collection of weak stable points, and survey the connection among weak stableness and shadowing property. We also consider the connection among weak stableness of a non-autonomous discrete dynamical system and its induced system. In conclusion, the results of this paper will provide some helps in the modelling of a general dynamical system.