Entropy

Research

14 pages, 3415 KiB

Open AccessArticle

Dynamics of Hopfield-Type Neural Networks with Modulo Periodic Unpredictable Synaptic Connections, Rates and Inputs

by Marat Akhmet, Madina Tleubergenova and Akylbek Zhamanshin

Entropy 2022, 24(11), 1555; https://doi.org/10.3390/e24111555 - 29 Oct 2022

Cited by 1 | Viewed by 1185

In this paper, we rigorously prove that unpredictable oscillations take place in the dynamics of Hopfield-type neural networks (HNNs) when synaptic connections, rates and external inputs are modulo periodic unpredictable. The synaptic connections, rates and inputs are synchronized to obtain the convergence of [...] Read more.

In this paper, we rigorously prove that unpredictable oscillations take place in the dynamics of Hopfield-type neural networks (HNNs) when synaptic connections, rates and external inputs are modulo periodic unpredictable. The synaptic connections, rates and inputs are synchronized to obtain the convergence of outputs on the compact subsets of the real axis. The existence, uniqueness, and exponential stability of such motions are discussed. The method of included intervals and the contraction mapping principle are applied to attain the theoretical results. In addition to the analysis, we have provided strong simulation arguments, considering that all the assumed conditions are satisfied. It is shown how a new parameter, degree of periodicity, affects the dynamics of the neural network. Full article

(This article belongs to the Special Issue Dynamical Systems, Differential Equations and Applications)

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18 pages, 2060 KiB

Open AccessArticle

Synchronization in Finite-Time of Delayed Fractional-Order Fully Complex-Valued Dynamical Networks via Non-Separation Method

by Qiaokun Kang, Qingxi Yang, Jing Yang, Qintao Gan and Ruihong Li

Entropy 2022, 24(10), 1460; https://doi.org/10.3390/e24101460 - 13 Oct 2022

Cited by 3 | Viewed by 1335

Abstract

The finite-time synchronization (FNTS) problem for a class of delayed fractional-order fully complex-valued dynamic networks (FFCDNs) with internal delay and non-delayed and delayed couplings is studied by directly constructing Lyapunov functions instead of decomposing the original complex-valued networks into two real-valued networks. Firstly, [...] Read more.

The finite-time synchronization (FNTS) problem for a class of delayed fractional-order fully complex-valued dynamic networks (FFCDNs) with internal delay and non-delayed and delayed couplings is studied by directly constructing Lyapunov functions instead of decomposing the original complex-valued networks into two real-valued networks. Firstly, a mixed delay fractional-order mathematical model is established for the first time as fully complex-valued, where the outer coupling matrices of the model are not restricted to be identical, symmetric, or irreducible. Secondly, to overcome the limitation of the use range of a single controller, two delay-dependent controllers are designed based on the complex-valued quadratic norm and the norm composed of its real and imaginary parts’ absolute values, respectively, to improve the synchronization control efficiency. Besides, the relationships between the fractional order of the system, the fractional-order power law, and the settling time (ST) are analyzed. Finally, the feasibility and effectiveness of the control method designed in this paper are verified by numerical simulation. Full article

(This article belongs to the Special Issue Dynamical Systems, Differential Equations and Applications)

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15 pages, 4740 KiB

Open AccessArticle

Multiple Reflections for Classical Particles Moving under the Influence of a Time-Dependent Potential Well

by Flávio Heleno Graciano, Diogo Ricardo da Costa, Edson D. Leonel and Juliano A. de Oliveira

Entropy 2022, 24(10), 1427; https://doi.org/10.3390/e24101427 - 7 Oct 2022

Viewed by 1214

Abstract

We study the dynamics of classical particles confined in a time-dependent potential well. The dynamics of each particle is described by a two-dimensional nonlinear discrete mapping for the variables energy

e_{n}

and phase

ϕ_{n}

of the periodic moving well. We obtain [...] Read more.

We study the dynamics of classical particles confined in a time-dependent potential well. The dynamics of each particle is described by a two-dimensional nonlinear discrete mapping for the variables energy

e_{n}

and phase

ϕ_{n}

of the periodic moving well. We obtain the phase space and show that it contains periodic islands, chaotic sea, and invariant spanning curves. We find the elliptic and hyperbolic fixed points and discuss a numerical method to obtain them. We study the dispersion of the initial conditions after a single iteration. This study allows finding regions where multiple reflections occur. Multiple reflections happen when a particle does not have enough energy to exit the potential well and is trapped inside it, suffering several reflections until it has enough energy to exit. We also show deformations in regions with multiple reflection, but the area remains constant when we change the control parameter

N_{C}

. Finally, we show some structures that appear in the

e_{0} e_{1}

plane by using density plots. Full article

(This article belongs to the Special Issue Dynamical Systems, Differential Equations and Applications)

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13 pages, 296 KiB

Open AccessArticle

Positive Periodic Solution for Second-Order Nonlinear Differential Equations with Variable Coefficients and Mixed Delays

by Zejian Dai and Bo Du

Entropy 2022, 24(9), 1286; https://doi.org/10.3390/e24091286 - 12 Sep 2022

Viewed by 1553

Abstract

In this paper, we study two types of second-order nonlinear differential equations with variable coefficients and mixed delays. Based on Krasnoselskii’s fixed point theorem, the existence results of positive periodic solution are established. It should be pointed out that the equations we studied [...] Read more.

In this paper, we study two types of second-order nonlinear differential equations with variable coefficients and mixed delays. Based on Krasnoselskii’s fixed point theorem, the existence results of positive periodic solution are established. It should be pointed out that the equations we studied are more general. Therefore, the results of this paper have better applicability. Full article

(This article belongs to the Special Issue Dynamical Systems, Differential Equations and Applications)

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9 pages, 406 KiB

Open AccessArticle

Laguerre Wavelet Approach for a Two-Dimensional Time–Space Fractional Schrödinger Equation

by Stelios Bekiros, Samaneh Soradi-Zeid, Jun Mou, Amin Yousefpour, Ernesto Zambrano-Serrano and Hadi Jahanshahi

Entropy 2022, 24(8), 1105; https://doi.org/10.3390/e24081105 - 11 Aug 2022

Cited by 1 | Viewed by 1308

Abstract

This article is devoted to the determination of numerical solutions for the two-dimensional time–spacefractional Schrödinger equation. To do this, the unknown parameters are obtained using the Laguerre wavelet approach. We discretize the problem by using this technique. Then, we solve the discretized nonlinear [...] Read more.

This article is devoted to the determination of numerical solutions for the two-dimensional time–spacefractional Schrödinger equation. To do this, the unknown parameters are obtained using the Laguerre wavelet approach. We discretize the problem by using this technique. Then, we solve the discretized nonlinear problem by means of a collocation method. The method was proven to give very accurate results. The given numerical examples support this claim. Full article

(This article belongs to the Special Issue Dynamical Systems, Differential Equations and Applications)

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13 pages, 855 KiB

Open AccessArticle

Axisymmetric Fractional Diffusion with Mass Absorption in a Circle under Time-Harmonic Impact

by Yuriy Povstenko and Tamara Kyrylych

Entropy 2022, 24(7), 1002; https://doi.org/10.3390/e24071002 - 20 Jul 2022

Cited by 1 | Viewed by 1567

Abstract

The axisymmetric time-fractional diffusion equation with mass absorption is studied in a circle under the time-harmonic Dirichlet boundary condition. The Caputo derivative of the order

0 < α \leq 2

is used. The investigated equation can be considered as the time-fractional generalization of [...] Read more.

The axisymmetric time-fractional diffusion equation with mass absorption is studied in a circle under the time-harmonic Dirichlet boundary condition. The Caputo derivative of the order

0 < α \leq 2

is used. The investigated equation can be considered as the time-fractional generalization of the bioheat equation and the Klein–Gordon equation. Different formulations of the problem for integer values of the time-derivatives

α = 1

and

α = 2

are also discussed. The integral transform technique is employed. The outcomes of numerical calculations are illustrated graphically for different values of the parameters. Full article

(This article belongs to the Special Issue Dynamical Systems, Differential Equations and Applications)

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18 pages, 328 KiB

Open AccessArticle

On an Approximate Solution of the Cauchy Problem for Systems of Equations of Elliptic Type of the First Order

by Davron Aslonqulovich Juraev, Ali Shokri and Daniela Marian

Entropy 2022, 24(7), 968; https://doi.org/10.3390/e24070968 - 13 Jul 2022

Cited by 7 | Viewed by 1386

Abstract

In this paper, on the basis of the Carleman matrix, we explicitly construct a regularized solution of the Cauchy problem for the matrix factorization of Helmholtz’s equation in an unbounded two-dimensional domain. The focus of this paper is on regularization formulas for solutions [...] Read more.

In this paper, on the basis of the Carleman matrix, we explicitly construct a regularized solution of the Cauchy problem for the matrix factorization of Helmholtz’s equation in an unbounded two-dimensional domain. The focus of this paper is on regularization formulas for solutions to the Cauchy problem. The question of the existence of a solution to the problem is not considered—it is assumed a priori. At the same time, it should be noted that any regularization formula leads to an approximate solution of the Cauchy problem for all data, even if there is no solution in the usual classical sense. Moreover, for explicit regularization formulas, one can indicate in what sense the approximate solution turns out to be optimal. Full article

(This article belongs to the Special Issue Dynamical Systems, Differential Equations and Applications)

16 pages, 316 KiB

Open AccessArticle

Heterogeneous Diffusion and Nonlinear Advection in a One-Dimensional Fisher-KPP Problem

by José Luis Díaz Palencia, Saeed ur Rahman and Antonio Naranjo Redondo

Entropy 2022, 24(7), 915; https://doi.org/10.3390/e24070915 - 30 Jun 2022

Cited by 7 | Viewed by 1474

Abstract

The goal of this study is to provide an analysis of a Fisher-KPP non-linear reaction problem with a higher-order diffusion and a non-linear advection. We study the existence and uniqueness of solutions together with asymptotic solutions and positivity conditions. We show the existence [...] Read more.

The goal of this study is to provide an analysis of a Fisher-KPP non-linear reaction problem with a higher-order diffusion and a non-linear advection. We study the existence and uniqueness of solutions together with asymptotic solutions and positivity conditions. We show the existence of instabilities based on a shooting method approach. Afterwards, we study the existence and uniqueness of solutions as an abstract evolution of a bounded continuous single parametric (t) semigroup. Asymptotic solutions based on a Hamilton–Jacobi equation are then analyzed. Finally, the conditions required to ensure a comparison principle are explored supported by the existence of a positive maximal kernel. Full article

(This article belongs to the Special Issue Dynamical Systems, Differential Equations and Applications)

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10 pages, 259 KiB

Open AccessArticle

The Retentivity of Four Kinds of Shadowing Properties in Non-Autonomous Discrete Dynamical Systems

by Yongxi Jiang, Tianxiu Lu, Jingmin Pi and Waseem Anwar

Entropy 2022, 24(3), 397; https://doi.org/10.3390/e24030397 - 12 Mar 2022

Cited by 5 | Viewed by 1644

Abstract

In this paper, four kinds of shadowing properties in non-autonomous discrete dynamical systems (NDDSs) are discussed. It is pointed out that if an NDDS has a

F

-shadowing property (resp. ergodic shadowing property,

\bar{d}

shadowing property,

\underset{̲}{d}

shadowing property), then the [...] Read more.

In this paper, four kinds of shadowing properties in non-autonomous discrete dynamical systems (NDDSs) are discussed. It is pointed out that if an NDDS has a

F

-shadowing property (resp. ergodic shadowing property,

\bar{d}

shadowing property,

\underset{̲}{d}

shadowing property), then the compound systems, conjugate systems, and product systems all have accordant shadowing properties. Moreover, the set-valued system

(K (X), {\bar{f}}_{1, \infty})

induced by the NDDS

(X, f_{1, \infty})

has the above four shadowing properties, implying that the NDDS

(X, f_{1, \infty})

has these properties. Full article

(This article belongs to the Special Issue Dynamical Systems, Differential Equations and Applications)

16 pages, 327 KiB

Open AccessFeature PaperArticle

Impulsive Reaction-Diffusion Delayed Models in Biology: Integral Manifolds Approach

by Gani Stamov, Ivanka Stamova and Cvetelina Spirova

Entropy 2021, 23(12), 1631; https://doi.org/10.3390/e23121631 - 3 Dec 2021

Cited by 7 | Viewed by 2399

Abstract

In this paper we study an impulsive delayed reaction-diffusion model applied in biology. The introduced model generalizes existing reaction-diffusion delayed epidemic models to the impulsive case. The integral manifolds notion has been introduced to the model under consideration. This notion extends the single [...] Read more.

In this paper we study an impulsive delayed reaction-diffusion model applied in biology. The introduced model generalizes existing reaction-diffusion delayed epidemic models to the impulsive case. The integral manifolds notion has been introduced to the model under consideration. This notion extends the single state notion and has important applications in the study of multi-stable systems. By means of an extension of the Lyapunov method integral manifolds’ existence, results are established. Based on the Lyapunov functions technique combined with a Poincarè-type inequality qualitative criteria related to boundedness, permanence, and stability of the integral manifolds are also presented. The application of the proposed impulsive control model is closely related to a most important problems in the mathematical biology—the problem of optimal control of epidemic models. The considered impulsive effects can be used by epidemiologists as a very effective therapy control strategy. In addition, since the integral manifolds approach is relevant in various contexts, our results can be applied in the qualitative investigations of many problems in the epidemiology of diverse interest. Full article

(This article belongs to the Special Issue Dynamical Systems, Differential Equations and Applications)

24 pages, 494 KiB

Open AccessArticle

Extension of Operational Matrix Technique for the Solution of Nonlinear System of Caputo Fractional Differential Equations Subjected to Integral Type Boundary Constrains

by Hammad Khalil, Murad Khalil, Ishak Hashim and Praveen Agarwal

Entropy 2021, 23(9), 1154; https://doi.org/10.3390/e23091154 - 2 Sep 2021

Cited by 5 | Viewed by 2538

Abstract

We extend the operational matrices technique to design a spectral solution of nonlinear fractional differential equations (FDEs). The derivative is considered in the Caputo sense. The coupled system of two FDEs is considered, subjected to more generalized integral type conditions. The basis of [...] Read more.

We extend the operational matrices technique to design a spectral solution of nonlinear fractional differential equations (FDEs). The derivative is considered in the Caputo sense. The coupled system of two FDEs is considered, subjected to more generalized integral type conditions. The basis of our approach is the most simple orthogonal polynomials. Several new matrices are derived that have strong applications in the development of computational scheme. The scheme presented in this article is able to convert nonlinear coupled system of FDEs to an equivalent S-lvester type algebraic equation. The solution of the algebraic structure is constructed by converting the system into a complex Schur form. After conversion, the solution of the resultant triangular system is obtained and transformed back to construct the solution of algebraic structure. The solution of the matrix equation is used to construct the solution of the related nonlinear system of FDEs. The convergence of the proposed method is investigated analytically and verified experimentally through a wide variety of test problems. Full article

(This article belongs to the Special Issue Dynamical Systems, Differential Equations and Applications)

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33 pages, 6636 KiB

Open AccessArticle

Secret Communication Systems Using Chaotic Wave Equations with Neural Network Boundary Conditions

by Yuhan Chen, Hideki Sano, Masashi Wakaiki and Takaharu Yaguchi

Entropy 2021, 23(7), 904; https://doi.org/10.3390/e23070904 - 16 Jul 2021

Cited by 8 | Viewed by 2665

Abstract

In a secret communication system using chaotic synchronization, the communication information is embedded in a signal that behaves as chaos and is sent to the receiver to retrieve the information. In a previous study, a chaotic synchronous system was developed by integrating the [...] Read more.

In a secret communication system using chaotic synchronization, the communication information is embedded in a signal that behaves as chaos and is sent to the receiver to retrieve the information. In a previous study, a chaotic synchronous system was developed by integrating the wave equation with the van der Pol boundary condition, of which the number of the parameters are only three, which is not enough for security. In this study, we replace the nonlinear boundary condition with an artificial neural network, thereby making the transmitted information difficult to leak. The neural network is divided into two parts; the first half is used as the left boundary condition of the wave equation and the second half is used as that on the right boundary, thus replacing the original nonlinear boundary condition. We also show the results for both monochrome and color images and evaluate the security performance. In particular, it is shown that the encrypted images are almost identical regardless of the input images. The learning performance of the neural network is also investigated. The calculated Lyapunov exponent shows that the learned neural network causes some chaotic vibration effect. The information in the original image is completely invisible when viewed through the image obtained after being concealed by the proposed system. Some security tests are also performed. The proposed method is designed in such a way that the transmitted images are encrypted into almost identical images of waves, thereby preventing the retrieval of information from the original image. The numerical results show that the encrypted images are certainly almost identical, which supports the security of the proposed method. Some security tests are also performed. The proposed method is designed in such a way that the transmitted images are encrypted into almost identical images of waves, thereby preventing the retrieval of information from the original image. The numerical results show that the encrypted images are certainly almost identical, which supports the security of the proposed method. Full article

(This article belongs to the Special Issue Dynamical Systems, Differential Equations and Applications)

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19 pages, 1684 KiB

Open AccessArticle

Solutions of the Multivariate Inverse Frobenius–Perron Problem

by Colin Fox, Li-Jen Hsiao and Jeong-Eun (Kate) Lee

Entropy 2021, 23(7), 838; https://doi.org/10.3390/e23070838 - 30 Jun 2021

Cited by 2 | Viewed by 1859

Abstract

We address the inverse Frobenius–Perron problem: given a prescribed target distribution

ρ

, find a deterministic map M such that iterations of M tend to

ρ

in distribution. We show that all solutions may be written in terms of a factorization that combines [...] Read more.

We address the inverse Frobenius–Perron problem: given a prescribed target distribution

ρ

, find a deterministic map M such that iterations of M tend to

ρ

in distribution. We show that all solutions may be written in terms of a factorization that combines the forward and inverse Rosenblatt transformations with a uniform map; that is, a map under which the uniform distribution on the d-dimensional hypercube is invariant. Indeed, every solution is equivalent to the choice of a uniform map. We motivate this factorization via one-dimensional examples, and then use the factorization to present solutions in one and two dimensions induced by a range of uniform maps. Full article

(This article belongs to the Special Issue Dynamical Systems, Differential Equations and Applications)

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14 pages, 561 KiB

Open AccessArticle

Dynamical Invariant and Exact Mechanical Analyses for the Caldirola–Kanai Model of Dissipative Three Coupled Oscillators

by Salim Medjber, Salah Menouar and Jeong Ryeol Choi

Entropy 2021, 23(7), 837; https://doi.org/10.3390/e23070837 - 30 Jun 2021

Cited by 8 | Viewed by 1876

Abstract

We study the dynamical invariant for dissipative three coupled oscillators mainly from the quantum mechanical point of view. It is known that there are many advantages of the invariant quantity in elucidating mechanical properties of the system. We use such a property of [...] Read more.

We study the dynamical invariant for dissipative three coupled oscillators mainly from the quantum mechanical point of view. It is known that there are many advantages of the invariant quantity in elucidating mechanical properties of the system. We use such a property of the invariant operator in quantizing the system in this work. To this end, we first transform the invariant operator to a simple one by using a unitary operator in order that we can easily manage it. The invariant operator is further simplified through its diagonalization via three-dimensional rotations parameterized by three Euler angles. The coupling terms in the quantum invariant are eventually eliminated thanks to such a diagonalization. As a consequence, transformed quantum invariant is represented in terms of three independent simple harmonic oscillators which have unit masses. Starting from the wave functions in the transformed system, we have derived the full wave functions in the original system with the help of the unitary operators. Full article

(This article belongs to the Special Issue Dynamical Systems, Differential Equations and Applications)

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36 pages, 495 KiB

Open AccessArticle

Simple Equations Method (SEsM): Algorithm, Connection with Hirota Method, Inverse Scattering Transform Method, and Several Other Methods

by Nikolay K. Vitanov, Zlatinka I. Dimitrova and Kaloyan N. Vitanov

Entropy 2021, 23(1), 10; https://doi.org/10.3390/e23010010 - 23 Dec 2020

Cited by 34 | Viewed by 3368

Abstract

The goal of this article is to discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations and to show that several well-known methods for obtaining exact solutions of such equations are connected to SEsM. In more detail, [...] Read more.

The goal of this article is to discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations and to show that several well-known methods for obtaining exact solutions of such equations are connected to SEsM. In more detail, we show that the Hirota method is connected to a particular case of SEsM for a specific form of the function from Step 2 of SEsM and for simple equations of the kinds of differential equations for exponential functions. We illustrate this particular case of SEsM by obtaining the three- soliton solution of the Korteweg-de Vries equation, two-soliton solution of the nonlinear Schrödinger equation, and the soliton solution of the Ishimori equation for the spin dynamics of ferromagnetic materials. Then we show that a particular case of SEsM can be used in order to reproduce the methodology of the inverse scattering transform method for the case of the Burgers equation and Korteweg-de Vries equation. This particular case is connected to use of a specific case of Step 2 of SEsM. This step is connected to: (i) representation of the solution of the solved nonlinear partial differential equation as expansion as power series containing powers of a “small” parameter

ϵ

; (ii) solving the differential equations arising from this representation by means of Fourier series, and (iii) transition from the obtained solution for small values of

ϵ

to solution for arbitrary finite values of

ϵ

. Finally, we show that the much-used homogeneous balance method, extended homogeneous balance method, auxiliary equation method, Jacobi elliptic function expansion method, F-expansion method, modified simple equation method, trial function method and first integral method are connected to particular cases of SEsM. Full article

(This article belongs to the Special Issue Dynamical Systems, Differential Equations and Applications)

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19 pages, 8549 KiB

Open AccessArticle

Dynamical Analysis of a New Chaotic Fractional Discrete-Time System and Its Control

by A. Othman Almatroud, Amina-Aicha Khennaoui, Adel Ouannas, Giuseppe Grassi, M. Mossa Al-sawalha and Ahlem Gasri

Entropy 2020, 22(12), 1344; https://doi.org/10.3390/e22121344 - 27 Nov 2020

Cited by 10 | Viewed by 2135

Abstract

This article proposes a new fractional-order discrete-time chaotic system, without equilibria, included two quadratic nonlinearities terms. The dynamics of this system were experimentally investigated via bifurcation diagrams and largest Lyapunov exponent. Besides, some chaotic tests such as the 0–1 test and approximate entropy [...] Read more.

This article proposes a new fractional-order discrete-time chaotic system, without equilibria, included two quadratic nonlinearities terms. The dynamics of this system were experimentally investigated via bifurcation diagrams and largest Lyapunov exponent. Besides, some chaotic tests such as the 0–1 test and approximate entropy (ApEn) were included to detect the performance of our numerical results. Furthermore, a valid control method of stabilization is introduced to regulate the proposed system in such a way as to force all its states to adaptively tend toward the equilibrium point at zero. All theoretical findings in this work have been verified numerically using MATLAB software package. Full article

(This article belongs to the Special Issue Dynamical Systems, Differential Equations and Applications)

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11 pages, 483 KiB

Open AccessArticle

The Complex Adaptive Delta-Modulator in Sliding Mode Theory

by Dhafer Almakhles

Entropy 2020, 22(8), 814; https://doi.org/10.3390/e22080814 - 25 Jul 2020

Viewed by 1967

Abstract

In this paper, we consider the stability and various dynamical behaviors of both discrete-time delta modulator (

Δ

-M) and adaptive

Δ

-M. The stability constraints and conditions of

Δ

-M and adaptive

Δ

-M are derived following the theory of quasi-sliding mode. [...] Read more.

In this paper, we consider the stability and various dynamical behaviors of both discrete-time delta modulator (

Δ

-M) and adaptive

Δ

-M. The stability constraints and conditions of

Δ

-M and adaptive

Δ

-M are derived following the theory of quasi-sliding mode. Furthermore, the periodic behaviors are explored for both the systems with steady-state inputs and certain parameter values. The results derived in this paper are validated using simulated examples which confirms the derived stability conditions and the existence of periodicity. Full article

(This article belongs to the Special Issue Dynamical Systems, Differential Equations and Applications)

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25 pages, 903 KiB

Open AccessArticle

On Leader-Following Consensus in Multi-Agent Systems with Discrete Updates at Random Times

by Ricardo Almeida, Ewa Girejko, Snezhana Hristova and Agnieszka Malinowska

Entropy 2020, 22(6), 650; https://doi.org/10.3390/e22060650 - 12 Jun 2020

Cited by 3 | Viewed by 2339

Abstract

This paper studies the leader-following consensus problem in continuous-time multi-agent networks with communications/updates occurring only at random times. The time between two consecutive controller updates is exponentially distributed. Some sufficient conditions are derived to design the control law that ensures the leader-following consensus [...] Read more.

This paper studies the leader-following consensus problem in continuous-time multi-agent networks with communications/updates occurring only at random times. The time between two consecutive controller updates is exponentially distributed. Some sufficient conditions are derived to design the control law that ensures the leader-following consensus is asymptotically reached (in the sense of the expected value of a stochastic process). The numerical examples are worked out to demonstrate the effectiveness of our theoretical results. Full article

(This article belongs to the Special Issue Dynamical Systems, Differential Equations and Applications)

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