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26 pages, 836 KiB

Open AccessArticle

A Dynamically Consistent Nonstandard Difference Scheme for a Discrete-Time Immunogenic Tumors Model

by Muhammad Salman Khan, Maria Samreen, Muhammad Asif Khan and Manuel De la Sen

Entropy 2022, 24(7), 949; https://doi.org/10.3390/e24070949 - 07 Jul 2022

Cited by 2 | Viewed by 1148

This manuscript deals with the qualitative study of certain properties of an immunogenic tumors model. Mainly, we obtain a dynamically consistent discrete-time immunogenic tumors model using a nonstandard difference scheme. The existence of fixed points and their stability are discussed. It is shown [...] Read more.

This manuscript deals with the qualitative study of certain properties of an immunogenic tumors model. Mainly, we obtain a dynamically consistent discrete-time immunogenic tumors model using a nonstandard difference scheme. The existence of fixed points and their stability are discussed. It is shown that a continuous system experiences Hopf bifurcation at one and only one positive fixed point, whereas its discrete-time counterpart experiences Neimark–Sacker bifurcation at one and only one positive fixed point. It is shown that there is no chance of period-doubling bifurcation in our discrete-time system. Additionally, numerical simulations are carried out in support of our theoretical discussion. Full article

(This article belongs to the Special Issue Nonlinear Dynamics and Analysis)

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20 pages, 3760 KiB

Open AccessArticle

Disturbance-Observer-Based U-Control (DOBUC) for Nonlinear Dynamic Systems

by Ruobing Li, Quanmin Zhu, Jun Yang, Pritesh Narayan and Xicai Yue

Entropy 2021, 23(12), 1625; https://doi.org/10.3390/e23121625 - 02 Dec 2021

Cited by 6 | Viewed by 1718

Abstract

U-model, which is a control-oriented model set with the property of generally facilitate nonlinearity dynamic inversion/cancellation, has been introduced to the Disturbance Observer-Based control (DOBC) methods to improve the performance of the nonlinear systems in this paper. A general DOB based U-Control (DOBUC) [...] Read more.

U-model, which is a control-oriented model set with the property of generally facilitate nonlinearity dynamic inversion/cancellation, has been introduced to the Disturbance Observer-Based control (DOBC) methods to improve the performance of the nonlinear systems in this paper. A general DOB based U-Control (DOBUC) framework is proposed to improve the disturbance attenuation capability of U-controller for both linear and nonlinear systems combined with (based on) the U-model-based dynamic inversion which expands the classical linear disturbance observer control to general nonlinear systems. The proposed two-step DOBUC design procedures in which the design of DOB and U-controller are totally independent and separated, enables the establishment of global exponential stability without being subject to disturbances and uncertainties. Comparative simulation experiments with Nonlinear DOBC in controlling Wind Energy Conversion Systems (WECS) and Permanent Magnet Synchronous Motors (PMSM) demonstrated the proposed method. Full article

(This article belongs to the Special Issue Nonlinear Dynamics and Analysis)

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9 pages, 256 KiB

Open AccessArticle

Some Integral Inequalities Involving Metrics

by Ravi P. Agarwal, Mohamed Jleli and Bessem Samet

Entropy 2021, 23(7), 871; https://doi.org/10.3390/e23070871 - 08 Jul 2021

Cited by 1 | Viewed by 1486

Abstract

In this work, we establish some integral inequalities involving metrics. Moreover, some applications to partial metric spaces are given. Our results are extension of previous obtained metric inequalities in the discrete case. Full article

(This article belongs to the Special Issue Nonlinear Dynamics and Analysis)

9 pages, 271 KiB

Open AccessArticle

Existence of a Unique Solution to a Fractional Partial Differential Equation and Its Continuous Dependence on Parameters

by Robert Stegliński

Entropy 2021, 23(7), 851; https://doi.org/10.3390/e23070851 - 01 Jul 2021

Viewed by 1604

Abstract

In the present paper we give conditions under which there exists a unique weak solution for a nonlocal equation driven by the integrodifferential operator of fractional Laplacian type. We argue for the optimality of some assumptions. Some Lyapunov-type inequalities are given. We also [...] Read more.

In the present paper we give conditions under which there exists a unique weak solution for a nonlocal equation driven by the integrodifferential operator of fractional Laplacian type. We argue for the optimality of some assumptions. Some Lyapunov-type inequalities are given. We also study the continuous dependence of the solution on parameters. In proofs we use monotonicity and variational methods. Full article

(This article belongs to the Special Issue Nonlinear Dynamics and Analysis)

17 pages, 40276 KiB

Open AccessArticle

SUPG-Stabilized Virtual Element Method for Optimal Control Problem Governed by a Convection Dominated Diffusion Equation

by Qiming Wang and Zhaojie Zhou

Entropy 2021, 23(6), 723; https://doi.org/10.3390/e23060723 - 05 Jun 2021

Cited by 1 | Viewed by 2269

Abstract

In this paper, the streamline upwind/Petrov Galerkin (SUPG) stabilized virtual element method (VEM) for optimal control problem governed by a convection dominated diffusion equation is investigated. The virtual element discrete scheme is constructed based on the first-optimize-then-discretize strategy and SUPG stabilized virtual element [...] Read more.

In this paper, the streamline upwind/Petrov Galerkin (SUPG) stabilized virtual element method (VEM) for optimal control problem governed by a convection dominated diffusion equation is investigated. The virtual element discrete scheme is constructed based on the first-optimize-then-discretize strategy and SUPG stabilized virtual element approximation of the state equation and adjoint state equation. An a priori error estimate is derived for both the state, adjoint state, and the control. Numerical experiments are carried out to illustrate the theoretical findings. Full article

(This article belongs to the Special Issue Nonlinear Dynamics and Analysis)

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10 pages, 273 KiB

Open AccessArticle

Stability of Non-Linear Dirichlet Problems with ϕ-Laplacian

by Michał Bełdziński, Marek Galewski and Igor Kossowski

Entropy 2021, 23(6), 647; https://doi.org/10.3390/e23060647 - 22 May 2021

Viewed by 1537

Abstract

We study the stability and the solvability of a family of problems

- {(ϕ (x^{'}))}^{'} = g (t, x, x^{'}, u) + f^{*}

with Dirichlet boundary conditions, where

ϕ

, [...] Read more.

We study the stability and the solvability of a family of problems

- {(ϕ (x^{'}))}^{'} = g (t, x, x^{'}, u) + f^{*}

with Dirichlet boundary conditions, where

ϕ

, u,

f^{*}

are allowed to vary as well. Applications for boundary value problems involving the p-Laplacian operator are highlighted. Full article

(This article belongs to the Special Issue Nonlinear Dynamics and Analysis)

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25 pages, 4730 KiB

Open AccessArticle

Hybrid Control of Digital Baker Map with Application to Pseudo-Random Number Generator

by Yuhui Shi and Yashuang Deng

Entropy 2021, 23(5), 578; https://doi.org/10.3390/e23050578 - 08 May 2021

Cited by 6 | Viewed by 1867

Abstract

Dynamical degradation occurs when chaotic systems are implemented on digital devices, which seriously threatens the security of chaos-based cryptosystems. The existing solutions mainly focus on the compensation of dynamical properties rather than on the elimination of the inherent biases of chaotic systems. In [...] Read more.

Dynamical degradation occurs when chaotic systems are implemented on digital devices, which seriously threatens the security of chaos-based cryptosystems. The existing solutions mainly focus on the compensation of dynamical properties rather than on the elimination of the inherent biases of chaotic systems. In this paper, a unidirectional hybrid control method is proposed to improve the dynamical properties and to eliminate the biases of digital chaotic maps. A continuous chaotic system is introduced to provide external feedback control of the given digital chaotic map. Three different control modes are investigated, and the influence of control parameter on the properties of the controlled system is discussed. The experimental results show that the proposed method can not only improve the dynamical degradation of the digital chaotic map but also make the controlled digital system produce outputs with desirable performances. Finally, a pseudorandom number generator (PRNG) is proposed. Statistical analysis shows that the PRNG has good randomness and almost ideal entropy values. Full article

(This article belongs to the Special Issue Nonlinear Dynamics and Analysis)

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11 pages, 273 KiB

Open AccessArticle

New Criteria on Oscillatory and Asymptotic Behavior of Third-Order Nonlinear Dynamic Equations with Nonlinear Neutral Terms

by Said R. Grace, Jehad Alzabut and Abdullah Özbekler

Entropy 2021, 23(2), 227; https://doi.org/10.3390/e23020227 - 15 Feb 2021

Cited by 4 | Viewed by 1689

Abstract

In the paper, we provide sufficient conditions for the oscillatory and asymptotic behavior of a new type of third-order nonlinear dynamic equations with mixed nonlinear neutral terms. Our theorems not only improve and extend existing theorems in the literature but also provide a [...] Read more.

In the paper, we provide sufficient conditions for the oscillatory and asymptotic behavior of a new type of third-order nonlinear dynamic equations with mixed nonlinear neutral terms. Our theorems not only improve and extend existing theorems in the literature but also provide a new approach as far as the nonlinear neutral terms are concerned. The main results are illustrated by some particular examples. Full article

(This article belongs to the Special Issue Nonlinear Dynamics and Analysis)

10 pages, 756 KiB

Open AccessArticle

Nonlinear Neutral Delay Differential Equations of Fourth-Order: Oscillation of Solutions

by Ravi P. Agarwal, Omar Bazighifan and Maria Alessandra Ragusa

Entropy 2021, 23(2), 129; https://doi.org/10.3390/e23020129 - 20 Jan 2021

Cited by 72 | Viewed by 2692

Abstract

The objective of this paper is to study oscillation of fourth-order neutral differential equation. By using Riccati substitution and comparison technique, new oscillation conditions are obtained which insure that all solutions of the studied equation are oscillatory. Our results complement some known results [...] Read more.

The objective of this paper is to study oscillation of fourth-order neutral differential equation. By using Riccati substitution and comparison technique, new oscillation conditions are obtained which insure that all solutions of the studied equation are oscillatory. Our results complement some known results for neutral differential equations. An illustrative example is included. Full article

(This article belongs to the Special Issue Nonlinear Dynamics and Analysis)

Review

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64 pages, 3912 KiB

Open AccessReview

A Survey of Function Analysis and Applied Dynamic Equations on Hybrid Time Scales

by Chao Wang and Ravi P. Agarwal

Entropy 2021, 23(4), 450; https://doi.org/10.3390/e23040450 - 11 Apr 2021

Cited by 5 | Viewed by 1790

Abstract

As an effective tool to unify discrete and continuous analysis, time scale calculus have been widely applied to study dynamic systems in both theoretical and practical aspects. In addition to such a classical role of unification, the dynamic equations on time scales have [...] Read more.

As an effective tool to unify discrete and continuous analysis, time scale calculus have been widely applied to study dynamic systems in both theoretical and practical aspects. In addition to such a classical role of unification, the dynamic equations on time scales have their own unique features which the difference and differential equations do not possess and these advantages have been highlighted in describing some complicated dynamical behavior in the hybrid time process. In this review article, we conduct a survey of abstract analysis and applied dynamic equations on hybrid time scales, some recent main results and the related developments on hybrid time scales will be reported and the future research related to this research field is discussed. The results presented in this article can be extended and generalized to study both pure mathematical analysis and real applications such as mathematical physics, biological dynamical models and neural networks, etc. Full article

(This article belongs to the Special Issue Nonlinear Dynamics and Analysis)

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