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Discrete and Continuous Dynamical Systems - S

February 2022 , Volume 15 , Issue 2

Issue on uncertain dynamical systems in science and engineering

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Hemen Dutta
2022, 15(2): i-iv doi: 10.3934/dcdss.2021170 +[Abstract](10357) +[HTML](222) +[PDF](89.5KB)
Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species
Fathalla A. Rihan and Hebatallah J. Alsakaji
2022, 15(2): 245-263 doi: 10.3934/dcdss.2020468 +[Abstract](2117) +[HTML](534) +[PDF](1108.33KB)

Environmental factors and random variation have strong effects on the dynamics of biological and ecological systems. In this paper, we propose a stochastic delay differential model of two-prey, one-predator system with cooperation among prey species against predator. The model has a global positive solution. Sufficient conditions of existence and uniqueness of an ergodic stationary distribution of the positive solution are provided, by constructing suitable Lyapunov functionals. Sufficient conditions for possible extinction of the predator populations are also obtained. The conditions are expressed in terms of a threshold parameter \begin{document}$ {\mathcal R}_0^s $\end{document} that relies on the environmental noise. Illustrative examples and numerical simulations, using Milstein's scheme, are carried out to illustrate the theoretical results. A small scale of noise can promote survival of the species. While relative large noises can lead to possible extinction of the species in such an environment.

On properties of similarity boundary of attractors in product dynamical systems
Nitha Niralda P C and Sunil Mathew
2022, 15(2): 265-281 doi: 10.3934/dcdss.2021004 +[Abstract](1037) +[HTML](418) +[PDF](541.79KB)

Fractals in higher dimensional dynamical systems have significant roles in physics and other applied sciences. In this paper, one of the key property of fractals, called self similarity in product systems, is studied using the concept of similarity boundary. The relationship between similarity boundary of an attractor in a product space to one of its projection spaces is discussed. The impact of inverse invariance of similarity boundary on its coordinate iterated function system is analyzed. Fractals satisfying the strong open set condition, restricted to attractors in product spaces, are characterized. The relationship between similarity boundary of attractors in product spaces and their overlapping sets is also obtained. The equivalency of the restricted open set condition (ROSC) and the strong open set condition in product spaces, is proved. Self similarity of an attractor in a product system is characterized using the Hausdorff measure of its similarity boundary. Also, the Hausdorff dimensions of the overlapping set and similarity boundary of attractors for different types of iterated function systems are obtained.

Finite- and multi-dimensional state representations and some fundamental asymptotic properties of a family of nonlinear multi-population models for HIV/AIDS with ART treatment and distributed delays
Divine Wanduku
2022, 15(2): 283-313 doi: 10.3934/dcdss.2021005 +[Abstract](1279) +[HTML](462) +[PDF](1282.07KB)

A multipopulation HIV/AIDS deterministic epidemic model is studied. The population structure is a multihuman behavioral structure composed of humans practicing varieties of distinct HIV/AIDS preventive measures learnt from information and education campaigns (IEC) in the community. Antiretroviral therapy (ART) treatment is considered, and the delay from HIV exposure until the onset of ART is considered. The effects of national and multilateral support providing official developmental assistance (ODAs) to combat HIV are represented. A separate dynamics for the IEC information density in the community is derived. The epidemic model is a system of differential equations with random delays. The basic reproduction number (BRN) for the dynamics is obtained, and stability analysis of the system is conducted, whereby other disease control conditions are obtained in a multi- and a finite dimensional phase space. Numerical simulation results are given.

A novel semi-analytical method for solutions of two dimensional fuzzy fractional wave equation using natural transform
Muhammad Arfan, Kamal Shah, Aman Ullah, Soheil Salahshour, Ali Ahmadian and Massimiliano Ferrara
2022, 15(2): 315-338 doi: 10.3934/dcdss.2021011 +[Abstract](1334) +[HTML](691) +[PDF](9361.63KB)

In this research article, the techniques for computing an analytical solution of 2D fuzzy wave equation with some affecting term of force has been provided. Such type of achievement for the aforesaid solution is obtained by applying the notions of a Caputo non-integer derivative in the vague or uncertainty form. At the first attempt the fuzzy natural transform is applied for obtaining the series solution. Secondly the homotopy perturbation (HPM) technique is used, for the analysis of the proposed result by comparing the co-efficient of homotopy parameter \begin{document}$ q $\end{document} to get hierarchy of equation of different order for \begin{document}$ q $\end{document}. For this purpose, some new results about Natural transform of an arbitrary derivative under uncertainty are established, for the first time in the literature. The solution has been assumed in term of infinite series, which break the problem to a small number of equations, for the respective investigation. The required results are then determined in a series solution form which goes rapidly towards the analytical result. The solution has two parts or branches in fuzzy form, one is lower branch and the other is upper branch. To illustrate the ability of the considered approach, we have proved some test problems.

A novel collocation approach to solve a nonlinear stochastic differential equation of fractional order involving a constant delay
Seddigheh Banihashemi, Hossein Jafaria and Afshin Babaei
2022, 15(2): 339-357 doi: 10.3934/dcdss.2021025 +[Abstract](1609) +[HTML](480) +[PDF](2358.22KB)

In present work, a step-by-step Legendre collocation method is employed to solve a class of nonlinear fractional stochastic delay differential equations (FSDDEs). The step-by-step method converts the nonlinear FSDDE into a non-delay nonlinear fractional stochastic differential equation (FSDE). Then, a Legendre collocation approach is considered to obtain the numerical solution in each step. By using a collocation scheme, the non-delay nonlinear FSDE is reduced to a nonlinear system. Moreover, the error analysis of this numerical approach is investigated and convergence rate is examined. The accuracy and reliability of this method is shown on three test examples and the effect of different noise measures is investigated. Finally, as an useful application, the proposed scheme is applied to obtain the numerical solution of a stochastic SIRS model.

Hyers-Ulam-Rassias stability of high-dimensional quaternion impulsive fuzzy dynamic equations on time scales
Chao Wang, Zhien Li and Ravi P. Agarwal
2022, 15(2): 359-386 doi: 10.3934/dcdss.2021041 +[Abstract](1223) +[HTML](421) +[PDF](492.77KB)

In this paper, the Hyers-Ulam-Rassias stability of high-dimensional quaternion fuzzy dynamic equations with impulses is first considered on time scales. Some fundamental calculus results of the high-dimensional fuzzy quaternion functions in fuzzy quaternion space are established. Based on it, some sufficient conditions are obtained to guarantee the Hyers-Ulam-Rassias stability of the quaternion impulsive fuzzy dynamic equations in high-dimensional case. Moreover, several examples are provided to show the feasibility of our main results on various types of time scales.

Controllability of Sobolev type fuzzy differential equation with non-instantaneous impulsive condition
Muslim Malik, Anjali Rose and Anil Kumar
2022, 15(2): 387-407 doi: 10.3934/dcdss.2021068 +[Abstract](1438) +[HTML](447) +[PDF](343.28KB)

In this manuscript, we investigate the existence, uniqueness and controllability results of a Sobolev type fuzzy differential equation with non-instantaneous impulsive conditions. Non-linear functional analysis, Banach fixed point theorem and fuzzy theory are the main techniques used to establish these results. In support, an example is given to validate the obtained analytical findings.

On the random wave equation within the mean square context
Julia Calatayud, Juan Carlos Cortés and Marc Jornet
2022, 15(2): 409-425 doi: 10.3934/dcdss.2021082 +[Abstract](781) +[HTML](312) +[PDF](496.51KB)

This paper deals with the random wave equation on a bounded domain with Dirichlet boundary conditions. Randomness arises from the velocity wave, which is a positive random variable, and the two initial conditions, which are regular stochastic processes. The aleatory nature of the inputs is mainly justified from data errors when modeling the motion of a vibrating string. Uncertainty is propagated from these inputs to the output, so that the solution becomes a smooth random field. We focus on the mean square contextualization of the problem. Existence and uniqueness of the exact series solution, based upon the classical method of separation of variables, are rigorously established. Exact series for the mean and the variance of the solution process are obtained, which converge at polynomial rate. Some numerical examples illustrate these facts.

Link theorem and distributions of solutions to uncertain Liouville-Caputo difference equations
Hari Mohan Srivastava, Pshtiwan Othman Mohammed, Juan L. G. Guirao and Y. S. Hamed
2022, 15(2): 427-440 doi: 10.3934/dcdss.2021083 +[Abstract](761) +[HTML](353) +[PDF](447.51KB)

We consider a class of initial fractional Liouville-Caputo difference equations (IFLCDEs) and its corresponding initial uncertain fractional Liouville-Caputo difference equations (IUFLCDEs). Next, we make comparisons between two unique solutions of the IFLCDEs by deriving an important theorem, namely the main theorem. Besides, we make comparisons between IUFLCDEs and their \begin{document}$ \varrho $\end{document}-paths by deriving another important theorem, namely the link theorem, which is obtained by the help of the main theorem. We consider a special case of the IUFLCDEs and its solution involving the discrete Mittag-Leffler. Also, we present the solution of its \begin{document}$ \varrho $\end{document}-paths via the solution of the special linear IUFLCDE. Furthermore, we derive the uniqueness of IUFLCDEs. Finally, we present some test examples of IUFLCDEs by using the uniqueness theorem and the link theorem to find a relation between the solutions for the IUFLCDEs of symmetrical uncertain variables and their \begin{document}$ \varrho $\end{document}-paths.

A stochastic population model of cholera disease
Peter J. Witbooi, Grant E. Muller, Marshall B. Ongansie, Ibrahim H. I. Ahmed and Kazeem O. Okosun
2022, 15(2): 441-456 doi: 10.3934/dcdss.2021116 +[Abstract](632) +[HTML](169) +[PDF](366.98KB)

A cholera population model with stochastic transmission and stochasticity on the environmental reservoir of the cholera bacteria is presented. It is shown that solutions are well-behaved. In comparison with the underlying deterministic model, the stochastic perturbation is shown to enhance stability of the disease-free equilibrium. The main extinction theorem is formulated in terms of an invariant which is a modification of the basic reproduction number of the underlying deterministic model. As an application, the model is calibrated as for a certain province of Nigeria. In particular, a recent outbreak (2019) in Nigeria is analysed and featured through simulations. Simulations include making forward projections in the form of confidence intervals. Also, the extinction theorem is illustrated through simulations.

A new approach based on inventory control using interval differential equation with application to manufacturing system
Md Sadikur Rahman, Subhajit Das, Amalesh Kumar Manna, Ali Akbar Shaikh, Asoke Kumar Bhunia, Ali Ahmadian and Soheil Salahshour
2022, 15(2): 457-480 doi: 10.3934/dcdss.2021117 +[Abstract](498) +[HTML](283) +[PDF](1513.09KB)

Now-a-days, uncertainty conditions play an important role in modelling of real-world problems. In this regard, the aim of this study is two folded. Firstly, the concept of system of interval differential equations and its solution procedure in the parametric approach have been proposed. To serve this purpose, using parametric representation of interval and its arithmetic, system of linear interval differential equations is converted to the system of differential equations in parametric form. Then, a mixing problem with three liquids is considered and the mixing process is governed by system of interval differential equations. Thereafter, the mixing liquid is used in the production process of a manufacturing firm. Secondly, using this concept, a production inventory model for single item has been developed by employing mixture of liquids and the proposed production system is formulated mathematically by using system of interval differential equations.The corresponding interval valued average profit of the proposed model has been obtained in parametric form and it is maximized by centre-radius optimization technique. Then to validate the proposed model, two numerical examples have been solved using MATHEMATICA software. In addition, we have shown the concavity of the objective function graphically using the code of 3D plot in MATHEMATICA. Finally, the post optimality analyses are carried out with respect to different system parameters.

Existence and regularity results for stochastic fractional pseudo-parabolic equations driven by white noise
Tran Ngoc Thach, Devendra Kumar, Nguyen Hoang Luc and Nguyen Huy Tuan
2022, 15(2): 481-499 doi: 10.3934/dcdss.2021118 +[Abstract](563) +[HTML](289) +[PDF](489.7KB)

Solutions of a direct problem for a stochastic pseudo-parabolic equation with fractional Caputo derivative are investigated, in which the non-linear space-time-noise is assumed to satisfy distinct Lipshitz conditions including globally and locally assumptions. The main aim of this work is to establish some existence, uniqueness, regularity, and continuity results for mild solutions.

2020 Impact Factor: 2.425
5 Year Impact Factor: 1.490
2021 CiteScore: 3.6

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