Stability Analysis of Fixed Points of Various Nonlinear Mathematical Models with Chaos Control

Abstract

Several types of research on the qualitative study of nonlinear models have appeared in applied mathematics in past few decades. Moreover, these investigations are mainly focused on theoretical results instead of applications. In this thesis, we are aimed to contribute some research in the qualitative study of nonlinear mathematical models in applied mathematics. In our work, the adopted approach for the qualitative study is simple and easy to understand for readers interested in the qualitative study of nonlinear models, which we encounter in mathematical biology, mathematical ecology, mathemat ical chemistry and engineering. We have also provided some new theoretical results on the boundedness of solutions, existence of fixed points, stability analysis, bifurcation and chaos control. These significant results were not available in the literature before. It is necessary to analyze the qualitative behaviour of various mathematical models to under stand complexities aligned with the qualitative study of dynamical systems. These models may be discrete-time or continuous-time models, depending upon the situation. Many biological and chemical interactions such as prey-predator interactions, plant-herbivore interactions, host-parasite interactions and several interactions are described using dis crete and continuous-time models. For example, these models are often used to describe the properties of an epidemic when it arrives in a community and to forecast under which circumstances it will be continued or wiped out from that community. Discrete time mathematical models are easy to analyze and have rich dynamics compared to their continuous-time counterparts. Hence, it is interesting to analyze the dynamics of such discrete-time mathematical models. In this work, we consider some nonlinear mathe matical models in continuous form and then discuss the qualitative behaviour of their discrete-time counterparts. Furthermore, we have used four different discretization techniques: piecewise argu ments, Euler’s forward method, nonstandard finite difference scheme, and fractional order discretization. The boundedness character of every positive solution of obtained systems is discussed, and the existence of a positive fixed point is discussed for every system. It is shown that each system undergoes the Neimark-Sacker bifurcation about its positive fixed point. Moreover, some mathematical models also show the existence of period-doubling bifurcation and chaos. It is shown that whenever a discrete-time mathematical model is obtained using a nonstandard finite difference scheme, it is dynamically consistent and exhibits similar dynamics as its continuous-time counterpart. In addition, two gener alized hybrid control techniques are presented to overcome the chaos and inconsistent behaviour of mathematical systems. A comparison of generalized hybrid techniques with the old hybrid method shows the effectiveness and wide range of generalized hybrid con 3 DRSML QAU trol techniques. Finally, some surprising numerical examples are provided to show some exceptional dynamics of discrete-time mathematical models and to justify our theoretical discussion. Let us briefly describe the content of every chapter. Chapter 1. Introduction and Preliminaries. In this chapter, we focus on the basic theory of the dynamical system. Moreover, some beneficial results for the qualitative study of dynamical systems are provided in this chapter. Chapter 2. Bifurcation analysis of a discrete-time four-dimensional cubic autocatalator chemical reaction model with coupling through uncatalyzed re actant. The continuous four-dimensional cubic autocatalator chemical reaction model is examined in this chapter. Using Routh-Hurwitz stability criteria, our major goals are to develop parametric requirements for the local stability of the continuous system and assess if a positive fixed point exists. Using Euler’s forward approach and a nonstandard difference scheme, we discretize the continuous model to provide a discrete-time equiva lent of the four-dimensional model. We also look at the prerequisites for the positive fixed point’s local asymptotic stability in the discrete-time system. Using a generic method for Neimark-Sacker bifurcation analysis, our research shows that the system exhibits a Neimark-Sacker bifurcation at the positive fixed point. In addition, depending on the bifurcation parameter values, we find chaotic dynamics in the discrete-time version of a simplified four-dimensional system. We provide a generalized hybrid control technique that combines parameter perturbation and feedback control to address the Neimark Sacker bifurcation and chaos. Finally, we offer a number of numerical illustrations to support the conclusions drawn from theory in this chapter. The findings of this chapter have recently been published in an extreme international journal. Chapter 3. Bifurcation analysis of a discrete-time phytoplankton-zooplankton model with Holling type-II response and toxicity. The interaction between phytoplankton and zooplankton plays a vital role in ecology. In this field, discrete-time mathematical models are commonly employed to understand the dynamics of phytoplankton-zooplankton interactions, where generations do not over lap, and new age groups replace older ones at regular intervals. This chapter focuses on converting a continuous-time phytoplankton-zooplankton model into a discrete-time counterpart using a dynamically consistent nonstandard difference scheme to ensure the models’ dynamical consistency. Additionally, the chapter explores boundedness condi tions for all solutions and establishes the existence of a exceptional positive fixed point. The linearized stability of the obtained system is analyzed with respect to all of its fixed points, and the presence of a Neimark-Sacker bifurcation around the one and only positive equilibrium is demonstrated under specific mathematical circumstances. To control the Neimark-Sacker bifurcation, a comprehensive hybrid control method is applied. Several numerical examples are provided to illustrate the abstract outcomes and match up to the 4 DRSML QAU dynamics of the discrete-time model with its continuous counterpart. Moreover, the numerical study shows that both the obtained system and its continuous time counterpart exhibit stability for the same parameter values and instability for the identical parametric values. This consistency in dynamical behavior is evident from the numerical study. Finally, the chapter concludes with a comparison between the modified hybrid method and the previous method. We have published this chapter in a high-quality international journal. Chapter 4. Bifurcation analysis of a discrete-time compartmental model for hypertensive or diabetic patients exposed to COVID-19. In this chapter, we investigate a mathematical model concerning hypertensive or diabetic patients who are exposed to COVID-19. The model is constructed using a set of first-order nonlinear differential equations. To discretize the continuous system, we utilize piecewise constant arguments. We derive local stability conditions for the equilibrium points of the resulting discrete-time mathematical system. Furthermore, we explore the occurrence of period doubling bifurcation and chaos in the absence of an isolated population. Notably, our system exhibits instability and chaos when the quarantined compartment is empty, which holds biological significance. Additionally, we study the existence of Neimark-Sacker bifurcation at the endemic equilibrium point. Moreover, through numerical simulations, we observe that the discrete-time mathe matical system undergoes period-doubling bifurcation around the endemic equilibrium. To control both the period-doubling bifurcation and Neimark-Sacker bifurcation, we em ploy a generalized hybrid control methodology. The model serves to emphasize the epi demiological importance of quarantine by illustrating chaos and oscillation within the context of the COVID-19 environment. The results of this chapter are published in an international journal of high quality. Chapter 5. Dynamics of a discrete-time fractional-order phytoplankton zooplankton model with Holling type-II response. This chapter is devoted to the qualitative study of a continuous-time phytoplankton-zooplankton model with Holling type-II response. Furthermore, we obtained its discrete-time counterpart by using a fractional-order discretization method. The local stability of the obtained system about all of its equilibrium points is discussed. It is proved that the system experience Neimark Sacker bifurcation about positive equilibrium point under some mathematical conditions. The Neimark-Sacker bifurcation is controlled using two modified hybrid control tech niques. Finally, at the end of this chapter, some interesting numerical examples are provided to support our theoretical discussion and explore the effectiveness and feasibil ity of newly designed control strategies. In addition, a comparison of modified hybrid techniques with the existing hybrid approach is given. This chapter is submitted in an international peer-review journal. 5 DRSML QAU Chapter 6. Bifurcation analysis and control of chaos in a discrete-time two trophic plant-herbivore model and dynamical consistency of a nonstandard difference scheme.In this final chapter, we explore the dynamics of two discrete-time plant-herbivore models. The discrete-time model is obtained by applying Euler’s forward method to convert a continuous-time plant-herbivore model. To ensure the dynamic consistency of the discrete-time model, we employ a nonstandard difference scheme. Furthermore, we discuss the local stability of the system and establish the existence of bifurcation around the positive equilibrium under specific mathematical conditions. In order to effectively manage the occurrence of bifurcation and chaos, we develop a mod ified hybrid technique. To validate our findings and demonstrate the reliability of the nonstandard difference scheme, we provide several numerical examples at the conclusion of the chapter. The results of this chapter are published in an international journal of high quality. Every chapter encloses some theorems on a qualitative study of different models, and we have delivered proofs for most of them. The explanations of the literature of each chapter are summarized at the end of each chapter. Moreover, all references for our study are provided in the bibliography section. This bibliography aims mainly to provide a reader with information on further reading. The symbol ✷ marks the conclusion of a theorem’s proof.