Robust stability and boundedness of uncertain conformable fractional-order delay systems under input saturation

In this article, a class of uncertain conformable fractional-order delay systems under input saturation is considered. By establishing the Lyapunov boundedness theorem for conformable fractional-order delay systems, some sufficient conditions for robust stability and boundedness of the systems are obtained. Examples are given to illustrate the obtained theory

Well-posedness and dynamics of impulsive fractional stochastic evolution equations with unbounded delay

This paper is concerned with the well-posedness and dynamics of delay impulsive fractional stochastic evolution equations with time fractional differential operator α ∈ (0, 1). After establishing the well-posedness of the problem, and a result ensuring the existence and uniqueness of mild solutions globally defined in future, the existence of a minimal global attracting set is investigated in the mean-square topology, under general assumptions not ensuing the uniqueness of solutions. Furthermore, in the case of uniqueness, it is possible to provide more information about the geometrical structure of such global attracting set. In particular, it is proved that the minimal compact globally attracting set for the solutions of the problem becomes a singleton. It is remarkable that the attraction property is proved in the usual forward sense, unlike the pullback concept used in the context of random dynamical systems, but the main point is that the model under study has not been proved to generate a random dynamical system.National Natural Science Foundation of ChinaFondo Europeo de Desarrollo Regional (FEDER)Ministerio de Economía y CompetitividadConsejería de Innovación, Ciencia y Empresa (Junta de Andalucía

Nonlinear Systems

Open Mathematics is a challenging notion for theoretical modeling, technical analysis, and numerical simulation in physics and mathematics, as well as in many other fields, as highly correlated nonlinear phenomena, evolving over a large range of time scales and length scales, control the underlying systems and processes in their spatiotemporal evolution. Indeed, available data, be they physical, biological, or financial, and technologically complex systems and stochastic systems, such as mechanical or electronic devices, can be managed from the same conceptual approach, both analytically and through computer simulation, using effective nonlinear dynamics methods. The aim of this Special Issue is to highlight papers that show the dynamics, control, optimization and applications of nonlinear systems. This has recently become an increasingly popular subject, with impressive growth concerning applications in engineering, economics, biology, and medicine, and can be considered a veritable contribution to the literature. Original papers relating to the objective presented above are especially welcome subjects. Potential topics include, but are not limited to: Stability analysis of discrete and continuous dynamical systems; Nonlinear dynamics in biological complex systems; Stability and stabilization of stochastic systems; Mathematical models in statistics and probability; Synchronization of oscillators and chaotic systems; Optimization methods of complex systems; Reliability modeling and system optimization; Computation and control over networked systems

Variational approach to p-Laplacian fractional differential equations with instantaneous and non-instantaneous impulses

In this paper, we examine the existence of solutions of p-Laplacian fractional differential equations with instantaneous and non-instantaneous impulses. New criteria guaranteeing the existence of infinitely many solutions are established for the considered problem. The problem is reduced to an equivalent form such that the weak solutions of the problem are defined as the critical points of an energy functional. The main result of the present work is established by using a variational approach and a mountain pass lemma. Finally, an example is given to illustrate our main result

Convergence properties and fixed points of two general iterative schemes with composed maps in banach spaces with applications to guaranteed global stability

This paper investigates the boundedness and convergence properties of two general iterative processes which involve sequences of self-mappings on either complete metric or Banach spaces. The sequences of self-mappings considered in the first iterative scheme are constructed by linear combinations of a set of self-mappings, each of them being a weighted version of a certain primary self-mapping on the same space. The sequences of self-mappings of the second iterative scheme are powers of an iteration-dependent scaled version of the primary self-mapping. Some applications are also given to the important problem of global stability of a class of extended nonlinear polytopic-type parameterizations of certain dynamic systems

Una contribución al análisis de las ecuaciones en derivadas parciales estocásticas funcionales con derivadas fraccionarias en tiempo y aplicaciones

On the one hand, the classical heat equation∂tu= ∆udescribes heatpropagation in a homogeneous medium, while the time fractional diffusionequation∂αtu= ∆uwith 0< α <1 has been widely used to model anoma-lous diffusion exhibiting subdiffusive behavior. On the other side, when weconsider a physical system in the real world, we have to consider some in-fluences of internal, external, or environmental noises. Besides, the wholebackground of physical system may be difficult to describe deterministical-ly. Therefore, in this thesis, we will construct three models to show theapplications of the time fractional stochastic functional partial differentialequations.In Chapter 2, we study a stochastic lattice system with Caputo fractionalsubstantial time derivative, the asymptotic behavior of this kind of problemis investigated. In particular, the existence of a global forward attractingset in the weak mean-square topology is established. A general theorem onthe existence of solutions for a fractional SDE in a Hilbert space under theassumption that the nonlinear term is weakly continuous in a given sense isestablished and applied to the lattice system. The existence and uniquenessof solutions for a more general fractional SDEs is also obtained under aLipschitz condition.In Chapter 3, the local and global existence and uniqueness of mild solu-tions to a kind of stochastic time fractional impulsive differential equationsare studied by means of a fixed point theorem, and with the help of theproperty ofα-order fractional solution operatorTα(t) and the resolvent op-eratorSα(t). Moreover, the exponential decay to zero of the mild solutionsto this model is also proved. However, the lack of compactness of theα-order resolvent operatorSα(t) does not allow us to establish the existenceand structure of attracting sets, which is a key concept for understandingthe dynamical properties.Therefore, the second model of Chapter 3 is concerned with the well-posedness and dynamics of delay impulsive fractional stochastic evolutionequations with time fractional differential operatorα∈(0,1). After estab-lishing the well-posedness of the problem, and a result ensuring the existenceand uniqueness of mild solutions globally defined in future, the existence ofa minimal global attracting set is investigated in the mean-square topology,under general assumptions not ensuing the uniqueness of solutions. Further-more, in the case of uniqueness, it is possible to provide more informationabout the geometrical structure of such global attracting set. In particular,it is proved that the minimal compact globally attracting set for the solution-1 s of the problem becomes a singleton. It is remarkable that the attractionproperty is proved in the usual forward sense, unlike the pullback conceptused in the context of random dynamical systems, but the main point is thatthe model under study has not been proved to generate a random dynamicalsystem.Chapter 4 is devoted to the well-posedness of stochastic time fractional2D-Stokes equations of orderα∈(0,1) containing finite or infinite delay withmultiplicative noise is established, respectively, in the spacesC([−h,0];L2(Ω);L2σ)) andC((−∞,0];L2(Ω;L2σ)). The existence and uniqueness of mild so-lution to such kind of equations are proved by using a fixed-point argument.Also the continuity with respect to initial data is shown. Finally, we con-clude with several comments on future research concerning the challengingmodel: time fractional stochastic delay 2D-Navier-Stokes equations withmultiplicative noise

STOCHASTIC DELAY DIFFERENTIAL EQUATIONS WITH APPLICATIONS IN ECOLOGY AND EPIDEMICS

Mathematical modeling with delay differential equations (DDEs) is widely used for analysis and predictions in various areas of life sciences, such as population dynamics, epidemiology, immunology, physiology, and neural networks. The memory or time-delays, in these models, are related to the duration of certain hidden processes like the stages of the life cycle, the time between infection of a cell and the production of new viruses, the duration of the infectious period, the immune period, and so on. In ordinary differential equations (ODEs), the unknown state and its derivatives are evaluated at the same time instant. In DDEs, however, the evolution of the system at a certain time instant depends on the past history/memory. Introduction of such time-delays in a differential model significantly improves the dynamics of the model and enriches the complexity of the system. Moreover, natural phenomena counter an environmental noise and usually do not follow deterministic laws strictly but oscillate randomly about some average values, so that the population density never attains a fixed value with the advancement of time. Accordingly, stochastic delay differential equations (SDDEs) models play a prominent role in many application areas including biology, epidemiology and population dynamics, mostly because they can offer a more sophisticated insight through physical phenomena than their deterministic counterparts do. The SDDEs can be regarded as a generalization of stochastic differential equations (SDEs) and DDEs.This dissertation, consists of eight Chapters, is concerned with qualitative and quantitative features of deterministic and stochastic delay differential equations with applications in ecology and epidemics. The local and global stabilities of the steady states and Hopf bifurcations with respect of interesting parameters of such models are investigated. The impact of incorporating time-delays and random noise in such class of differential equations for different types of predator-prey systems and infectious diseases is studied. Numerical simulations, using suitable and reliable numerical schemes, are provided to show the effectiveness of the obtained theoretical results.Chapter 1 provides a brief overview about the topic and shows significance of the study. Chapter 2, is devoted to investigate the qualitative behaviours (through local and global stability of the steady states) of DDEs with predator-prey systems in case of hunting cooperation on predators. Chapter 3 deals with the dynamics of DDEs, of multiple time-delays, of two-prey one-predator system, where the growth of both preys populations subject to Allee effects, with a direct competition between the two-prey species having a common predator. A Lyapunov functional is deducted to investigate the global stability of positive interior equilibrium. Chapter 4, studies the dynamics of stochastic DDEs for predator-prey system with hunting cooperation in predators. Existence and uniqueness of global positive solution and stochastically ultimate boundedness are investigated. Some sufficient conditions for persistence and extinction, using Lyapunov functional, are obtained. Chapter 5 is devoted to investigate Stochastic DDEs of three-species predator prey system with cooperation among prey species. Sufficient conditions of existence and uniqueness of an ergodic stationary distribution of the positive solution to the model are established, by constructing a suitable Lyapunov functional. Chapter 6 deals with stochastic epidemic SIRC model with time-delay for spread of COVID-19 among population. The basic reproduction number ℛs0 for the stochastic model which is smaller than ℛ0 of the corresponding deterministic model is deduced. Sufficient conditions that guarantee the existence of a unique ergodic stationary distribution, using the stochastic Lyapunov functional, and conditions for the extinction of the disease are obtained. In Chapter 7, some numerical schemes for SDDEs are discussed. Convergence and consistency of such schemes are investigated. Chapter 8 summaries the main finding and future directions of research. The main findings, theoretically and numerically, show that time-delays and random noise have a significant impact in the dynamics of ecological and biological systems. They also have an important role in ecological balance and environmental stability of living organisms. A small scale of white noise can promote the survival of population; While large noises can lead to extinction of the population, this would not happen in the deterministic systems without noises. Also, white noise plays an important part in controlling the spread of the disease; When the white noise is relatively large, the infectious diseases will become extinct; Re-infection and periodic outbreaks can also occur due to the time-delay in the transmission terms

Nonlinear dynamics of plankton ecosystem with impulsive control and environmental fluctuations

It is well known that the density of plankton populations always increases and decreases or keeps invariant for a long time, and the variation of plankton density is an important factor influencing the real aquatic environments, why do these situations occur? It is an interesting topic which has become the common interest for many researchers. As the basis of the food webs in oceans, lakes, and reservoirs, plankton plays a significant role in the material circulation and energy flow for real aquatic ecosystems that have a great effect on the economic and social values. Planktonic blooms can occur in some environments, however, and the direct or indirect adverse effects of planktonic blooms on real aquatic ecosystems, such as water quality, water landscape, aquaculture development, are sometimes catastrophic, and thus planktonic blooms have become a challenging and intractable problem worldwide in recent years. Therefore, to understand these effects so that some necessary measures can be taken, it is important and meaningful to investigate the dynamic growth mechanism of plankton and reveal the dynamics mechanisms of formation and disappearance of planktonic blooms. To this end, based on the background of the ecological environments in the subtropical lakes and reservoirs, this dissertation research takes mainly the planktonic algae as the research objective to model the mechanisms of plankton growth and evolution. In this dissertation, some theories related to population dynamics, impulsive control dynamics, stochastic dynamics, as well as the methods of dynamic modeling, dynamic analysis and experimental simulation, are applied to reveal the effects of some key biological factors on the dynamics mechanisms of the spatial-temporal distribution of plankton and the termination of planktonic blooms, and to predict the dynamics evolutionary processes of plankton growth. The main results are as follows: Firstly, to discuss the prevention and control strategies on planktonic blooms, an impulsive reaction-diffusion hybrid system was developed. On the one hand, the dynamic analysis showed that impulsive control can significantly influence the dynamics of the system, including the ultimate boundedness, extinction, permanence, and the existence and uniqueness of positive periodic solution of the system. On the other hand, some experimental simulations were preformed to reveal that impulsive control can lead to the extinction and permanence of population directly. More precisely, the prey and intermediate predator populations can coexist at any time and location of their inhabited domain, while the top predator population undergoes extinction when the impulsive control parameter exceeds some a critical value, which can provide some key arguments to control population survival by means of some reaction-diffusion impulsive hybrid systems in the real life. Additionally, a heterogeneous environment can affect the spatial distribution of plankton and change the temporal-spatial oscillation of plankton distribution. All results are expected to be helpful in the study of dynamic complex of ecosystems. Secondly, a stochastic phytoplankton-zooplankton system with toxic phytoplankton was proposed and the effects of environmental stochasticity and toxin-producing phytoplankton (TPP) on the dynamics mechanisms of the termination of planktonic blooms were discussed. The research illustrated that white noise can aggravate the stochastic oscillation of plankton density and a high-level intensity of white noise can accelerate the extinction of plankton and may be advantageous for the disappearance of harmful phytoplankton, which imply that the white noise can help control the biomass of plankton and provide a guide for the termination of planktonic blooms. Additionally, some experimental simulations were carried out to reveal that the increasing toxin liberation rate released by TPP can increase the survival chance of phytoplankton population and reduce the biomass of zooplankton population, but the combined effects of those two toxin liberation rates on the changes in plankton are stronger than that of controlling any one of the two TPP. All results suggest that both white noise and TPP can play an important role in controlling planktonic blooms. Thirdly, we established a stochastic phytoplankton-toxic producing phytoplankton-zooplankton system under regime switching and investigated how the white noise, regime switching and TPP affect the dynamics mechanisms of planktonic blooms. The dynamical analysis indicated that both white noise and toxins released by TPP are disadvantageous to the development of plankton and may increase the risk of plankton extinction. Also, a series of experimental simulations were carried out to verify the correctness of the dynamical analysis and further reveal the effects of the white noise, regime switching and TPP on the dynamics mechanisms of the termination of planktonic blooms. On the one hand, the numerical study revealed that the system can switch from one state to another due to regime shift, and further indicated that the regime switching can balance the different survival states of plankton density and decrease the risk of plankton extinction when the density of white noise are particularly weak. On the other hand, an increase in the toxin liberation rate can increase the survival chance of phytoplankton but reduce the biomass of zooplankton, which implies that the presence of toxic phytoplankton may have a positive effect on the termination of planktonic blooms. These results may provide some insightful understanding on the dynamics of phytoplankton-zooplankton systems in randomly disturbed aquatic environments. Finally, a stochastic non-autonomous phytoplankton-zooplankton system involving TPP and impulsive perturbations was studied, where the white noise, impulsive perturbations and TPP are incorporated into the system to simulate the natural aquatic ecological phenomena. The dynamical analysis revealed some key threshold conditions that ensure the existence and uniqueness of a global positive solution, plankton extinction and persistence in the mean. In particular, we determined if there is a positive periodic solution for the system when the toxin liberation rate reaches a critical value. Some experimental simulations also revealed that both white noise and impulsive control parameter can directly influence the plankton extinction and persistence in the mean. Significantly, enhancing the toxin liberation rate released by TPP increases the possibility of phytoplankton survival but reduces the zooplankton biomass. All these results can improve our understanding of the dynamics of complex of aquatic ecosystems in a fluctuating environment

Epidemic Models with Pulse Vaccination and Time Delay

In this thesis we discuss deterministic compartmental epidemic models. We study the asymp- totic stability of the disease-free solution of models with pulse vaccination campaigns. The main contributions of this thesis are to extend the literature of pulse vaccination models with delay. We take results for ordinary differential equation models and extend them to models with delay differential equations. Model generalizations include the use of a general incidence term as an upper bound for the actual incidence, and the use of switch parameters to approximate time-varying parameters. In particular, we look at contact rate parameters which are piecewise constant or time-varying. We extend literature results for non-delay general incidence models to find uniform asymptotic stability of the disease-free solution which helps us to add delay. We find an upper bound for the susceptible population under pulse vaccination and use this bound to tighten results for eradication thresholds: that is, we use this upper bound to find sufficient conditions for the uniform asymptotic stability of the disease-free solution of delayed pulse vaccination models. We extend literature results for constant contact rate bilinear incidence delay models to models with periodic time-varying contact rate, and determine conditions under which the disease-free solution is uniformly asymptotically stable for small delay. We also find conditions for disease permanence in the corresponding non-delay, time-varying-parameter pulse vaccination model. For piecewise- constant contact rate bilinear incidence models we again find thresholds which guarantee uniform asymptotic stability under small delay. We additionally discuss the effects of time-varying total population on our results, through a change of variables to population fractions. The total population is commonly held constant in the literature, for analytical simplicity, so we survey the methods for time-varying total population and the effects of such variation on the pulse vaccination schemes. We retain thresholds for eradication by considering the compartment populations as fractions of the total, instead of population numbers. The result is also applied to constant-population delay systems. When changing from standard incidence to bilinear incidence in delay systems, we discuss a way to estimate the effect of time-varying N. We support our theory with simulation results