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In this paper, we present some new stability criteria in the sense of Ulam for the solutions of fractional differential equations involving the conformable fractional derivative. Our results are based on a fixed point alternative which is developed for generalized metric spaces. This study improves and extends the literature in this topic since there is no previous progress on the problem we consider. We also provide examples to illustrate our results in a separate section. [ABSTRACT FROM AUTHOR]

In this study, our aim is to explore the dynamics of COVID-19 or 2019-nCOV in Argentina considering the parameter values based on the real data of this virus from March 03, 2020 to March 29, 2021 which is a data range of more than one complete year. We propose a Atangana–Baleanu type fractional-order model and simulate it by using predictor–corrector (P-C) method. First we introduce the biological nature of this virus in theoretical way and then formulate a mathematical model to define its dynamics. We use a well-known effective optimization scheme based on the renowned trust-region-reflective (TRR) method to perform the model calibration. We have plotted the real cases of COVID-19 and compared our integer-order model with the simulated data along with the calculation of basic reproductive number. Concerning fractional-order simulations, first we prove the existence and uniqueness of solution and then write the solution along with the stability of the given P-C method. A number of graphs at various fractional-order values are simulated to predict the future dynamics of the virus in Argentina which is the main contribution of this paper. [ABSTRACT FROM AUTHOR]

In this paper, a toxin producing phytoplankton-zooplankton system with the delay is investigated. Firstly, the nonnegativity and boundedness of solutions are given. Then the local and global asymptotic stabilities of the boundary equilibrium are investigated, and the existence of local Hopf bifurcations is established as the delay crosses a threshold value at the positive equilibrium. Furthermore, there exists at least one positive periodic solution as the delay varies in some regions by using the global Hopf bifurcation result of Wu for functional differential equations. In addition, the impacts of the toxic substances are also investigated. At last, an explicit algorithm is derived for the stability and direction of the bifurcating periodic solution by using center manifold theory and normal form method. [ABSTRACT FROM AUTHOR]

Rayleigh-Bénard convection in nanoliquids is studied in the presence of volumetric heat source. The present analytical work concerns twenty nanoliquids. Carrier liquids considered are water, ethylene glycol, engine oil and glycerine and with them five different nanoparticles considered are copper, copper oxide, silver, alumina and titania. Expression for the thermophysical properties of the nanoliquids is chosen from phenomenological laws or mixture theory. Heat source is characterized by an internal nanoliquid Rayleigh number RInl. Heat source adds to the energy of the system and hence an advanced onset is observed in this case compared to the problem with no heat source. In the case of heat sink, however, heat is drawn from the system leading to delay in onset. The individual effect of all the nanoparticles is to advance convection. Enhanced heat transport situation is observed in each of the nanoliquids with engine-oil-silver transporting maximum heat and water-titania the least. Additional Fourier modes are found not to have any profound effect on the results predicted by minimal modes. The connection between the Lorenz model and the Ginzburg-Landau model is clearly shown in the paper. [ABSTRACT FROM AUTHOR]

In this paper, the finite-time stability of a class of fractional q-difference equations with time-delay is investigated. First, we state and prove a generalized q-Gronwall inequality. In addition, we introduce a delayed q-Mittag-Leffler type matrix function and explore explicit formula of solutions to the considered system. Then two criteria to guarantee finite-time stability are obtained by utilizing the proposed delayed q-Mittag-Leffler type matrix and generalized q-Gronwall inequality, respectively. Finally, an example and its simulation are presented to illustrate our main results. [ABSTRACT FROM AUTHOR]

The fractional-order Newton-Leipnik system (FoNLs) was studied in this paper from the point of view of two strange attractors. Within the same parameter condition of FoNLs, the different of initial conditions are numerically studied. The ranges are relatively wide for the different initial conditions used in this analysis and the synchronization of two identical FoNLs is investigated in this paper. An active control is the technique used for achieving the synchronization between chaotic FoNLs. Based on the theory of Laplace transformation (LTF), the conditions for achieving synchronisation of fractional order systems are studied. In order to verify the efficiency and viability of this process, numerical simulations are given, so the outcome of the process is simultaneously validated with theoretical models and computational methods. The above concept is also applied by this paper to secure communications using the chaotic masking system. [ABSTRACT FROM AUTHOR]

In this paper, we study the nonlinear viscoelastic plate equation modeling the deformation of a suspension bridge, given by u tt (x , y , t) + Δ 2 u (x , y , t) - ∫ 0 t g (t - τ) Δ 2 u (τ) d τ + a (x , y) | u t | m u t + | u | ρ u = 0 , in the bounded domain Ω = (0 , π) × (- d , d) , which is assumed to be hinged on its vertical edges and free on its remaining horizontal edges. We prove, with general conditions on the relaxation function, that the decay rate of energy is similar to that of the relaxation function regardless of the presence or the absence of the frictional damping. [ABSTRACT FROM AUTHOR]

In the literature, several mathematical models have been formulated and developed to describe the within-host dynamics of either human immunodeficiency virus (HIV) or human T-lymphotropic virus type I (HTLV-I) monoinfections. In this paper, we formulate and analyze a novel within-host dynamics model of HTLV-HIV coinfection taking into consideration the response of cytotoxic T lymphocytes (CTLs). The uninfected CD 4 + T cells can be infected via HIV by two mechanisms, free-to-cell and infected-to-cell. On the other hand, the HTLV-I has two modes for transmission, (i) horizontal, via direct infected-to-cell touch, and (ii) vertical, by mitotic division of active HTLV-infected cells. It is well known that the intracellular time delays play an important role in within-host virus dynamics. In this work, we consider six types of distributed-time delays. We investigate the fundamental properties of solutions. Then, we calculate the steady states of the model in terms of threshold parameters. Moreover, we study the global stability of the steady states by using the Lyapunov method. We conduct numerical simulations to illustrate and support our theoretical results. In addition, we discuss the effect of multiple time delays on stability of the steady states of the system. [ABSTRACT FROM AUTHOR]

A prey–predator model with constant-effort harvesting on the prey and predators is investigated in this paper. First, we discuss the number and type of the equilibria by analyzing the equations of equilibria and the distribution of eigenvalues. Second, with the rescaled harvesting efforts as bifurcation parameters, a subcritical Hopf bifurcation is exhibited near the multiple focus and a Bogdanov–Takens bifurcation is also displayed near the BT singularity by analyzing the versal unfolding of the model. With the variation of bifurcation parameters, the system shows multi-stable structure, and the attractive domains for different attractors are constituted by the stable and unstable manifolds of saddles and the limit cycles bifurcated from Hopf and Bogdanov–Takens bifurcations. Finally, a cusp point and two generalized Hopf points are found on the saddle-node bifurcation curve and the Hopf bifurcation curves, respectively. Several phase diagrams for parameters near one of the generalized Hopf points are exhibited through the generalized Hopf bifurcation. [ABSTRACT FROM AUTHOR]

In this paper, we study a Lotka-Volterra prey-predator system with feedback control. We establish sufficient conditions under which a unique positive equilibrium is globally stable. Further, we show that a suitable feedback control on predator species can make prey species that is on the brink of extinction become globally stable, but under the conditions of small feedback control on predator, the prey species still extinct, whereas the predator species is stable at certain values. Several examples are presented to show the feasibility of the main results. [ABSTRACT FROM AUTHOR]

In this paper, the problem of anti-disturbance control for a class of multi-input and multi-output (MIMO) nonlinearly parameterized systems with mismatched general periodic disturbances is investigated via a composite adaptive anti-disturbance control scheme. The composite adaptive anti-disturbance control method is presented by using disturbance observer technique, back-stepping method and adaptive control approach. A novel disturbance observer is designed to estimate the disturbances generated by a linear system with nonlinear output function. Rigorous stability analysis for the augmented closed-loop system is developed by direct Lyapunov stability theory. It is shown that the system outputs asymptotically converge to zero in the presence of mismatched general periodic disturbances. Finally, a simulation example is given to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper studies the globally asymptotic stability of neutral-type impulsive neural networks with discrete and bounded continuously distributed delays. By using the Lyapunov functional method, quadratic convex combination approach, a novel Gu's Lemma, Jensen integral inequality and linear convex combination technique, several novel sufficient conditions are derived to ensure the globally asymptotic stability of the equilibrium point of the networks. The proposed results, which do not require the differentiability and monotonicity of the activation functions, can be easily checked via Matlab software. Finally, two numerical examples are given to demonstrate the effectiveness of our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, an adaptive neural network (NN) switching control strategy is proposed for the trajectory tracking problem of robotic manipulators. The proposed system comprises an adaptive switching neural controller and the associated robust compensation control law. Based on the Lyapunov stability theorem and average dwell-time approach, it is shown that the proposed control scheme can guarantee tracking performance of the robotic manipulators system, in the sense that all variables of the closed-loop system are bounded and the effect due to the external disturbance and approximate error of radical basis function (RBF) NNs on the tracking error can be converged to zero in an infinite time. Finally, simulation results on a two-link robotic manipulator show the feasibility and validity of the proposed control scheme. [ABSTRACT FROM AUTHOR]

An impulsive differential equation with time varying delay is proposed in this paper. By using some analysis techniques with combination of coincidence degree theory, sufficient conditions for the permanence, the existence and global attractivity of positive periodic solution are established. The results of this paper improve and generalize some previously known results. [ABSTRACT FROM AUTHOR]

In this paper, a HIV-TB co-infection model is explored which incorporates a non-linear treatment rate for TB. We begin with presenting a HIV-TB co-infection model and analyze both HIV and TB sub-models separately. The basic reproduction numbers corresponding to HIV-only, TB-only and the HIV-TB full model are computed. The disease-free equilibrium point of the HIV sub-model is shown to be locally as well as globally asymptotically stable when its corresponding reproduction number is less than unity. The HIV-only model exhibits a transcritical bifurcation. On the other hand, for the TB sub-model, the disease-free equilibrium point is locally asymptotically stable but may not be globally asymptotically stable. We have also analyzed the full HIV-TB co-infection model. Numerical simulations are performed to investigate the effect of treatment rate in the presence of resource limitation for TB infected individuals, which emphasize the fact that to reduce co-infection from the population programs to accelerate the treatment of TB should be implemented. [ABSTRACT FROM AUTHOR]

We discuss two problems which grew out of an introductory differential equations class but were solved only later, each after having been put into a different context. First, how do you find a rather complicated Lyapunov function with your bare hands, without using a fully developed theory (while reconstructing the steps leading up to such a theory)? Second, how can you obtain a global picture of the phase-portrait of a dynamical system (thereby invoking ideas from projective geometry)? Since classroom experiences played an important part in the making of this paper, didactical aspects will also be discussed. [ABSTRACT FROM AUTHOR]

In this paper a model of Hopfield's graded response neural network is investigated. A network whose neurons are subject to a certain impulsive state displacement at random times is considered. The model is set up and studied. The presence of random moments of impulses in the model leads to a change of the solutions to stochastic processes. Also, we use the Riemann–Liouville fractional derivative to model adequately the long-term memory and the nonlocality in the neural networks. We set up in an appropriate way both the initial conditions and the impulsive conditions at random moments. The application of the Riemann–Liouville fractional derivative leads to a new definition of the equilibrium point. We define mean-square Mittag-Leffler stability in time of the equilibrium point of the model and study this type of stability. Some sufficient conditions for this type of stability are obtained. The general case with time varying self-regulating parameters of all units and time varying functions of the connection between two neurons is studied. [ABSTRACT FROM AUTHOR]

Cigarette smoking on college campuses has become a significant public health issue, which in turn led to an increasing focus on establishing programs to reduce its prevalence. In this paper, a compartmental model depicting the spread and cessation of the smoking habit on college campuses, obtained using theoretical principles often employed in mathematical epidemiology, is proposed and analysed. The existence and stability of the habitual smoking-free and habitual smoking-persistent equilibria, respectively, are explored in terms of a threshold parameter, hereby called the smokers generation number and denoted by R c . A sensitivity analysis indicates that R c is the most sensitive to the contact rate between habitual-smokers and occasional-smokers and to the rate of successfully quitting smoking. Numerical simulations of the proposed optimal control strategies reveal that the most effective approach to reduce the prevalence of cigarette smoking and possibly achieve a smoking-free campus should combine both control measures, namely allocating mandatory smoking rooms together with educating the public on the harmful effects of smoking and providing large scale guidance, counselling and support therapy to help students quit smoking. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study a kind of fractional-order multiple point boundary value problem involving noninstantaneous impulse and abstract bounded operator. The existence and uniqueness is obtained by the Banach contraction principle. And by applying direct analysis methods, we establish some conditions of the Ulam–Hyers stability for this problem. Finally, an interesting application example is given to illustrate the validity of the results. [ABSTRACT FROM AUTHOR]

Results reported in this paper study the existence and stability of a class of implicit generalized proportional fractional integro-differential Langevin equations with nonlocal fractional integral conditions. The main theorems are proved with the help of Banach's, Krasnoselskii's, and Schaefer's fixed point theorems and Ulam's approach. Finally, an example is given to demonstrate the applicability of our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonautonomous predator-prey system with Beddington-DeAngelis functional response and explore the global stability of boundary solution. Based on the dynamics of logistic equation, some new sufficient conditions on the global asymptotic stability of boundary solution are presented for general time-dependence case. Our main results indicate that (i) the long-term ineffective predation behaviour or high mortality of predator species will lead the predator species to extinction, even if the intraspecies competition of predator species is weak or no intraspecies competition; (ii) the long-term intense intraspecific competition may lead the predator species to extinction, even though the long-term accumulative predation benefit is higher than the death lose. When all parameters are periodic functions with common period, a necessary and sufficient condition on the global stability of boundary periodic solution is obtained. In addition, some numerical simulations are performed to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a stage-structured predator–prey model with mutual interference, in which includes two discrete delays. By theoretical analysis, we establish the stability of the unique positive equilibrium and the existence of Hopf bifurcation when the maturation delay for predators is used as the bifurcation parameter. Our results exhibit that the maturation delay for preys does not affect the stability of the positive equilibrium. However, the maturation delay for predator is able to destabilize the positive equilibrium and causes periodic solutions. Numerical simulations are carried out to illustrate the theoretical results and display the differential impacts of two type delays and mutual interference. [ABSTRACT FROM AUTHOR]

In this paper, the asymptotic behavior of solutions of a class of non-autonomous third order neutral delay differential equations is studied. Results on the stability, boundedness, and the square integrability of solutions and their derivatives are obtained. An example to illustrate the results is included. [ABSTRACT FROM AUTHOR]

In this paper, we consider a three species plankton–fish system that incorporates external toxicity and nonlinear harvesting. We consider that the growth of species are affected directly or indirectly by an external toxic substance and the feeding of the predator on the affected prey is considered as Holling type II functional response. All the possible biological feasible equilibrium points are determined analytically as well as numerically and performed stability analysis around these equilibrium points. It is shown that the system undergoes for Hopf bifurcation when the growth rate of prey passes some threshold value. Furthermore, Pontryagin's maximum principle has been applied to obtain optimal control of harvesting to maximize the benefit as well as the conservation of the ecosystem. We perform numerical simulations to justify and illustrate our analytical results. Some numerical tools such as phase portraits, time evaluation and bifurcation diagrams are presented to ensure the complex dynamics in the system. Period doubling cascade route to chaos is examined and validated through the numerical calculation of Lyapunov exponents and sensitivity analysis. [ABSTRACT FROM AUTHOR]

A continuous attractor of recurrent neural network is a connected set of stable equilibrium points. It's widely used to interpret a lot of brain activities. A large number of literatures have studied discrete attractors and single continuous attractor. In this paper, the coexisting continuous attractors for nonlinear neural networks is studied and the coexisting conditions of two continuous attractors for network with and without external input in two-dimensional space are obtained by analysing the eigenvalues of the matrix. Finally, all the results are verified by simulation. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence and uniqueness of solution for random fractional integro-differential equation. The existence of at least mean square continuous solution for this problem is discussed. Using the fixed point theorem, Ulam–Hyers stability and Ulam–Hyers–Rassias stability of random fractional integro-differential equation are studied. Finally, we give an example to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we mainly consider the existence and Hyers-Ulam stability of solutions for a class of fractional differential equations involving time-varying delays and non-instantaneous impulses. By the Krasnoselskii's fixed point theorem, we present the new constructive existence results for the addressed equation. In addition, we deduce that the equations have Hyers-Ulam stable solutions by utilizing generalized Grönwall's inequality. Some results in this literature are new and improve some early conclusions. [ABSTRACT FROM AUTHOR]

Quasi-polynomial systems are strongly related to mass action systems. In this paper, we extend mass action systems for the first. Then, we prove that any quasi-polynomial system with nonnegative exponents is realizable, i.e., there exists an extended mass action system can realize its dynamics. An algorithm is designed to get the realization. The input of the algorithm is the matrices characterizing the mathematical structure of the quasi-polynomial system, while the output is the realization. Since extended complex balanced mass action systems are locally asymptotically stable, quasi-polynomial systems with complex balanced realizations are also locally asymptotically stable. Based on the initial realization produced by the algorithm, we propose a systematic method for computing complex balanced realization of quasi-polynomial system. Lastly, we illustrate the efficiency of the method with two numerical examples. [ABSTRACT FROM AUTHOR]

This paper considers the delay-dependent stability problem of recurrent neural networks with interval time-varying delays. An appropriate Lyapunov–Krasovskii functional is constructed and the combination method of Wirtinger inequality and reciprocally convex optimization technique is employed. Combing a new activation function segmentation method of the boundary condition and the orthogonal complement lemma, three further improved delay-dependent stability criteria are established. Finally, two numerical examples show the effectiveness of our proposed method by comparison with the recent existing works. [ABSTRACT FROM AUTHOR]

This paper considers the stability problem for some coupled systems of fractional differential equations on networks (CSFDENs). We provide a systematic method for constructing a Lyapunov function for CSFDENs by using graph theory and the Lyapunov method. Consequently, some sufficient conditions for stability, uniform stability and uniform asymptotic stability of CSFDENs are obtained. Finally, an example and some numerical simulations are presented to verify the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

It is astonishing that after more than half a century intensive research in the area of non-conservative systems of second order differential equations still new interesting results appear, see . In that paper an old stability criterion by Metelitsyn and Frik was reinvented. We shortly repeat this result in order to emphasize that the criterion is sufficient but not necessary for stability. Afterwards we concentrate on circulatory systems with purely imaginary eigenvalues and investigate the influence of indefinite damping. Finally the possibility of stabilizing circulatory systems by gyroscopic forces will be commented. Examples will demonstrate the developed theory. [ABSTRACT FROM AUTHOR]

A two species non-autonomous competitive phytoplankton system with nonlinear inter-inhibition terms and one toxin producing phytoplankton is studied in this paper. Sufficient conditions which guarantee the extinction of a species and the global attractivity of the other one are obtained. Some parallel results corresponding to Yue (Adv. Differ. Equ. 2016:1, 2016, doi:) are established. Numeric simulations are carried out to show the feasibility of our results. [ABSTRACT FROM AUTHOR]

In this paper, the problem of asymptotical stability for static recurrent neural networks is investigated. Based on delay partitioning approach and a new Lyapunov–Krasvoskii functional, delay-independent conditions are proposed to ensure the asymptotic stability of the static recurrent neural networks. The delay-independent conditions are less conservative than the existing ones. Expressed in linear matrix inequalities, the stability conditions can be checked using the standard numerical software. Two numerical examples are provided to illustrate the effectiveness and the reduced conservatism of the proposed results. [ABSTRACT FROM PUBLISHER]

In this paper, the problem of stability analysis for uncertain genetic regulatory networks with time-varying delays is investigated. The time-varying delay function in this paper is not required to be either continuously differentiable, or its derivative less than one. By choosing an appropriate Lyapunov-Krasovskii functional and employing some free-weighting matrices, some new delay-dependent and delay-derivative-dependent stability criteria are presented in terms of linear matrix inequality. And the new criteria are applicable to both fast and slow time-varying delays. Finally, three numerical examples are used to demonstrate the usefulness of the main results and less conservativeness of the proposed method. [ABSTRACT FROM AUTHOR]