Conditions For Global Asymptotic Stability Research Articles

Neutral-type of delayed inertial neural networks and their stability analysis using the LMI Approach

A theoretical investigation of neutral-type of delayed inertial neural networks using the Lyapunov stability theory and Linear Matrix Inequality (LMI) approach is presented. Based on a suitable variable transformation, an inertial neural network consisting of second-order differential equations can be converted into a first-order differential model. The sufficient conditions of the delayed inertial neural network are derived by constructing suitable Lyapunov functional candidates, introducing new free weighting matrices, and utilizing the Writinger integral inequality. Through the LMI solution, we analyse the global asymptotic stability condition of the resulting delayed inertial neural network. Simulation examples are presented to demonstrate the effectiveness of the derived analytical results.

Global stability of a price model with multiple delays

Consider the delay differential equation \begin{equation*} \dot{x}(t)=a \Bigg(\sum_{i=1}^n b_i\big[x(t-s_i)- x(t-r_i)\big]\Bigg)-g(x(t)), \end{equation*} where $a>0$, $b_i>0$ and $0\leq s_i 0} g(\xi)/\xi>2a$. This equation can be interpreted as a price model, where $x(t)$ represents the price of an asset (e.g. price of share or commodity, currency exchange rate etc.) at time $t$. The first term on the right-hand side represents the positive response for the recent tendencies of the price and $-g(x(t))$ is responsible for the instantaneous negative feedback to the deviation from the equilibrium price. We study the local and global stability of the unique, non-hyperbolic equilibrium point. The main result gives a sufficient condition for global asymptotic stability of the equilibrium. The region of attractivity is also estimated in case of local asymptotic stability.

Existence, uniqueness, and global asymptotic stability analysis for delayed complex-valued Cohen–Grossberg BAM neural networks

This paper investigates the existence, uniqueness, and global asymptotic stability of equilibrium point for a complex-valued Cohen–Grossberg delayed bidirectional associative memory neural networks. The two types of complex-valued behaved functions, amplification functions and activation functions, are considered. By using homeomorphism theory and inequality technique, the sufficient conditions for the existence of unique equilibrium point are obtained. Then, by constructing a suitable Lyapunov–Krasovskii functional, the global asymptotic stability condition of the proposed neural networks is derived in terms of linear matrix inequalities. This linear matrix inequality can be efficiently solved via the standard numerical packages. Finally, the numerical examples are given to validate the effectiveness of theoretical results.

The Role of Hyalomma Truncatum on the Dynamics of Rift Valley Fever: Insights from a Mathematical Epidemic Model.

To date, our knowledge of Rift Valley fever (RVF) disease spread and maintenance is still limited, as flooding, humid weather and presence of biting insects such as mosquitoes, have not completely explained RVF outbreaks. We propose a model that includes livestock, mosquitoes and ticks compartments structured according to their questing and feeding behaviour in order to study the possible role of ticks on the dynamics of RVF. To quantify disease transmission at the initial stage of the epidemic, we derive an explicit formula of the basic reproductive number, [Formula: see text]. Using the concept of Metzler matrix, we state necessary conditions for global asymptotic stability of the disease-free equilibrium. Results suggest that although host-ticks interactions may serve as disease reservoirs or disease amplifiers, the Aedes reproductive number should be kept under unity if disease post-epizootics activities are to be controlled. Results of both local and global sensitivity analysis of selected model parameters indicate that [Formula: see text] is more sensitive to the ticks attachment and detachment rates, probability of transmission from ticks to host and from host to ticks, length of infection in livestock and ticks death rate. Furthermore, when comparing the mean value of [Formula: see text] with that from previous studies which did not include ticks we found that our [Formula: see text] is very much larger resulting in an increase in the exponential phase of an outbreak. These findings suggest that if ticks are capable of transmitting the virus, they may be contributing to disease outbreaks and endemicity.

Global Properties of a HBV Infection Model with HBV DNA-Containing Capsids and CTL Immune Response

In this paper, the global dynamical behavior of a hepatitis B virus (HBV) infection model with HBV DNA-containing capsids and cytotoxic T lymphocytes (CTLs) immune response is investigated. We derive the conditions for global asymptotic stability of the steady states of the model in terms of the basic reproduction number $$R_{0}$$ and the immune response reproduction number $$R_{{ CTL}}$$ . By constructing appropriate Lyapunov functions, it is shown that the disease-free steady state is globally asymptotically stable when $$R_{0}\le 1$$ , the immune-free steady state is globally asymptotically stable when $$R_{{ CTL}}\le 1<R_{0}$$ and the endemic steady state is globally asymptotically stable when $$R_{{ CTL}}>1$$ . Further, we incorporate two discrete delays in the model to account for the intracellular delays in the production of productively infected hepatocytes and capsids. We also derive the global properties of this two-delay model in terms of $$R_{0}$$ and $$R_{{ CTL}}$$ . Finally, illustrative numerical simulations are presented to support our theoretical findings.

Asymptotic stability analysis of autonomous systems by applying the method of localization of compact invariant sets

The asymptotic stability and global asymptotic stability of equilibria in autonomous systems of differential equations are analyzed. Conditions for asymptotic stability and global asymptotic stability in terms of compact invariant sets and positively invariant sets are proved. The functional method of localization of compact invariant sets is proposed for verifying the fulfillment of these conditions. Illustrative examples are given.

Global asymptotic stabilization of periodic nonlinear systems with stable free dynamics

We consider a nonlinear control system with periodic coefficients. We study the problem of asymptotic stabilization of the equilibrium x=0 of the closed-loop system by state feedback. We assume that the free dynamic system possesses a periodic Lyapunov function ensuring Lyapunov stability of the equilibrium x=0. We have developed a method for constructing a damping control for affine systems with periodic coefficients based on a generalization of weak Jurdjevic–Quinn conditions, using an extension of the notion of the commutator to non-stationary vector fields. Using this approach, we obtain sufficient conditions for uniform local and global asymptotic stabilization of general nonlinear systems, in particular, affine control systems, with periodic coefficients. Stabilization results generalize known results for time-invariant systems to time-varying periodic systems.

A Model of Perfect Pediatric Vaccination of Dengue with Delay and Optimal Control

A delayed mathematical model of Dengue dynamical transmission between vector mosquitoes and human, incorporating a control strategy of perfect pediatric vaccination is proposed in this paper. By some analytical skills, we obtain the existence of disease-free equilibria and endemic equilibrium, the necessary conditions of global asymptotical stability about two disease-free equilibria. Further, by Pontryagin’s maximum principle, we obtain the optimal control of the disease. Finally, numerical simulations are carried out to verify the correctness of the theoretical results and feasibility of the control measure.

Stability Analysis of the Periodic Solutions of Some Kinds of Predator-Prey Dynamical Systems

Analysis of predator-prey dynamical systems that have the functional response which generalizes the other types of functional responses in two dimensions is mainly studied in this paper. The main problems for this study are to detect the if and only if conditions for attaining the periodic solution of the considered system and to find the condition for global asymptotic stability of this solution for some different types of predator-prey systems that are obtained from that system. To get the desired results, some aspects of semigroup theory for stability analysis and coincidence degree theory are used.

Sufficient and necessary conditions for global stability of genetic regulator networks with time delays

This paper is concerned with the global stability of the nonlinear model for genetic regulator networks (GRNs) with time delays. Four new sufficient and necessary conditions for global asymptotic stability and global exponential stability of the equilibrium point of GRNs are derived. Specifically, using comparing theorem and Dini derivation method, three weak sufficient conditions for global stability of GRNs with constant time delays are proposed. Finally, a general GRN model is used to illustrate the effectiveness of the proposed theoretical results. Compared with the previous results, some sufficient and necessary conditions for Lyapunov stability of GRNs are proposed, which are not seen before.

Compound-combination synchronization of chaos in identical and different orders chaotic systems

Abstract This paper proposes a new synchronization scheme called compound-combination synchronization. The scheme is investigated using six chaotic Josephson junctions evolving from different initial conditions based on the drive-response configuration via the active backstepping technique. The technique is applied to achieve compound-combination synchronization of: (i) six identical third order resistive-capacitive-inductive-shunted Josepshon junctions (RCLSJJs) (with three as drive and three as response systems); (ii) three third order RCLSJJs (as drive systems) and three second order resistive-capacitive-shunted Josepshon junctions (RCSJJs (as response systems). In each case, sufficient conditions for global asymptotic stability for compound-combination synchronization to any desired scaling factors are achieved. Numerical simulations are employed to verify the feasibility and effectiveness of the compound-combination synchronization scheme. The result shows that this scheme could be used to vary the junction signal to any desired level and also give a better insight into synchronization in biological systems wherein different organs of different dynamical structures and orders are involved. The scheme could also provide high security in information transmission due to the complexity of its dynamical formulation.

Global analysis of an infection age model with a class of nonlinear incidence rates

SIR infection age models with a very general class of nonlinear incidence rates f(S,J) are investigated. We give a necessary and sufficient condition for global asymptotic stability of the free-equilibrium related to the basic reproduction number. Furthermore, additional conditions allow us to prove an exponential stability of this disease-free equilibrium. Finally, by using a Lyapunov functional, we show the global asymptotic stability of the endemic equilibrium whenever it exists.

Role of forcing in large-time behavior of vorticity equation solutions on a sphere

The nonlinear barotropic vorticity equation (BVE) describing the vortex dynamics of viscous incompressible and forced fluid on a rotating sphere is considered. The asymptotic behavior of solutions of nonstationary BVE as t → ∞ is studied. Particular forms of the external vorticity source are given that guarantee the existence of a bounded attractive set in the phase space of solutions. The asymptotic behavior of the BVE solutions is shown to depend on both the structure and the smoothness of external forcing. Three types of sufficient conditions for global asymptotic stability of smooth and weak BVE solutions are also given. Simple attractive sets of a viscous incompressible fluid on a sphere under quasi-periodic polynomial forcing are considered. Each attractive set represents a BVE quasi-periodic solution of the complex (2n + 1)-dimensional subspace Hn of homogeneous spherical polynomials of degree n. The Hausdorff dimension of its trajectory, being an open spiral densely wound around a 2n-dimensional torus in Hn, equals to 2n. As the generalized Grashof number G becomes small enough then the domain of attraction of such spiral solution is expanded from Hn to the entire BVE phase space. It is shown that for a given G, there exists an integer nG such that each spiral solution generated by a forcing of Hn with n ≥ nG is globally asymptotically stable. Thus we demonstrate the difference in the asymptotic behavior of solutions in the cases, then Grashof number G is fixed and bounded, but the forcing is stationary or non-stationary. Whereas the dimension of the fluid attractor under a stationary forcing is limited above by G, the dimension of the spiral attractive solution (equal to 2n) may become arbitrarily large as the degree n of the quasi-periodic polynomial forcing grows. Since the small-scale quasi-periodic functions, unlike the stationary ones, more adequately depict the forcing in the barotropic atmosphere, this result is of meteorological interest and shows that the dimension of attractive sets depends not only on the forcing amplitude, but also on its spatial and temporal structure. This example also shows that the search of a finite-dimensional global attractor in the barotropic atmosphere is not well justified.

Global Dynamics of a SEIRS Epidemic Model with Saturated Disease Transmission Rate and Vaccination Control

In this paper we take a sound approach to construct a susceptible-exposed-infected-recovered-susceptible type epidemic model with explicit vaccination control. The disease transmission from susceptible to infected class is taken in saturated form. The system is investigated both for constant control and optimal control cases. Basic reproduction number and its consequences to the system has been established. Geometric approach is used to obtain the global asymptotic stability condition of the system. Finally, with the help of some numerical illustrations, we check the validity of our theoretical results.

Stability and synchronization of memristor-based fractional-order delayed neural networks

Global asymptotic stability and synchronization of a class of fractional-order memristor-based delayed neural networks are investigated. For such problems in integer-order systems, Lyapunov–Krasovskii functional is usually constructed, whereas similar method has not been well developed for fractional-order nonlinear delayed systems. By employing a comparison theorem for a class of fractional-order linear systems with time delay, sufficient condition for global asymptotic stability of fractional memristor-based delayed neural networks is derived. Then, based on linear error feedback control, the synchronization criterion for such neural networks is also presented. Numerical simulations are given to demonstrate the effectiveness of the theoretical results.

A necessary and sufficient condition for global asymptotic stability of time-varying Lotka–Volterra predator–prey systems

The purpose of this paper is to present a necessary and sufficient condition which guarantees that an interior equilibrium of a certain predator–prey system is globally asymptotically stable. This ecological system is a model of Lotka–Volterra type whose prey population receives time-variation of the environment. We assume that the time-varying coefficient is weakly integrally positive and has a weaker property than uniformly continuous. Our necessary and sufficient condition is expressed by an improper double integral on the time-varying coefficient. Our work is inspired by the study of the stability theory for damped linear oscillators.

Global attitude stabilization of rigid spacecraft with unknown input delay

Global attitude stabilization of a rigid spacecraft with unknown actuator delay time is an important problem that has rarely been studied. In this paper, first we investigate a Lyapunov-based controller for attitude regulation of a rigid spacecraft with delayed inputs. Simple conditions for global asymptotical stability are obtained by assuming that the true delay value is unknown, but approximation of its upper bound is available. It is also shown that a proper design of the controller prevents the unwinding phenomenon. Then, we extend the results for the system while taking disturbance effects into account. Based on Lyapunov–Krasovskii methodology, it is proven that the proposed controller can drive the closed-loop trajectories to a small region in the neighborhood of the origin in the presence of external disturbances and model uncertainties. Various numerical simulations are carried out to illustrate the effectiveness of the proposed control system.

Stability tests for second order linear and nonlinear delayed models

For the nonlinear second order Lienard-type equations with time-varying delays $$\ddot{x}(t)+\sum_{k=1}^m f_k(t,x(t),\dot{x}(g_k(t)))+\sum_{k=1}^ls_k(t,x(h_k(t)))=0,$$ global asymptotic stability conditions are obtained. The results are based on the new sufficient stability conditions for relevant linear equations and are applied to derive explicit stability conditions for the nonlinear Kaldor–Kalecki business cycle model. We also explore multistability of the sunflower non-autonomous equation and its modifications.

Global stability of a stochastic predator–prey model with Allee effect

In this paper, we study a stochastic predator–prey model with Beddington–DeAngelis functional response and Allee effect, and show that there is a unique global positive solution to the system with the positive initial value. Sufficient conditions for global asymptotic stability are established. Some simulation figures are introduced to support the analytical findings.

Sufficient and necessary conditions for Lyapunov stability of genetic networks with SUM regulatory logic

In this paper, a nonlinear model for genetic regulator networks (GRNs) with SUM regulatory logic is presented. Four sufficient and necessary conditions of global asymptotical stability and global exponential stability for the equilibrium point of the GRNs are proposed, respectively. Specifically, three weak sufficient conditions and corresponding corollaries are derived by using comparing theorem and Dini derivative method. Then, a famous GRN model is used as the example to illustrate the effectiveness of our theoretical results. Comparing to the results in the previous literature, some novel ideas, study methods and interesting results are explored.