Equilibrium Point Analysis Research Articles

Stability analysis of distributed-order nonlinear dynamic systems

ABSTRACTThe problem of asymptotic stability analysis of equilibrium points in nonlinear distributed-order dynamic systems with non-negative weight functions is considered in this paper. The Lyapunov direct method is extended to be used for this stability analysis. To this end, at first, a discretisation scheme with convergence property is introduced for distributed-order dynamic systems. Then, on the basis of this tool, Lyapunov theorems are proved for asymptotic stability analysis of equilibrium points in distributed-order systems. As the order weight function assumed for the distributed-order systems is general enough, the results are applicable to a wide range of nonlinear distributed-order systems such as fractional-order systems with multiple fractional derivatives. To verify the applicability of the obtained results, these results are applied for the stability analysis of a distributed-order diffusion system and control of a fractional-order Lorenz system with multiple fractional derivatives.

Optimal predator control policy and weak Allee effect in a delayed prey–predator system

In this article, a system of delay differential equations to represent the predator–prey dynamics with weak Allee effect in the growth of predator population is discussed. The delay parameter regarding the time lag corresponds to the predator gestation period. Mathematical features such as uniform persistence, permanence, stability, Hopf bifurcation at the interior equilibrium point of the system are analyzed and verified by numerical simulations. Bistability between different equilibrium points is properly discussed. The chaotic behaviors of the system are recognized through bifurcation diagram, Poincare section, and maximum Lyapunov exponent. By constructing a suitable Lyapunov functional for the time-delayed model, global asymptotic stability analysis of the positive equilibrium points has been performed separately. It can be observed that the Allee parameter $$\theta $$ can destabilize the non-delay system, whereas $$\theta $$ and the attack rate of predator can stabilize the time-delayed model and can control the chaotic oscillations through period-halving bifurcation. The optimal predator control policy with Allee parameter ( $$\theta $$ ) as the control parameter is also discussed.

Pattern formation by super-diffusion in FitzHugh–Nagumo model

The aim of this paper is to examine the Turing instability and pattern formation in the FitzHugh–Nagumo model with super-diffusion in two dimensional numerical simulation. We also studied the effects of the super-diffusive exponent on pattern formation concluding that with the presence of super-diffusion the stable homogenous steady state becomes unstable. By using the stability analysis of local equilibrium point, we procure the conditions which ensure that the Turing and Hopf bifurcations occur. For pattern selection, the weak nonlinear multi-scale analysis is used to derive the amplitude equations of the stationary patterns. We then apply amplitude equations and observe that this model has very rich dynamical behaviors, such as stripes, spots and hexagon patterns. The complexity of the dynamics in this system is theoretically discussed and graphically displayed in numerical simulation. The simulation helps us to show the effectiveness of theoretical analysis and patterns which appear numerically.

Modulated compact-like pulse signals in a nonlinear electrical transmission line: A specific case studied

The nonlinear electrical transmission line (NLTL) with intersite nonlinearity, recently introduced by Kenmogne and Yemélé is reexplored, with a particular attention carried on their obtained extended nonlinear Schrödinger (ENLS) equation, that is the basic NLS equation with additional nonlinear dispersive terms. Through the dynamical studies and analysis of equilibrium points it is shown that in the absence of linear dispersion term, a new solution for the ENLS equation is found, that is the envelope bright solitary wave with compact support. Analytical criterion of existence of this compact bright enveloped soliton is derived, and its stability is confirmed through numerical simulations of the exact equations of the network. Finally, some potential applications in communication systems are suggested.

A MATHEMATICAL APPROACH TO STUDY THE EFFECT OF POLLUTANTS/TOXICANTS IN AQUATIC ENVIRONMENT

Acid lowers the pH levels in water bodies below what is required for survival of aquatic life and increases the toxicity of metals. For this effect, a mathematical model has been proposed using a system of non-linear ordinary differential equations with four state variables. The dependent variables are amount of acid and metal in water, density of favorable resources (phytoplankton), density of fish population and nutrient concentration under the assumption that the amount of metal present in water is less than the amount of acid present in water. Conditions for local stability and feasible equilibrium points have been determined. Nonlinear stability analysis of the non-trivial equilibrium points has been discussed and it was found that system of the differential equations show more feasible results if the crowding effect is incorporated for fish population. Further it was also observed that, nutrients play important role for the growth and survival of the species. Conditions for the existence of the equilibrium points have been drawn and the criteria for the survival or the extinction of the species have been obtained using numerical simulation. Stability of the system is explained analytically as well as graphically.

Saturating Stiffness Control of Robot Manipulators with Bounded Inputs

Abstract A saturating stiffness control scheme for robot manipulators with bounded torque inputs is proposed. The control law is assumed to be a PD-type controller, and the corresponding Lyapunov stability analysis of the closed-loop equilibrium point is presented. The interaction between the robot manipulator and the environment is modeled as spring-like contact forces. The proper behavior of the closed-loop system is validated using a three degree-of-freedom robotic arm.

Multi-piecewise quadratic nonlinearity memristor and its 2N-scroll and 2N + 1-scroll chaotic attractors system.

In this paper, two kinds of novel ideal active flux-controlled smooth multi-piecewise quadratic nonlinearity memristors with multi-piecewise continuous memductance function are presented. The pinched hysteresis loop characteristics of the two memristor models are verified by building a memristor emulator circuit. Using the two memristor models establish a new memristive multi-scroll Chua's circuit, which can generate 2N-scroll and 2N+1-scroll chaotic attractors without any other ordinary nonlinear function. Furthermore, coexisting multi-scroll chaotic attractors are found in the proposed memristive multi-scroll Chua's circuit. Phase portraits, Lyapunov exponents, bifurcation diagrams, and equilibrium point analysis have been used to research the basic dynamics of the memristive multi-scroll Chua's circuit. The consistency of circuit implementation and numerical simulation verifies the effectiveness of the system design.

Analysis of a hand–foot–mouth disease model

In this paper, we have introduced a six-compartmental epidemic model with hand, foot and mouth disease (HFMD) infection. The total population is divided into six subclasses, namely susceptible, exposed, infective in asymptomatic phase, infective in symptomatic phase, quarantined and recovered class. Some basic properties such as boundedness and non-negativity of solutions are discussed. The basic reproduction number ([Formula: see text]) of the system is obtained using next generation matrix method. Then the deterministic dynamical behaviors of the system are studied. Our study includes the existence and stability analysis of equilibrium points of the system. The sensitivity analysis of our system helps us to find out the parameters of greater interest. Next, we deal with the epidemic model with three controls (two treatment controls with quarantine control). We show that there exists an optimal control, which is effective in controlling the disease outbreak in a cost effective way. Numerical simulation is presented with the help of MATLAB, which shows the reliability of our model from the practical point of view.

Effects of system parameter and fractional order on dynamic behavior evolution in fractional-order Genesio-Tesi system

The object of this paper is to reveal the relationship between dynamics of the fractional-order Genesio-Tesi system and its parameter, especially the order of fractional derivatives. At first, the transcritical bifurcation is carried out based on the stability analysis of equilibrium points. Then, effects of system parameter and fractional order on the dynamics have been investigated deeply and systematically. For commensurate fractional-order system, period-doubling bifurcation, reverse period-doubling bifurcation, period window, transient chaos and steady chaos are found with parameter and fractional order varying simultaneously. For incommensurate fractional-order system, there also exists the route to chaos by period-doubling bifurcation. Furthermore, the effect of different fractional orders on dynamical behavior evolution has been compared. These new findings contribute to successful selecting the most efficient control function, which has been validated by some numerical tools.

Pengembangan Model Matematika SIRD (Susceptibles-Infected-Recovery-Deaths) Pada Penyebaran Virus Ebola

Ebola is a deadly disease caused by a virus and is spread through direct contact with blood or body fluids such as urine, feces, breast milk, saliva and semen. In this case, direct contact means that the blood or body fluids of patients were directly touching the nose, eyes, mouth, or a wound someone open.
 In this paper examined two mathematical models SIRD (Susceptibles-Infected-Recovery-Deaths) the spread of the Ebola virus in the human population. Both the mathematical model SIRD on the spread of the Ebola virus is a model by Abdon A. and Emile F. D. G. and research development model. This study was conducted to determine the point of disease-free equilibrium and endemic equilibrium point and stability analysis of the dots, knowing the value of the basic reproduction number (R0) and a simulation model using Matlab software version 6.1.0.450.
 From the analysis of the two models, obtained the same point for disease-free equilibrium point with the stability of different points and different points for endemic equilibrium point with the stability of both the same point and the same value to the value of the basic reproduction number (R0). After simulating the model using Matlab software version 6.1.0.450, it can be seen changes in the behavior of the population at any time.

Bifurcation analysis of a single-group asset flow model

We study the stability and Hopf bifurcation analysis of an asset pricing model that based on the model introduced by Caginalp and Balenovich in 1999, under the assumption of a xed amount of cash and stock in the system. First, we study stability analysis of equilibrium points. Choosing the momentum coe¢ cient as a bifurcation parameter, we also show that Hopf bifurcation occurs when the bifurcation parameter passes through a critical value. Analytical results are supported by numerical simulations. A key conclusion for economics and nance is the existence of periodic solutions for an interval of the bifurcation parameter, which is the trend-based (or momentum) coe¢ cient. Key words. Asset price dynamics, stability of price dynamics, Hopf bifurcation, price trend, momentum, market dynamics, liquidity, periodic solutions. AMS subject classi cation. : 91B25, 91B26, 91B50, 91G80, 91G99, 34D20, 34C60, 37G15, 37N40 1. Introduction. A central theme in classical nance is that market participants all have access to the same information, and all seek to optimize their returns so that a unique equilibrium price is established (see, for example, [3], [18], [20]). The approach to equilibrium is often assumed to be a process involving some randomness or noise, but otherwise smooth and rapid. Aside from noise, one can expect that prices will not overshoot the equilibrium price since the equation governing the change in price, P , is a rst order in time, i.e., P 0 = F (D=S) where S and D are supply and demand that depend on price but not on the recent price derivative history. As such, there is no mechanism for oscillations or cyclic behavior within this setting. A well known example of cyclic behavior in economics is called the cobweb theorem whereby prices oscillate periodically due to the time lag between supply and demand decisions. Agricultural commodities provide a simple example with a delay between planting and harvesting (see [21] (pages 133-134 gives two agricultural examples: rubber and corn) and [36]). In nancial markets, however, the prevailing theory (at least during latter part of the 20th century), called the e¢ cient market hypothesis (EMH), maintains the existence of in nite arbitrage capital that would quickly exploit any deviations between the trading price and the intrinsic or fundamental value of the asset, which are necessarily unique since the participants have the same information and calculation of future returns. The absence of any delay in information or trading excludes, mathematically, the possibilities of overshooting the equilibrium price or oscillating about it. While policy makers often discuss instabilities in asset prices, classical nance tends to treat these as rare occurrences within a stochastic setting. In particular, much of classical nance is based on the concept that an asset’s price, P (t), is governed by Current address: Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA. Permanent address: Department of Mathematics, TOBB University of Economics and Technology, 06560-Ankara, TURKEY. H. Merdan was supported by TUBITAK (The Scienti c and Technological Research Council of Turkey) yDepartment of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA zDepartment of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA

Modeling and Analysis of Malt-Drug Resistance Tuberculosis in Densly Populated Areas

Tuberculosis is an airborne disease caused by the bacterium called mycobacterium tuberculosis. We have compartmentalized the population based on the exposed level to the disease and described the flow using a flowchart. Mathematical model is developed to describe the population dynamics of the compartments. The migration of people from infected class to exposed class, due to failure of continuing the medicine for any reason, is called here as Malt – drug resistance tuberculosis. The equilibrium points identified are disease free, endemic and epidemic. Equilibrium point analysis is made and has been included. Formula for reproduction number is derived. Numerical simulation study of the Mathematical model is conducted using ode45 function of MATLAB software. It is shown that the propagation of the disease is more in the more populated areas and less in the less populated areas.

NVT 分子動力学シミュレーションを用いたメタンハイドレート/水/メタン系における三相平衡点の解析

NVT 分子動力学シミュレーションを用いたメタンハイドレート/水/メタン系における三相平衡点の解析

Improving Performance of Ieee 802.11 by a Dynamic Control Backoff Algorithm Under Unsaturated Traffic Loads

The IEEE 802.11 backoff algorithm is very important for controlling system throughput over contentionbased wireless networks. For this reason, there are many studies on wireless network performance focus on developing backoff algorithms. However, most existing models are based on saturated traffic loads, which are not a real representation of actual network conditions. In this paper, a dynamic control backoff time algorithm is proposed to enhance both delay and throughput performance of the IEEE 802.11 distributed coordination function. This algorithm considers the distinction between high and low traffic loads in order to deal with unsaturated traffic load conditions. In particular, the equilibrium point analysis model is used to represent the algorithm under various traffic load conditions. Results of extensive simulation experiments illustrate that the proposed algorithm yields better performance throughput and a better average transmission packet delay than related algorithms.

Modeling and Stability of Prey - Predator System Involving Infectious Disease in Each Population with Harvesting of the Prey

In this paper, a mathematical model consisting of the prey- predator model involving disease in both the population with harvesting of prey is proposed and analyzed. The existence, uniqueness and boundedness of the solution are discussed. The existences and the stability analysis of all possible equilibrium points are studied. Finally the effects both of the disease and harvest on the dynamical behavior the model are discussed numerically.

Additional Food Induced Turing Patterns for a Diffusive Predator–Prey Model

We present a diffusive ratio-dependent predator–prey model with additional food supply to predators and investigate the spatiotemporal dynamics induced as a result of this additional food supply. We firstly, consider the co-existential equilibrium for the temporal model and discuss its local stability properties. We then carry out the local stability analysis of this co-existential equilibrium point for the spatiotemporal model. Secondly, we derive the parametric conditions under which Turing instability (i.e., the co-existential equilibrium point is stable for the temporal model but is unstable with respect to the spatiotemporal model) can occur as a result of additional food being added to the system. We also show that Turing bifurcation takes place and a spatially heterogeneous pattern emerges over time. Finally, the corresponding Turing patterns are revealed by way of simulations. In conclusion, we observe the important role of the quality and quantity of additional food in order for the Turing instability to occur even for parameter values for which the co-existential equilibrium of the original ratio-dependent model either does not exist or is unstable.

The Effect of Disease and Harvest in the Eco-Epidemical Food Chain Model

This paper describes a three species food web model with linear functional response and incidence disease. This model consisting of a prey, intermediate predator and top predator where harvesting of top predator species and SIS disease spread in prey are taken into consideration. The stability analysis of all possible equilibrium points are carried out. We discussed the effect both of harvest and disease on the stability of this model. Finally, we used the numerical simulations to verify the analytical results.

Modeling the spread of Rubella disease using the concept of with local derivative with fractional parameter

Our aim in this work was to examine the model underpinning the spread of the Rubella virus using the novel derivative called beta‐derivative. The study of the equilibrium points together with the analysis of the disease free equilibrium points was presented. Due to the complexity of the modified equation, we introduced a new operator based on the Sumudu transform. The properties of this operator were proposed and proved in detail. We made used of this operator together with the idea of perturbation method to derive a special solution of the extended model. The stability of the method for solving this model was presented. The uniqueness of the special solution was presented, and numerical simulations were done. The graphical representations show that the model depends on both parameters and the fractional order. © 2015 Wiley Periodicals, Inc. Complexity 21: 442–451, 2016

A delayed eco-epidemiological system with infected prey and predator subject to the weak Allee effect

We consider a system of delay differential equations to represent predator–prey eco-epidemic dynamics with weak Allee effect in the growth of predator population. The basic aim of the paper is to observe the dynamics of such system under the influence of gestation delay of predator and Allee parameter. We analyze essential mathematical features of the proposed model such as uniform persistence, stability and Hopf-bifurcation at the interior equilibrium point of the system. Global asymptotic stability analysis of the positive equilibrium points by constructing a suitable Lyapunov function for the delayed model is carried out separately. We perform several numerical simulations to illustrate the applicability of the proposed mathematical model and our analytical findings. We observe that the system exhibits chaotic oscillation due to increase of the delay parameter τ. We also observe that there is a threshold of Allee parameter above which the predator population will be washed away from the system.

The Effect of Two Infectious Diseases and Harvesting in the Eco-Epidemiological Model

In this paper, an Eco-Epidemiological model with different infectious diseases in prey population and the optimal harvesting in predator population is proposed and studied. Linear type used to describe the functional response. Sufficient and necessary conditions for existence each equilibrium points of the system are established, the bounded and stability analysis of all possible equilibrium points are studied. The effect of harvest on the stability of this system is investigated. Finally, the dynamical behavior of system is discussed using Numerical simulation.