Equilibrium Point Analysis Research Articles

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.

A mathematical study of a predator–prey model with disease circulating in the both populations

Disease in ecological systems plays an important role. In the present investigation we propose and analyze a predator–prey mathematical model in which both species are affected by infectious disease. The parasite is transmitted directly (by contact) within the prey population and indirectly (by consumption of infected prey) within the predator population. We derive biologically feasible and insightful quantities in terms of ecological as well as epidemiological reproduction numbers that allow us to describe the dynamics of the proposed system. Our observations indicate that predator–prey system is stable without disease but high infection rate drive the predator population toward extinction. We also observe that predation of vulnerable infected prey makes the disease to eradicate into the community composition of the model system. Local stability analysis of the interior equilibrium point near the disease-free equilibrium point is worked out. To study the global dynamics of the system, numerical simulations are performed. Our simulation results show that for higher values of the force of infection in the prey population the predator population goes to extinction. Our numerical analysis reveals that predation rates specially on susceptible prey population and recovery of infective predator play crucial role for preventing the extinction of the susceptible predator and disease propagation.

Modeling and Stability Analysis for Measles Metapopulation Model with Vaccination

In this paper, a metapopulation model is formulated as a system of ordinary differential equations to study the impact of vaccination on the spread of measles. The disease-free equilibrium is computed and proved to be locally and globally asymptotically stable if RC 1. We show that when there are no movements between the two patches, there exists at least one endemic equilibrium for all RCi >1 and bifurcation analysis of endemic equilibrium point proves that forward (supercritical) bifurcation occurs in each patch. Numerical simulation results are also presented to validate analytical results and to show the impact of vaccination on the incidence and prevalence of measles in a metapopulation.

Dynamics in Bank Crisis Model

Bank crisis is grabbing more serious attention as several financial turmoils have broken out in the past several decades, which leads to a number of researches in this field. Comparing with researches carried out on basis of degree distribution in complex networks, this paper puts forward a mathematical model constructed upon dynamic systems, for which we mainly focus on the stability of critical point. After the model is constructed to describe the evolution of the banking market system, we devoted ourselves to find out the critical point and analyze its stability. However, to refine the stability of the critical point, we add some impulsive terms in the former model. And we discover that the bank crisis can be controlled according to the analysis of equilibrium points of the modified model, which implies the interference from outside may modify the robustness of the bank network.

Chaos to Order: Role of Additional Food to Predator in a Food Chain Model

The chaos and its controllability is one of the main objectives of mathematical ecology today. In this paper, we propose a new chaos control technique for a three species predator prey system by supplying additional food to predators. A thorough mathematical analysis in this direction revealed the vital role of quality and quantity of the additional food in the controllability of the predator–prey system. The local stability analysis of boundary and interior equilibrium points of the system are done. We have determined the persistence conditions of the system. Bifurcation analysis of the proposed model is done with respect to quality and quantity of additional food. A Hopf point bifurcation phenomena is introduced to detect periodic behaviour of the system. By varying quality and quantity of additional food one can control the chaotic dynamics of a food chain. We establish that chaotic population dynamics of predator–prey system can be controlled by supplying additional food to predators. This study introduce a new chaos control mechanism in a predator–prey system which has applications in fishery management and biological conservation of prey predator species. This model has a special importance in the systems, such as caves, some small marine islands etc.

Analysis of Vehicle Dynamic Equilibrium Points with 3-DOF Based on Genetic Algorithm

To further study the stability of vehicle dynamics, a vehicle handling stability’s nonlinear model (including longitudinal, lateral and yaw movement three degrees of freedom) was established. Genetic algorithm was proposed for the vehicle dynamics system’s equilibrium points with 3-DOF. This algorithm solves the problem that cannot be solved through the traditional analytic algorithms and numerical methods. Comparing with the existing research results, the feasibility of solving the equilibrium point by the genetic algorithm is verified. It provides the theoretical foundation for dynamic modification and optimization design of powertrain.

On stability of equilibrium points in nonlinear fractional differential equations and fractional Hamiltonian systems

In this article, a brief stability analysis of equilibrium points in nonlinear fractional order dynamical systems is given. Then, based on the first integral concept, a definition of planar Hamiltonian systems with fractional order introduced. Some interesting properties of these fractional Hamiltonian systems are also presented. Finally, we illustrate two examples to see the differences between fractional Hamiltonian systems with their classical order counterparts.© 2014 Wiley Periodicals, Inc. Complexity 21: 93–99, 2015

Consensus Equilibrium Point Analysis for Heterogeneous Multi-Agent System in Undirected Graph

The consensus problem of heterogeneous multi-agent systems is investigated. A linear consensus protocol is proposed for this system. Aiming at consensus condition can be only obtained, but the consensus equilibrium point cannot be obtained in this study, the consensus equilibrium point is derived by applying graph theory and matrix theory in this paper. Finally, a numerical simulation is provided to illustrate the effectiveness of the theoretical result.

DYNAMIC CHARACTER ANALYSIS OF A LDR, MEMRISTOR-BASED CHAOTIC SYSTEM

Based on the achievements of realization of the light dependent resistor (LDR) memristor (MR) analog model, a chaotic system has been proposed in this paper. The construction of such a nonlinear system is firstly shown in this paper, by dissipative characteristic analysis, equilibrium point and stability analysis, the strange attractor is possibly existing at each equilibrium point is confirmed. Also the dynamic characters are discussed, which indicates the nonlinear circuit we built based on LDR MR is a chaotic system.

A Parameter Setting Method for Spiral Optimization from Stability Analysis of Dynamic Equilibrium Point

In recent years, the authors have proposed an effective metaheuristic method for continuous optimization problems based on an analogy of spiral phenomena in nature. This method is called Spiral Opt...

Vaccination Dynamics of Chickenpox in Agona West Municipality of Ghana

The study showed that chickenpox cases in recent times have increased in Agona West Municipality with = 2 . We estimated the reproductive number of chickenpox with vaccination to be R = 0.3 < 1. This showed that after vaccination, the disease will die out. It was estimated that 46.2% corresponding to 53, 241 of the people needs to be vaccinated in order to control the chickenpox. The stability analysis of disease free and endemic equilibrium point of chickenpox transmission with vaccination was estimated to be unstable and asymptotically stable. Sensitivity analysis of the SEIR model with vaccination showed that when vaccination rate coefficient is increased chickenpox will be under control.

Mathematical Modelling of Population Growth: The Case of Logistic and Von Bertalanffy Models

In this paper, some theoretical mathematical aspects of the known predator-prey problem are considered by relaxing the assumptions that interaction of a predation leads to little or no effect on growth of the prey population and the prey growth rate parameter is a positive valued function of time. The predator growth model is derived considering that the prey follows a known growth models viz., Logistic and Von Bertalanffy. The result shows that the predator’s population growth models look to be new functions. For either models, the predator population size either converges to a finite positive limit or to 0 or diverges to +∞. It is shown algebraically and illustrated pictorially that there is a condition at which the predator-prey population models both converge to the same finite limit. Derivations and simulation studies are provided in the paper. Analysis of equilibrium points and stability is also included.

1P1-O02 Analysis and compensation of disturbances for wheeled inverted pendulum assistant robot for load carrying and ramp climbing : Equilibrium point analysis in the presence of disturbances(Mobile Robot with Special Mechanism (1))

In this paper the analysis of the disturbances that enter the system of a wheeled inverted pendulum type assistant robot is presented. Using null space techniques the equilibrium conditions under the presence of disturbances and the subsequent effects on the system is determined. A new controller based on the Extended State Observer is implemented to counteract these effects. Results of experiments comparing the new and previous control for both load carrying and ramp climbing are discussed and presented.

Analysis of dynamical behaviors in a nonlinear switching circuit system

A nonlinear circuit model with periodic switching is established. The fold bifurcation and Hopf bifurcation sets of the subsystems are derived via the analysis of the relevant equilibrium points as well as the stabilities. Complex dynamical behaviors caused by periodic switching in various equilibrium states of subsystems are investigated. The results show that there exist two types of destabilizing cases, i.e., period-doubling bifurcation and saddle-node bifurcation, in the variation of periodic solution to the switching system with parameter, leading to different forms of chaotic oscillations correspondingly. Furthermore, by analyzing the the phase trajectory and its corresponding bifurcation, the mechanisms for different types of oscillations are presented, which can explain some phenomena of the switched dynamical system.

Performance Modeling and Stability of Semi-Persistent Scheduling with Initial Random Access in LTE

In this paper we examine the feasibility of semi-persistent scheduling (SPS) for voice over IP (VoIP) by random access and evaluate its performance in terms of throughput of random access and traffic channels, and random access delay. We further investigate system stability issues and present methods to stabilize the system. To see the VoIP capacity gain, we show the maximum number of acceptable VoIP terminals without exceeding some front-end packet dropping (i.e., voice clipping) probability. In addition, we examine the effect of the parameter called implicit release after in the LTE standard on the system performance, which is used for silence period detection. Our performance evaluation model based on Equilibrium Point Analysis is compared to simulations.

Dynamical behaviors of a system with switches between the Rössler oscillator and Chua circuits

The behaviors of a system that alternates between the Rössler oscillator and Chua's circuit is investigated to explore the influence of the switches on the dynamical evolution. Switches related to the state variables are introduced, upon which a typical switching dynamical model is established. Bifurcation sets of the subsystems are derived via analysis of the related equilibrium points, which divide the parameters into several regions corresponding to different types of attractors. The dynamics behave typically in period orbits with the variation of the parameters. The focus/cycle periodic switching phenomenon is explored in detail to present the mechanism of the movement. The period-doubling bifurcation to chaos can be observed via the doubling increase of the turning points related to the switches. Furthermore, period-decreasing sequences have been obtained, which can be explained by the variation of the eigenvalues associated with the equilibrium points of the subsystems.

MODELING H1N1 FLU EPIDEMIC WITH CONTACT TRACING AND QUARANTINE

A nonlinear mathematical model is proposed and analyzed to study the dynamics of 2009 H1N1 flu epidemic in a homogeneous population with constant immigration of susceptibles. The effect of contact tracing and quarantine (isolation) strategies in reducing the spread of H1N1 flu is incorporated. The model monitors the dynamics of five sub-populations (classes), namely susceptible with high infection risk, susceptible with reduction of infection risk, infective, quarantined and recovered individuals. The model analysis includes the determination of equilibrium points and carrying out their stability analysis in terms of the threshold parameter R0. Moreover, the numerical simulation of the proposed model is also performed by using fourth order Runge–Kutta method along with the sensitivity analysis of the endemic equilibrium point. The analysis and numerical simulation results demonstrate that the maximum implementation of contact tracing and quarantine strategies help in reducing endemic infective class size and hence act as effective intervention strategy to control the disease. This gives a theoretical interpretation to the practical experiences that the early contact tracing and quarantine strategies are critically important to control the outbreak of epidemics.

Dynamical stability analysis of multiple equilibrium points in time-varying delayed recurrent neural networks with discontinuous activation functions

This paper is concerned with the dynamical stability analysis of multiple equilibrium points in recurrent neural networks with time-varying delays and discontinuous activation functions. Based on the decomposition of state space, some sufficient conditions for the existence of multiple equilibrium points are established, which ensure that n-dimensional recurrent neural networks with k-level discontinuous activation functions can have kn equilibrium points. Under these conditions, the equilibrium points are locally exponentially stable. Moreover, some conditions for the existence of sets of stable equilibrium points and unstable equilibrium points are derived for recurrent neural networks without delay and with discontinuous activation functions. Finally, three examples are given to illustrate the effectiveness of the results.

Estimating the Attraction Domain for the Boost Inverter

AbstractThis work presents an approach for estimating the domain of attraction for polynomial systems with state and control‐signal constraints, including saturation. In many problems, it is possible to derive global stability properties for such systems, neglecting constraints. Consideration of the constraints usually makes the problem much more complicated. In this paper, the stability analysis performed for the unconstrained case is used for the problem as a whole. For application of the method, there are powerful computational tools that can be employed in cases of polynomial systems. The technique is not only valid for the analysis of equilibrium points, but also for other attractors, such as limit cycles.As examples, the domain of attraction for given control laws is estimated for both a nonlinear DC‐DC boost converter and for a boost inverter.