Showing papers on "Center manifold published in 2021"

Bifurcation and Control for a Predator-Prey System With Two Delays

Abstract: This brief mainly studies the dynamic analysis and control problem for the Leslie-Gower predator-prey system, which is established by a delay-differential equation. We use the Euler scheme to derive the discrete form of Leslie-Gower model. By discussing the associated characteristic equation, its dynamics properties, including stability analysis and Neimark-Sacker bifurcation, are investigated. Furthermore, we use the center manifold reduction and normal form theory to show the directions of Neimark-Sacker bifurcation, and we also analyze the stability of periodic bifurcation solution. In order to achieve effective control of the above bifurcation, we proposed a novel delayed feedback control scheme. Finally, an simulation example is given to verify the effectiveness of the main conclusion.

Bifurcations in a modified Leslie–Gower predator–prey discrete model with Michaelis–Menten prey harvesting

Abstract: In this paper, a modified Leslie–Gower predator–prey discrete model with Michaelis–Menten type prey harvesting is investigated. It is shown that the model exhibits several bifurcations of codimension 1 viz. Neimark–Sacker bifurcation, transcritical bifurcation and flip bifurcation on varying one parameter. Bifurcation theory and center manifold theory are used to establish the conditions for the existence of these bifurcations. Moreover, existence of Bogdanov–Takens bifurcation of codimension 2 (i.e. two parameters must be varied for the occurrence of bifurcation) is exhibited. The non-degeneracy conditions are determined for occurrence of Bogdanov–Takens bifurcation. The extensive numerical simulation is performed to demonstrate the analytical findings. The system exhibits periodic solutions including flip bifurcation and Neimark–Sacker bifurcation followed by the wide range of dense chaos.

Stability and Hopf Bifurcation Analysis for a Two-Species Commensalism System with Delay

Abstract: This paper is devoted to studying the dynamics of a two-species commensalism system with delay. By analyzing the characteristic equation and regarding the time delay as the bifurcation parameter, we investigate the local asymptotic stability of the positive equilibrium and show the existence of periodic solutions bifurcating from the positive equilibrium. Then, we derive the precise formulae to determine the Hopf bifurcation direction and the stability of the bifurcating periodic solutions by using the normal form theory and the center manifold theorem. Numerical simulation results are also included to support our theoretical analysis.

Complex dynamics of a predator–prey system with herd and schooling behavior: with or without delay and diffusion

Abstract: In this paper, we propose and analyze a predator–prey system incorporating a new type of functional response function accounting for both the herd behavior for prey and the schooling behavior for predator. In the absence of spatial diffusion and time delay, we prove the existence of extinction region and the stability of all feasible nonnegative equilibria. In addition, we obtain conditions under which the system undergoes saddle-node, Hopf and Bogdanov–Takens types of bifurcations. For the spatiotemporal system, we investigate the existence of Hopf bifurcation induced by time delay near one of the positive equilibria and derive conditions for determining the property of the bifurcating periodic solution by using the normal form theory and center manifold reduction. Further, we investigate the effects of time delay or/and diffusion on the dynamics of the system. Our numerical and theoretical results show that both time delay and spatial diffusion play significant roles in determining the dynamics of the system.

Bifurcation of Symmetric Domain Walls for the Bénard–Rayleigh Convection Problem

Abstract: We prove the existence of domain walls for the Benard-Rayleigh convection problem. Our approach relies upon a spatial dynamics formulation of the hydrodynamic problem, a center manifold reduction, and a normal form analysis of a reduced system. Domain walls are constructed as heteroclinic orbits of this reduced system.

Qualitative Analysis and Bifurcation in a Neuron System With Memristor Characteristics and Time Delay

Abstract: This article focuses on the hybrid effects of memristor characteristics, time delay, and biochemical parameters on neural networks. First, we propose a novel neuron system with memristor and time delays in which the memristor is characterized by a smooth continuous cubic function. Second, the existence of equilibria of this type of neuron system is examined in the parameter space. Sufficient conditions that ensure the stability of equilibria and occurrence of pitchfork bifurcation are given for the memristor-based neuron system without delay. Third, some novel criteria of the addressed neuron system are constructed for guaranteeing the delay-dependent and delay-independent stability. The specific conditions are provided for Hopf bifurcations, and the properties of Hopf bifurcation are ascertained using the center manifold reduction and the normal form theory. Moreover, there exists a phenomenon of bistability for the delayed memristor-based neuron system having three equilibria. Finally, the effectiveness of the theoretical results is demonstrated by numerical examples.

Stability and bifurcation analysis of a discrete prey–predator model with sigmoid functional response and Allee effect

Abstract: In this paper, we have considered a discrete-time predator–prey model with sigmoid functional response and Allee effect. We have discussed the conditions of existence of the feasible fixed point. The stability criterion of the fixed point carried out algebraically. Analytically we have shown that the system undergoes flip and Neimark–Sacker bifurcations. We also analyzed Neimark–Sacker bifurcation by center manifold theorem taking discretization factor(generation gap) as a bifurcation parameter. Under a parametric condition, all the bifurcation phenomena and chaotic features have been justified numerically. Finally, we use the hybrid control strategy to control flip and Neimark–Sacker bifurcations.

Dynamics of a delayed predator-prey system with fear effect, herd behavior and disease in the susceptible prey

Abstract: In this article, a delayed predator-prey system with fear effect, disease and herd behavior in prey incorporating refuge is established. Firstly, the positiveness and boundedness of the solutions is proved, and the basic reproduction number $ R_0 $ is calculated. Secondly, by analyzing the characteristic equations of the system, the local asymptotic stability of the equilibria is discussed. Then taking time delay as the bifurcation parameters, the existence of Hopf bifurcation of the system at the positive equilibrium is given. Thirdly, the global asymptotic stability of the equilibria is discussed by constructing a suitable Lyapunov function. Next, the direction of Hopf bifurcation and the stability of the periodic solution are analyzed based on the center manifold theorem and normal form theory. What's more, the impact of the prey refuge, fear effect and capture rate on system is given. Finally, some numerical simulations are performed to verify the correctness of the theoretical results.

Study of Stability and Bifurcation of Three Species Food Chain Model with Non-monotone Functional Response

Abstract: In this research article, we consider a tri-trophic food chain model with one prey and two predators such as- prey, intermediate predator and top predator. In this model, the prey and intermediate predator follows non-monotonic functional response; top predator consumes prey and intermediate predator following Holling type I functional response. The positivity, boundedness of solutions of the proposed model and stability conditions of different equilibrium points are discussed here. Then using Center Manifold theorem, the nature of non-hyperbolic type equilibrium points are discussed. After that, different local bifurcations such as- Saddle-node, Transcritical and Hopf bifurcations are studied theoretically as well as numerically by considering half-saturation constant and death rate of intermediate predator as the bifurcation parameters. Finally, the dynamics of the proposed model has been illustrated with the help of some numerical simulations.

Spatiotemporal patterns in a diffusive predator–prey system with Leslie–Gower term and social behavior for the prey

Abstract: In this paper, we deal with a new approximation of a diffusive predator--prey model with Leslie--Gower term and social behavior for the prey subject to Neumann boundary conditions. A new approach for a predator-prey interaction in the presence of prey social behavior has been considered. Our main topic in this work is to study the influence of the prey's herd shape on the predator-prey interaction in the presence of Leslie--Gower term. First of all, we examine briefly the system without spatial diffusion. By analyzing the distribution of the eigenvalues associated with the constant equilibria, the local stability of the equilibrium points and the existence of Hopf bifurcation have been investigated. Then, the spatiotemporal dynamics introduced by self diffusion was determined, where the existence of the positive solution, Hopf bifurcation, Turing driven instability, Turing-Hopf bifurcation point have been derived. Further, the effect of the prey's herd shape rate on the prey and predator equilibrium densities as well as on the Hopf bifurcating points has been discussed. Finally, by using the normal form theory on the center manifold, the direction and stability of the bifurcating periodic solutions have also been obtained. To illustrate the theoretical results, some graphical representations are given.

Hopf bifurcation of a VEIQS worm propagation model in mobile networks with two delays

Abstract: The spread of worm virus has brought great loss to our production and life. In this paper, a new Vulnerable-Exposed-Infectious-Quarantined-Secured (VEIQS) worm propagation model with a saturated incidence and two delays is proposed. The local stability of the worm-existence equilibrium and the occurrence of Hopf bifurcation at the critical values of the two delays are obtained by regarding different combinations of time delays as bifurcation parameters. It shows that the model is ideal stable when the time delay is below the critical value and a Hopf bifurcation occurs when the time delay is above the critical value. In particular, direction and stability of the Hopf bifurcation are determined by using the center manifold theorem. Finally, some numerical simulations are presented in order to verify the analytical results.

Modeling and analysis of a predator–prey type eco-epidemic system with time delay

Abstract: In this research work, a delay-induced eco-epidemic model using a reconstructed Leslie–Gower-type growth rate is formulated and analyzed. An extended qualitative nature of the solutions of the model system like boundedness, strong uniform persistence, and permanence is examined to secure the longstanding viability of the system. The stability of the system is investigated at different stationary points, and sufficient conditions are obtained for the local as well as global stability. The dynamics of the delay-induced model system, including the Hopf bifurcation phenomenon, is rigorously studied around the coexisting equilibrium using the normal form method and center manifold theorem. Also, the length of the delay to preserve the stability of the coexisting equilibrium is evaluated. It is observed that the effect of infection on the total harvest is negligible, but the effort to harvest can reduce the infection and preserve the system’s stability. The results may help to determine the point of reference for disease persistence and extinction. Based on our analytical results, several numerical simulations are also performed.

Mathematical Modeling and Analysis of Transmission Dynamics and Control of Schistosomiasis

Abstract: This paper presents a mathematical model that describes the transmission dynamics of schistosomiasis for humans, snails, and the free living miracidia and cercariae. The model incorporates the treated compartment and a preventive factor due to water sanitation and hygiene (WASH) for the human subpopulation. A qualitative analysis was performed to examine the invariant regions, positivity of solutions, and disease equilibrium points together with their stabilities. The basic reproduction number, , is computed and used as a threshold value to determine the existence and stability of the equilibrium points. It is established that, under a specific condition, the disease-free equilibrium exists and there is a unique endemic equilibrium when . It is shown that the disease-free equilibrium point is both locally and globally asymptotically stable provided , and the unique endemic equilibrium point is locally asymptotically stable whenever using the concept of the Center Manifold Theory. A numerical simulation carried out showed that at , the model exhibits a forward bifurcation which, thus, validates the analytic results. Numerical analyses of the control strategies were performed and discussed. Further, a sensitivity analysis of was carried out to determine the contribution of the main parameters towards the die out of the disease. Finally, the effects that these parameters have on the infected humans were numerically examined, and the results indicated that combined application of treatment and WASH will be effective in eradicating schistosomiasis.

Dynamical Behavior Analysis of a Time-Delay SIRS-L Model in Rechargeable Wireless Sensor Networks

Abstract: The traditional SIRS virus propagation model is used to analyze the malware propagation behavior of wireless rechargeable sensor networks (WRSNs) by adding a new concept: the low-energy status nodes. The SIRS-L model has been developed in this article. Furthermore, the influence of time delay during the charging behavior of the low-energy status nodes needs to be considered. Hopf bifurcation is studied by discussing the time delay that is chosen as the bifurcation parameter. Finally, the properties of the Hopf bifurcation are explored by applying the normal form theory and the center manifold theorem.

Cosmological dynamics and bifurcation analysis of the general non-minimally coupled scalar field models

Abstract: Non-minimally coupled scalar field models are well-known for providing interesting cosmological features. These include a late time dark energy behavior, a phantom dark energy evolution without singularity, an early time inflationary universe, scaling solutions, convergence to the standard $\Lambda$CDM, etc. While the usual stability analysis helps us determine the evolution of a model geometrically, bifurcation theory allows us to precisely locate the parameters' values describing the global dynamics without a fine-tuning of initial conditions. Using the center manifold theory and bifurcation analysis, we show that the general model undergoes a transcritical bifurcation, which predicts us to tune our models to have certain desired dynamics. We obtained a class of models and a range of parameters capable of describing a cosmic evolution from an early radiation era towards a late time dark energy era over a wide range of initial conditions. There is also a possible scenario of crossing the phantom divide line. We also find a class of models where the late time attractor mechanism is indistinguishable from that of a structurally stable general relativity based model; thus, we can elude the big rip singularity generically. Therefore, bifurcation theory allows us to select models that are viable with cosmological observations.

A Discrete-Time Model for Consumer–Resource Interaction with Stability, Bifurcation and Chaos Control

Abstract: Keeping in mind the interactions between budmoths and the quality of larch trees located in the Swiss Alps (a mountain range in Switzerland), a discrete-time model is proposed and studied. The novel model is proposed with implementation of a nonlinear functional response that incorporates plant quality. The proposed functional response is validated with real observed data of larch budmoth interactions. Furthermore, we investigate the qualitative behavior of the proposed discrete-time system with interactions between budmoths and the quality of larch trees. Proofs of the boundedness of solutions, and the existence of fixed points and their topological classification are carried out. It is proved that the system experiences period-doubling bifurcation at its positive fixed point using the center manifold theorem and normal forms theory. Moreover, existence and direction for the torus bifurcation are also investigated for larch budmoth interactions. Bifurcating and fluctuating behaviors of the system are controlled through utilization of chaos control strategies. Numerical simulations are presented to demonstrate the theoretical findings. At the end, theoretical investigations are validated with field and experimental data.

Turing patterns and spatiotemporal patterns in a tritrophic food chain model with diffusion

Abstract: In this paper, the temporal, spatial, and spatiotemporal patterns of a tritrophic food chain reaction–diffusion model with Holling type II functional response are studied. Firstly, for the model with or without diffusion, we perform a detailed stability and Hopf bifurcation analysis and derive criteria for determining the direction and stability of the bifurcation by the center manifold and normal form theory. Moreover, diffusion-driven Turing instability occurs, which induces spatial inhomogeneous patterns for the reaction–diffusion model. Then, the existence of positive non-constant steady-states of the reaction–diffusion model is established by the Leray–Schauder degree theory and some a priori estimates. Finally, numerical simulations are presented to visualize the complex dynamic behavior.

Memristor-based oscillatory behavior in the FitzHugh–Nagumo and Hindmarsh–Rose models

Abstract: The neural firing activities related to information coding maintaining the information transmission vary qualitatively considering the electromagnetic induction. The firing of a single neuron can be investigated by Hopf bifurcation analysis. In this paper, with the help of the center manifold theory and algebraic invariants method, general parameter conditions are obtained for the existence and stability of Hopf bifurcation in the memristive FitzHugh–Nagumo (FHN) and Hindmarsh–Rose (HR) models. By studying the roots of the characteristic polynomial, the parameter ranges that cause the systems to undergo oscillations have been studied. With the help of the Lyapunov functions, the general form of the first Lyapunov coefficient is obtained for the memristive FHN model. By using the center manifold theory and algebraic invariants method, the memristive HR model is reduced to a more manageable model which inherits the local dynamics of the original model. The effects of electromagnetic induction on neural oscillations are studied for general parameter conditions. The oscillatory bursting firing regimes for the models are illustrated by numerical simulations.

Bogdanov–Takens and Triple Zero Bifurcations for a Neutral Functional Differential Equations with Multiple Delays

Abstract: In this paper, a neutral functional differential equation with multiple delays is considered. In a first step, we assumed some sufficient hypotheses to guarantee the existence of the Bogdanov–Takens and the triple-zero bifurcations. In a second step, the normal form of the two bifurcations is obtained by using the reduction on the center manifold and the theory of the normal form. Finally, we applied our study to a class of three-neuron bidirectional associative memory networks, its dynamic behaviors are studied and proved by an example and its numerical simulations.

Bifurcation analysis of a special delayed predator-prey model with herd behavior and prey harvesting

Abstract: In this paper, we propose a predator-prey system with square root functional response, two delays and prey harvesting, in which an algebraic equation stands for the economic interest of the yield of the harvest effort. Firstly, the existence of the positive equilibrium is discussed. Then, by taking two delays as bifurcation parameters, the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained. Next, some explicit formulas determining the properties of Hopf bifurcation are analyzed based on the normal form method and center manifold theory. Furthermore, the control of Hopf bifurcation is discussed in theory. What's more, the optimal tax policy of system is given. Finally, simulations are given to check the theoretical results.

Hopf bifurcations in a class of reaction-diffusion equations including two discrete time delays: An algorithm for determining Hopf bifurcation, and its applications

Abstract: We analyze Hopf bifurcation and its properties of a class of system of reaction-diffusion equations involving two discrete time delays. First, we discuss the existence of periodic solutions of this class under Neumann boundary conditions, and determine the required conditions on parameters of the system at which Hopf bifurcation arises near equilibrium point. Bifurcation analysis is carried out by choosing one of the delay parameter as a bifurcation parameter and fixing the other in its stability interval. Second, some properties of periodic solutions such as direction of Hopf bifurcation and stability of bifurcating periodic solution are studied through the normal form theory and the center manifold reduction for functional partial differential equations. Moreover, an algorithm is developed in order to determine the existence of Hopf bifurcation (and its properties) of variety of system of reaction-diffusion equations that lie in the same class. The benefit of this algorithm is that it puts a very complex and long computations of existence of Hopf bifurcation for each equation in that class into a systematic schema. In other words, this algorithm consists of the conditions and formulae that are useful for completing the existence analysis of Hopf bifurcation by only using coefficients in the characteristic equation of the linearized system. Similarly, it is also useful for determining the direction analysis of the Hopf bifurcation merely by using the coefficients of the second degree Taylor polynomials of functions in the right hand side of the system. Finally, the existence of Hopf bifurcation for three different problems whose governing equations stay in that class is given by utilizing the algorithm derived, and thus the feasibility of the algorithm is presented.

Modeling the importance of temporary hospital beds on the dynamics of emerged infectious disease

Abstract: To explore the impact of available and temporarily arranged hospital beds on the prevention and control of an infectious disease, an epidemic model is proposed and investigated. The stability analysis of the associated equilibria is carried out, and a threshold quantity basic reproduction number ( R0) that governs the disease dynamics is derived and observed whether it depends both on available and temporarily arranged hospital beds. We have used the center manifold theory to derive the normal form and have shown that the proposed model undergoes different types of bifurcations including transcritical (backward and forward), Bogdanov–Takens, and Hopf-bifurcation. Bautin bifurcation is obtained at which the first Lyapunov coefficient vanishes. We have taken advantage of Sotomayor’s theorem to establish the saddle-node bifurcation. Numerical simulations are performed to support the theoretical findings.

A deep dive into the $2g+h$ resonance: separatrices, manifolds and phase space structure of navigation satellites

Abstract: Despite extended past studies, several questions regarding the resonant structure of the medium-Earth orbit (MEO) region remain hitherto unanswered. This work describes in depth the effects of the $2g+h$ lunisolar resonance. In particular, (i) we compute the correct forms of the separatrices of the resonance in the inclination-eccentricity space for fixed semi-major axis. This allows to compute the change in the width of the $2g+h$ resonance as the altitude increases. (ii) We discuss the crucial role played by the value of the inclination of the Laplace plane, $i_{L}$. Since $i_L$ is comparable to the resonance's separatrix width, the parametrization of all resonance bifurcations has to be done in terms of the proper inclination $i_{p}$, instead of the mean one. (iii) The subset of circular orbits constitutes an invariant subspace embedded in the full phase space, the center manifold $\mathcal{C}$. Using $i_p$ as a label, we compute its range of values for which $\mathcal{C}$ becomes a normally hyperbolic invariant manifold (NHIM). The structure of invariant tori in $\mathcal{C}$ allows to explain the role of the initial phase $h$ noticed in several works. (iv) Through Fast Lyapunov Indicator (FLI) cartography, we portray the stable and unstable manifolds of the NHIM as the altitude increases. Manifold oscillations dominate in phase space between $a=24,000$ km and $a=30,000$ km as a result of the sweeping of the $2g+h$ resonance by the $h-\Omega_{\rm{Moon}}$ and $2h-\Omega_{\rm{Moon}}$ resonances. The noticeable effects of the latter are explained as a consequence of the relative inclination of the Moon's orbit with respect to the ecliptic. The role of the phases $(h,\Omega_{\rm{Moon}})$ in the structures observed in the FLI maps is also clarified. Finally,(v) we discuss how the understanding of the manifold dynamics could inspire end-of-life disposal strategies.