Showing papers on "Center manifold published in 2023"
TL;DR: In this paper , the bifurcation structure of co-existing traveling wave solutions of a three-component competition-diffusion system that models an exotic competing species W invading the native system of strongly competing species U and V was analyzed.
2 citations
TL;DR: In this article , the authors deduce a predator-prey model with discrete time in the interior of R+2 using a new discrete method to study its local dynamics and Neimark-Sacker bifurcation.
Abstract: In this paper, we deduce a predator–prey model with discrete time in the interior of R+2 using a new discrete method to study its local dynamics and Neimark–Sacker bifurcation. Compared with continuous models, discrete ones have many unique properties that help to understand the changing patterns of biological populations from a completely new perspective. The existence and stability of the three equilibria are analyzed, and the formation conditions of Neimark–Sacker bifurcation around the unique positive equilibrium point are established using the center manifold theorem and bifurcation theory. An attracting closed invariant curve appears, which corresponds to the periodic oscillations between predators and prey over a long period of time. Finally, some numerical simulations and their biological meanings are given to reveal the complex dynamical behavior.
2 citations
TL;DR: In this article , a discrete predator-prey model incorporating Allee effect and cannibalism is derived from its continuous version by semidiscretization method, and the existence and local stability of fixed points of the discret system are investigated, but more important, the sufficient conditions for the occurrence of its period-doubling bifurcation and Neimark-Sacker bifury are obtained using the center manifold theorem and local bifurbcation theory.
Abstract: In this paper, a discrete predator-prey model incorporating Allee effect and cannibalism is derived from its continuous version by semidiscretization method. Not only the existence and local stability of fixed points of the discret system are investigated, but more important, the sufficient conditions for the occurrence of its period-doubling bifurcation and Neimark-Sacker bifurcation are obtained using the center manifold theorem and local bifurcation theory. Finally some numerical simulations are given to illustrate the existence of Neimark-Sacker bifurcation. The outcome of the study reveals that this discrete system undergoes various bifurcations including period-doubling bifurcation and Neimark-Sacker bifurcation.
1 citations
TL;DR: In this article , a discrete delayed dynamic system of three marine species: prey, predator, and superpredator is considered, and the authors consider the negative fishing effect of these species.
Abstract: In this paper, we have a discrete delayed dynamic system of three marine species: prey, predator, and superpredator. In addition to the effect of prey toxicity, we consider the negative fishing effect of these species. The study of this model consists of the search for equilibria with eigenvalue analysis, the existence of Hopf bifurcations at interior equilibria, and the determination of direction and stability analysis of Hopf bifurcation using the theory of normal form and center manifold. Some examples are given with numerical simulations to illustrate the results in different cases of delay.
1 citations
TL;DR: In this paper , a Beddington-DeAngelis prey-predator model with fear effect, refuge and harvesting is investigated, and the positivity of solutions and boundedness of system are given.
Abstract: In this paper, a Beddington–DeAngelis prey–predator model with fear effect, refuge and harvesting is investigated. First, the positivity of solutions and boundedness of system are given. Then, the existence and local stability of equilibria of such system are obtained. Next, not only different codimension-one bifurcations, such as saddle-node bifurcation, transcritical bifurcation and Hopf bifurcation take place, but also Bogdanov–Takens bifurcation of codimension-two occurs as predicted by the center manifold theorem and bifurcation theory. Finally, some numerical simulations are carried out to confirm our theoretical results.
1 citations
TL;DR: In this article , a delayed diffusive phytoplankton-zooplankton fish model with a refuge and Crowley-Martin and Holling II functional responses is established.
Abstract: In our paper, a delayed diffusive phytoplankton-zooplankton-fish model with a refuge and Crowley-Martin and Holling II functional responses is established. First, for the model without delay and diffusion, we not only analyze the existence and stability of equilibria, but also discuss the occurrence of Hopf bifurcation by choosing the refuge proportion of phytoplankton as the bifurcation parameter. Then, for the model with delay, we set some sufficient conditions to demonstrate the existence of Hopf bifurcation caused by delay; we also discuss the direction of Hopf bifurcation and the stability of the bifurcation of the periodic solution by using the center manifold and normal form theories. Next, for a reaction-diffusion model with delay, we show the existence and properties of Hopf bifurcation. Finally, we use Matlab software for numerical simulation to prove the previous theoretical results.
1 citations
TL;DR: In this article , a novel dynamic delayed feedback scheme is put forward and applied to successfully control the tipping phenomenon in small-world networks, and the conditions for stability and the occurrence of bifurcation-induced tipping are given.
Abstract: Small-world networks have been of increasing interest because of their high similarity with complex networks in reality. Achievements about controlling the propagation and evolution of small world networks have been highlighted, but little is known about the tipping mechanism and control of such network. In this paper, a novel dynamic delayed feedback scheme is put forward and applied to successfully control the tipping phenomenon in small-world networks. Firstly, the linear characteristic equation of the controlled system is analyzed and the conditions for stability and the occurrence of bifurcation-induced tipping are given. Then, the direction of Hopf bifurcation, which reveals the further evolution mechanism of tipping, is analyzed by using the normal form theory and center manifold reduction. Finally, the numerical simulations are provided to support the theoretical analysis and to verify the effectiveness of the proposed dynamic delayed feedback controller for the small-world network.
1 citations
TL;DR: In this article , a Lotka-Volterra competition-diffusion-advection system with time delay was investigated, where the diffusion and advection rates of two competitors are different.
1 citations
1 citations
TL;DR: In this paper , the authors investigated the effect of the Allee effect on the stability of a discrete-time predator-prey model with Holling type-II functional response using equilibrium analysis, stability analysis and bifurcation theory.
Abstract: In recent years, the stability of the predator–prey model subject to the Allee effect has become an interesting issue. This study investigates the effect of Allee effect on the stability of a discrete-time predator–prey model with Holling type-II functional response. Using equilibrium analysis, stability analysis and bifurcation theory, the mathematical characteristics of the proposed model are examined. Model experiences flip bifurcation and Neimark–Sacker bifurcation based on the center manifold theorem and bifurcation theory. Our analytical results are demonstrated by numerical simulations.
TL;DR: In this article , the authors proposed a new approach to establish the existence and uniqueness of a center manifold in a system with delta functions in the $ L^2 $-framework and derived a reduced system with suitable nonlinearity from the original model.
Abstract: Camphor boat and disk are types of interfacial self-propellers that rely on the modification of forces present at the air-liquid interface. In a one-dimensional circuit, self-organized congestion and clustering phenomena are observed experimentally and numerically [11,43]. Although the center manifold theories proposed in [5,7] are useful in the analysis of collective motions, the requirement in the reduction process is not fulfilled because the linearized operator involves Dirac delta functions. To overcome such mathematical difficulties, a theory in the $ (H^1)^* $-framework was developed in [17]. However, the reduced equation does not include any nonlinearity. Thus, information on the collective motions could not be obtained. The objectives of the present study are to propose a new approach to establish the existence and uniqueness of a center manifold in a system with delta functions in the $ L^2 $-framework and to derive a reduced system with suitable nonlinearity from the original model in [33].
TL;DR: In this article , the existence, stability and bifurcation direction of periodic traveling waves for the Nicholson's blowflies model with delay and advection are investigated by applying the Hopf bifurlcation theorem, center manifold theorem as well as normal form theory.
Abstract: The existence, stability and bifurcation direction of periodic traveling waves for the Nicholson's blowflies model with delay and advection are investigated by applying the Hopf bifurcation theorem, center manifold theorem as well as normal form theory. Some numerical simulations are presented to illustrate our main results.
TL;DR: In this paper , a predator-prey system with two delays is considered, the prey is sea urchins and the predator is crabs, and the theory of normal form and the center manifold are used to determine the direction of the bifurcations.
Abstract: In this study, we take into account a predator-prey system with two delays, the prey is sea urchins and the predator is crabs. The focus is given to the Allee effect where the prey population undergoes, the poisoning of few predators, and a fishing effect on both species considered as selective for the prey. We aim to analyze the system’s stability around interior equilibrium using the theory of bifurcations and determine stability intervals related to delays. The theory of normal form and the center manifold are used to determine the direction of the bifurcations. Finally, numerical simulations are given by numerical methods in DDE-Biftool Matlab package to illustrate the theoretical results.
TL;DR: In this article , a delayed prey-predator eco-epidemiological model with the nonlinear media is considered, and the positivity and boundedness of solutions are given.
Abstract: In this paper, a delayed prey-predator eco-epidemiological model with the nonlinear media is considered. First, the positivity and boundedness of solutions are given. Then, the basic reproductive number is showed, and the local stability of the trivial equilibrium and the disease-free equilibrium are discussed. Next, by taking the infection delay as a parameter, the conditions of the stability switches are given due to stability switching criteria, which concludes that the delay can generate instability and oscillation of the population through Hopf bifurcation. Further, by using normal form theory and center manifold theory, some explicit expressions determining direction of Hopf bifurcation and stability of periodic solutions are obtained. What's more, the correctness of the theoretical analysis is verified by numerical simulation, and the biological explanations are also given. Last, the main conclusions are included in the end.
17 Feb 2023
TL;DR: In this article , the authors proved the existence of a special coordinate system on the center manifold that will allow them to describe the local dynamics on the centre manifold in terms of these periodic normal forms.
Abstract: A recent work arXiv:2207.02480 by the authors on the existence of a periodic smooth finite-dimensional center manifold near a nonhyperbolic cycle in delay differential equations motivates derivation of periodic normal forms. In this paper, we prove the existence of a special coordinate system on the center manifold that will allow us to describe the local dynamics on the center manifold in terms of these periodic normal forms. Furthermore, we characterize the center eigenspace by proving the existence of time periodic smooth Jordan chains for the original and the adjoint system.
TL;DR: In this paper , a mathematical model describing the interaction of malignant glioma cells, macrophages and CD8+T cells is discussed and the biologically feasible equilibria and corresponding local stability are deduced.
TL;DR: In this paper , the authors considered the discrete delay as a bifurcation parameter, demonstrating that the system undergoes Hopf bifurbation at a critical value of the delay parameter and established the global stability of the model at axial and positive equilibrium points.
Abstract: This work investigates a prey–predator model featuring a Holling-type II functional response, in which the fear effect of predation on the prey species, as well as prey refuge, are considered. Specifically, the model assumes that the growth rate of the prey population decreases as a result of the fear of predators. Moreover, the detection of the predator by the prey species is subject to a delay known as the fear response delay, which is incorporated into the model. The paper establishes the preliminary conditions for the solution of the delayed model, including positivity, boundedness and permanence. The paper discusses the existence and stability of equilibrium points in the model. In particular, the paper considers the discrete delay as a bifurcation parameter, demonstrating that the system undergoes Hopf bifurcation at a critical value of the delay parameter. The direction and stability of periodic solutions are determined using central manifold and normal form theory. Additionally, the global stability of the model is established at axial and positive equilibrium points. An extensive numerical simulation is presented to validate the analytical findings, including the continuation of the equilibrium branch for positive equilibrium points.
TL;DR: In this article , a prey-predator model for plankton interactions in aquatic environments and the influence of planktivorous fish predation is presented, and the stability characteristics of models with and without gestation delay are investigated.
Abstract: This work presents a prey-predator model for plankton interactions in aquatic environments and the influence of planktivorous fish predation. Considering that some phytoplankton and zooplankton can be harvested due to their economic value, the time lag required for zooplankton gestation (representing the time delay) is included. The current study's main objective is to analyze planktivorous fish's influence on harvesting activity in both populations and the corresponding discrete delay (delay in zooplankton gestation) in the model. The stability characteristics of models with and without gestation delay are investigated, and the gestation delays threshold value is determined. The periodic solution's stability and the Hopf bifurcations' direction are determined by employing the center manifold theory. We use Lyapunov's method to investigate the global stability of the model without time delay. Bioeconomic equilibrium is obtained in three cases, each with certain conditions. In addition, the optimal harvesting strategy from the model is obtained by using the Pontryagin Maximum Principle and considering harvesting effort as a control variable. Finally, numerical simulations are performed to illustrate our analytical findings, and the ecological implications of our analytical findings are also discussed.
TL;DR: In this article , a Neimark-Sacker bifurcation of the delay differential equation was derived by using the semidiscretization method, and some results for the existence and stability of the Neimarks-Sacks bifurbation were derived using the center manifold theorem and bifurlcation theory.
Abstract: In this paper, we revisit a delay differential equation. By using the semidiscretization method, we derive its discrete model. We mainly deeply dig out a Neimark–Sacker bifurcation of the discrete model. Namely, some results for the existence and stability of Neimark–Sacker bifurcation are derived by using the center manifold theorem and bifurcation theory. Some numerical simulations are also given to validate the existence of the Neimark–Sacker bifurcation derived.
20 Jan 2023
TL;DR: In this paper , the authors present a methodology for the computation of whole sets of heteroclinic connections between iso-energetic slices of center manifolds of center x center x saddle fixed points of autonomous Hamiltonian systems.
Abstract: This paper presents methodology for the computation of whole sets of heteroclinic connections between iso-energetic slices of center manifolds of center x center x saddle fixed points of autonomous Hamiltonian systems. It involves: (a) computing Taylor expansions of the center-unstable and center-stable manifolds of the departing and arriving fixed points through the parameterization method, using a new style that uncouples the center part from the hyperbolic one, thus making the fibered structure of the manifolds explicit; (b) uniformly meshing iso-energetic slices of the center manifolds, using a novel strategy that avoids numerical integration of the reduced differential equations and makes an explicit 3D representation of these slices as deformed solid ellipsoids; (c) matching the center-stable and center-unstable manifolds of the departing and arriving points in a Poincar\'e section. The methodology is applied to obtain the whole set of iso-energetic heteroclinic connections from the center manifold of L2 to the center manifold of L1 in the Earth-Moon circular, spatial Restricted Three-Body Problem, for nine increasing energy levels that reach the appearance of Halo orbits in both L1 and L2. Some comments are made on possible applications to space mission design.
TL;DR: In this paper , the authors derived sufficient criteria for the asymptotic stability of the equilibria and the existence of Hopf bifurcation to the delayed model.
Abstract: This work focuses on an HIV infection model with intracellular delay and immune response delay, in which the former delay refers to the time it takes for healthy cells to become infectious after infection, and the latter delay refers to the time when immune cells are activated and induced by infected cells. By investigating the properties of the associated characteristic equation, we derive sufficient criteria for the asymptotic stability of the equilibria and the existence of Hopf bifurcation to the delayed model. Based on normal form theory and center manifold theorem, the stability and the direction of the Hopf bifurcating periodic solutions are studied. The results reveal that the intracellular delay cannot affect the stability of the immunity-present equilibrium, but the immune response delay can destabilize the stable immunity-present equilibrium through the Hopf bifurcation. Numerical simulations are provided to support the theoretical results.
TL;DR: In this paper , a discrete predator-prey model with square root functional response describing prey herd behavior and nonlinear predator harvesting has been considered, and three equilibria of the system have been found and observed that two equilibrium points always exist and are feasible, but the interior equilibrium point is feasible under a parametric condition.
Abstract: Abstract A discrete predator–prey model with square root functional response describing prey herd behavior and nonlinear predator harvesting has been considered in the present work. Three equilibria of the system have been found and observed that two equilibrium points always exist and are feasible, but the interior equilibrium point is feasible under a parametric condition. The local stability of the three equilibria has been analyzed. The interior equilibrium point is locally asymptotically stable under a parametric condition. It is examined that a flip and Neimark–Sacker bifurcations have occurred in the system at the axial equilibrium point. The flip and Neimark–Sacker bifurcations have been analyzed by the center manifold theorem and bifurcation theory, considering the harvesting coefficient as the bifurcation parameter. The proposed discrete model with a nonlinear Michaelis–Menten type harvesting effect on the predator population exhibits rich dynamics; for instance, bifurcations, chaos, and more complex dynamical behaviors. The discrete-time model also produced few numerical simulation results that are more accurate than the continuous model. The proposed discrete model will be performed better than the continuous model in populations with non-overlapping generations or smaller densities. The harvesting coefficient’s optimal value has finally been identified, and an optimal harvesting policy has been introduced. To verify the results, further numerical simulations have been performed.
TL;DR: In this paper , the complex dynamics of a nonlinear discretized predator-prey model with the nonlinear Allee effect in prey and both populations are investigated, and the rigorous results are derived from the existence and stability of the fixed points of the model.
Abstract: The complex dynamics of a nonlinear discretized predator-prey model with the nonlinear Allee effect in prey and both populations are investigated. First, the rigorous results are derived from the existence and stability of the fixed points of the model. Second, we establish a model with the Allee effect in prey undergoing codimension-one bifurcations (flip bifurcation and Neimark–Sacker bifurcation) and codimension-two bifurcation associated with 1 : 2 strong resonance by using center manifold theorem and bifurcation theory, and the direction of bifurcations is also evaluated. In particular, chaos in the sense of Marotto is proved at some certain conditions. Third, numerical simulations are performed to illustrate the effectiveness of the theoretical results and other complex dynamical behaviors, such as the period-3, 4, 6, 8, 9, 30, and 43 orbits, attracting invariant cycles, coexisting chaotic sets, and so forth. Of most interest is the finding of coexisting attractors and multistability. Moreover, a moderate Allee effect in predators can stabilize the dynamical behavior. Finally, the hybrid feedback control strategy is implemented to stabilize chaotic orbits existing in the model.
TL;DR: In this article , a mathematical model of stem cells is discussed with its motivation to describe the tissue relationship by introducing a two compartments model, and the clear link between the proliferation phase of the stem cells and the circulating neutrophil phase is set forth after delay feedback control of the state variable of Stem cells.
Abstract: The mathematical model of stem cells is discussed with its motivation to describe the tissue relationship by technically introducing a two compartments model. The clear link between the proliferation phase of stem cells and the circulating neutrophil phase is set forth after delay feedback control of the state variable of stem cells. Hopf bifurcation is discussed with varying free parameters and time delays. Based on the center manifold theory, the normal form near the critical point is computed and the stability of bifurcating periodical solution is rigorously discussed. With the aids of the artificial tool on-hand which implies how much tedious work doing by DDE-Biftool software, the bifurcating periodic solution after Hopf point is continued by varying time delay.
03 Jul 2023
TL;DR: In this paper , a slow-fast system with two slow and one fast variables is studied and a normal form for the system in a neighbourhood of the pair "equilibrium-fold" is derived.
Abstract: We study a slow-fast system with two slow and one fast variables. We assume that the slow manifold of the system possesses a fold and there is an equilibrium of the system in a small neighbourhood of the fold. We derive a normal form for the system in a neighbourhood of the pair "equilibrium-fold" and study the dynamics of the normal form. In particular, as the ratio of two time scales tends to zero we obtain an asymptotic formula for the Poincar\'e map and calculate the parameter values for the first period-doubling bifurcation. The theory is applied to a generalization of the FitzHugh-Nagumo system.
01 Mar 2023
TL;DR: In this article , the authors revisited the Kepler problem with linear drag and showed that the corresponding invariant sets are smooth manifolds, and they used normal form theory and a blowup transformation to reveal that the invariant manifolds are nonhyperbolic stable sets of (limiting) periodic orbits.
Abstract: In this paper, we revisit the Kepler problem with linear drag. With dissipation, the energy and the angular momentum are both decreasing, but in \cite{margheri2017a} it was shown that the eccentricity vector has a well-defined limit in the case of linear drag. This limiting eccentricity vector defines a conserved quantity, and in the present paper, we prove that the corresponding invariant sets are smooth manifolds. These results rely on normal form theory and a blowup transformation, which reveals that the invariant manifolds are (nonhyperbolic) stable sets of (limiting) periodic orbits. Moreover, we identify a separate invariant manifold which corresponds to a zero limiting eccentricity vector. This manifold is obtained as a generalized center manifold over the zero eigenspace of a zero-Hopf point. Finally, we present a detailed blowup analysis, which provides a geometric picture of the dynamics. %We will use this to shed light on a degenerate case of the limiting eccentricity vector. %The aforementioned invariant manifolds only make up a subset hereof. We believe that our approach and results will have general interest in problems with blowup dynamics.