Center Manifold Research Articles

Bifurcations and limit cycle prediction of rotor systems with fluid-film bearings using center manifold reduction

Bifurcations and limit cycle prediction of rotor systems with fluid-film bearings using center manifold reduction

Periodic Center Manifolds for DDEs in the Light of Suns and Stars

Periodic Center Manifolds for DDEs in the Light of Suns and Stars

Stability and Bautin bifurcation of four-wheel-steering vehicle system with driver steering control.

In this paper, the stability and Bautin bifurcation of a four-wheel-steering (4WS) vehicle system, by considering driver steering control, are investigated. By using the central manifold theory and projection method, the first and second Lyapunov coefficients are calculated to predict the type of Hopf bifurcation of the vehicle system. The topological structure of Bautin bifurcation, a degenerate Hopf bifurcation of the 4WS vehicle system, is presented in parameter space, and it reveals the dynamics of the vehicle system of different choices of control parameters. The influences of system parameters on critical values of the bifurcation parameter are also analyzed. It is shown that with the increase in the frontal visibility distance of the driver control strategy coefficient and the cornering stiffness coefficients of rear wheels, the critical speed increases. Nevertheless, the critical speed decreases with the increase in the distance from the center of gravity of the vehicle to the front axles, Driver's perceptual time delay, and cornering stiffness coefficients of the front wheels.

Bifurcation Analysis and 0-1 Chaos Test of a Discrete T System

This study examines discrete-time T system. We begin by listing the topological divisions of the system's fixed points. Then, we analytically demonstrate that a discrete T system sits at the foundation of a Neimark Sacker(NS) bifurcation under specific parametric circumstances. With the use of the explicit Flip-NS bifurcation criterion, we establish the flip-NS bifurcation's reality. Center manifold theory is then used to establish the direction of both bifurcations. We do numerical simulations to validate our theoretical findings. Additionally, we employ the $0-1$ test for chaos to demonstrate whether or not chaos exists in the system. In order to stop the system's chaotic trajectory, we ultimately employ a hybrid control method.

Mathematical modeling and analysis of COVID-19 and TB co-dynamics

This study proposes a mathematical model for examining the COVID-19 and tuberculosis (TB) co-dynamics thoroughly. First, the single infection dynamics: COVID-19 infection and TB infection models are taken into consideration and examined. Following that, the co-dynamics with TB and COVID-19 is also investigated. In order to comprehend the developed model dynamics, the basic system attributes including the region of definition, theory of nonnegativity and boundedness of solution are investigated. Further, a qualitative analysis of the equilibria of the formulated model equations is performed. The equilibria of both infection models are globally asymptotically stable if their respective basic reproductive number is smaller than one. As the associated reproductive number reaches unity, they experience the forward bifurcation phenomenon. Additionally, it is demonstrated that the formulated co-dynamics model would not experience backward bifurcation by applying the center manifold theory. Moreover, model fitting is done by using daily reported COVID-19 cumulative data in Ethiopia between March 13, 2020, and May 31, 2022. For instance, the non-linear least squares approach of fitting a function to data was performed in the fitting process using scipy.optimize.curve_fit from the Python. Finally, to corroborate the analytical findings of the model equation, numerical simulations were conducted.

A Mathematical Modelling of a Plant-Herbivore Community with Additional Effects of Food on the Environment

By taking into account various food components in the ecosystem, the research intends to develop a set of difference equations to simulate a plant-herbivore interaction of Holling Type II. We determine the local stability of the equilibrium points for the scenarios of extinction, semi-extinction (extinction for one species), and coexistence using the Linearized Stability Theorem. For a suitable Lyapunov function, we investigate theoretical findings to determine the global stability of the coexisting equilibrium point. It is clear that the system exhibits both Flip and Neimark-Sacker bifurcation under particular circumstances using the central manifold theorem and the bifurcation theory. Numerical simulations are done by MATLAB which are used to validate our conclusions.

Global Stability and Bifurcation Analysis of a Virus Infection Model with Nonlinear Incidence and Multiple Delays

In order to investigate the impact of general nonlinear incidence, cellular infection, and multiple time delays on the dynamical behaviors of a virus infection model, a within-host model describing the virus infection is formulated and studied by taking these factors into account in a single model. Qualitative analysis of the global properties of the equilibria is carried out by utilizing the methods of Lyapunov functionals. The existence and properties of local and global Hopf bifurcations are discussed by regarding immune delay as the bifurcation parameter via the normal form, center manifold theory, and global Hopf bifurcation theorem. This work reveals that the immune delay is mainly responsible for the existence of the Hopf bifurcation and rich dynamics rather than the intracellular delays, and the general nonlinear incidence does not change the global stability of the equilibria. Moreover, ignoring the cell-to-cell infection may underevaluate the infection risk. Numerical simulations are carried out for three kinds of incidence function forms to show the rich dynamics of the model. The bifurcation diagrams and the identification of the stability region show that increasing the immune delay can destabilize the immunity-activated equilibrium and induce a Hopf bifurcation, stability switches, and oscillation solutions. The obtained results are a generalization of some existing models.

Dynamical Behaviors in a Stage-Structured Model with a Birth Pulse

This paper presents an exploitation model with a stage structure to analyze the dynamics of a fish population, where periodic birth pulse and pulse fishing occur at different fixed time. By utilizing the stroboscopic map, we can obtain an accurate cycle of the system and investigate the stability thresholds. Through the application of the center manifold theorem and bifurcation theory, our research has shown that the given model exhibits transcritical and flip bifurcation near its interior equilibrium point. The bifurcation diagrams, maximum Lyapunov exponents and phase portraits are presented to further substantiate the complexity. Finally, we present high-resolution stability diagrams that demonstrate the global structure of mode-locking oscillations. We also describe how these oscillations are interconnected and how their complexity unfolds as control parameters vary. The two parametric planes illustrate that the structure of Arnold’s tongues is based on the Stern–Brocot tree. This implies that the periodic occurrence of birth pulse and pulse fishing contributes to the development of more complex dynamical behaviors within the fish population.

Stability and Hopf bifurcation of a modified Leslie–Gower predator–prey model with Smith growth rate and B–D functional response

This paper is concerned with a modified Leslie–Gower predator–prey diffusive dynamics system with Smith growth subject to homogeneous Neumann boundary condition, in which the functional response is Beddington–DeAngelis (Denote it by B–D) functional response term. Firstly, by applying the theory of stability and the Hopf bifurcation, we discuss the local stability and the existence of the Hopf bifurcation at the positive constant equilibrium solution of the ODE model, which the model undergoes the Hopf bifurcation when bifurcation parameter δ crosses the bifurcation critical value b0. Moreover, stability of the bifurcation periodic solution is analyzed. Secondly, the Turing instability and the direction of Hopf bifurcation of the corresponding to PDE system are investigated by using Normal form theory and Centre manifold theory. Finally, we study the numerical simulations of this system to illustrate the theoretical analysis.

Dynamical behaviors of an epidemic model with partial immunity having nonlinear incidence and saturated treatment in deterministic and stochastic environments

This manuscript deals with an epidemic model with partial immunity having nonlinear incidence and saturated treatment. Positivity and boundedness of the solutions have been established here. We discuss local stability of all equilibria. The proposed system experiences various types of bifurcations, namely Transcritical, Saddle–node, Hopf, and Bogdanov–Takens bifurcation of co-dimension 2. The system is reduced to a two-dimensional system using center manifold theorem to deduce normal form for Bogdanov–Takens bifurcation of co-dimension 2 when two eigenvalues at the endemic equilibrium point becomes zero. Whether deterministic model overestimates the condition for disease propagation, to observe this we also analysis stochastic model. We derive the condition for extinction, persistence in mean and stationary distribution. All theoretical findings are justified by numerical simulations. Finally, to check validity of the model, we fit it with real reported influenza data of Canada.

Complex dynamics induced by multiple gestation delays in a Leslie–Gower‐type system with competitive interference

The dynamics of food consumption by higher trophic level is complicated as its availability, choice, and consequently their development. The top predator interference has a strong influence on the dynamic structure of the tritrophic food chain system. The proposed article deals with the characteristic features of a three‐tire modified Leslie–Gower model with a one‐prey two‐predator system with the assumption of multiple gestation delays. The intermediate predator of the food chain is taken as a specialist type, whereas the growth of top predator is assumed by sexual reproduction. Two different response functions named Monod–Haldane (M‐H) and Beddington–DeAngelis (B‐D) are considered for the modeling of system dynamics. The population densities of intermediate and top predators are assumed to be affected by two different gestation delays. The boundedness and positive equilibria and their stability conditions are examined theoretically for the delay‐free system. The local and global stability conditions of the delayed system are also determined. With the help of normal form theory and central manifold arguments, the properties of bifurcating periodic solutions are carried out. The numerical computation is performed to validate our analytical findings that show the different dynamical outcomes such as periodic and chaotic dynamics. The Hopf‐bifurcation scenario for different parameters of the model system is well studied. Further, the stability behavior of multiple delays for the different cases is investigated. The system shows chaotic behavior when the gestation delay of intermediate predators is large enough. It is also observed that the top predator density is highly fluctuating through the creation of chaos by the inclusion of time delay. Finally, the time delay destabilizing effect is noticed for three‐way interactions among prey, intermediate predator, and top predator, highlighting its significance in chaotic dynamics.

On periodically modulated rolls in the generalized Swift–Hohenberg equation: Galerkin’ approximations

We study in this paper the existence of periodically modulated in one variable and localized in another variable solutions to the cubic Swift–Hohenberg equation on the plane R2. In the first part we try to apply the method by Kirschgässner–Mielke to reduce the problem to the search of finite dimensional submanifolds with periodic orbits on them in some formal infinite-dimensional dynamical system generated by the stationary SH equation. It turns out that it cannot be done immediately due to properties of the spectrum for the linearized system at the localized roll (a one-dimensional pulse). In the second part we change roles of variables and formulate the problem as funding homoclinic orbits to an equilibrium of the formal infinite-dimensional system in the space of periodic functions in variable y. Staying apart the proof of the exact theorem on the existence of related center manifold, we exploit the Bubnov–Galerkin method to derive the Hamiltonian finite-dimensional ODEs with four or six degrees of freedom having the saddle type equilibrium whose homoclinic orbits correspond to approximate solution on needed type. The search for homoclinic orbits is performed by means of numerical methods.

Spatiotemporal dynamics optimization of a delayed reaction–diffusion mussel–algae model based on PD control strategy

The dynamic control over time evolution of ordinary differential systems has developed rapidly in recent years, but how to control the spatiotemporal evolution dynamics of partial differential systems is still an open question. Turing pattern is a major spatiotemporal evolution behavior in mussel-algal ecosystems and its control can pull the ecosystem back from the borderline of collapse and make it more stable. However, there has been relatively little research on the optimal control of Turing pattern in the mussel-algal system. In this paper, we first attempt to propose a proportional-derivative (PD) control strategy for the reaction–diffusion mussel–algae model under the influence of time delays. Turing instability driven by diffusion occurs in the diffusive mussel–algae model without control. The introduction of PD control can effectively suppress the occurrence of Turing instability and keep the ecological model stable in both time and space. When the time delay exceeds a critical value, the diffusive mussel–algae model will lose stability and undergo a Hopf bifurcation. PD controller can make Hopf bifurcation occur in advance or in hysteresis by changing the threshold point of Hopf bifurcation. The intrinsic mechanism of Hopf bifurcation is analyzed by means of the formal theory and the central manifold theorem. The numerical simulations demonstrate the efficiency and feasibility of the PD control scheme.

Modeling the dynamics of COVID-19 in the presence of Delta and Omicron variants with vaccination and non-pharmaceutical interventions

Since its inception in December 2019, many safe and effective vaccines have been invented and approved for use against COVID-19 along with various non-pharmaceutical interventions. But the emergence of numerous SARS-CoV-2 variants has put the effectiveness of these vaccines, and other intervention measures under threat. So it is important to understand the dynamics of COVID-19 in the presence of its variants of concern (VOC) in controlling the spread of the disease. To address these situations and to find a way out of this problem, a new mathematical model consisting of a system of non-linear differential equations considering the original COVID-19 strain with its two variants of concern (Delta and Omicron) has been proposed and formulated in this paper. We then analyzed the proposed model to study the transmission dynamics of this multi-strain model and to investigate the consequences of the emergence of multiple new SARS-CoV-2 variants which are more transmissible than the previous ones. The control reproduction number, an important threshold parameter, is then calculated using the next-generation matrix method. Further, we presented the discussion about the stability of the model equilibrium. It is shown that the disease-free equilibrium (DFE) of the model is locally asymptotic stable when the control reproduction is less than unity. It is also shown that the model has a unique endemic equilibrium (EEP) which is locally asymptotic stable when the control reproduction number is greater than unity. Using the Center Manifold theory it is shown that the model also exhibits the backward bifurcation phenomenon when the control reproduction number is less than unity. Again without considering the re-infection of the recovered individuals, it is proved that the disease-free equilibrium is globally asymptotically stable when the reproduction threshold is less than unity. Finally, numerical simulations are performed to verify the analytic results and to show the impact of multiple new SARS-CoV-2 variants in the population which are more contagious than the previous variants. Global uncertainty and sensitivity analysis has been done to identify which parameters have a greater impact on disease dynamics and control disease transmission. Numerical simulation suggests that the emergence of new variants of concern increases COVID-19 infection and related deaths. It also reveals that a combination of non-pharmaceutical interventions with vaccination programs of new more effective vaccines should be continued to control the disease outbreak. This study also suggests that more doses of vaccine should provide to combat new and deadly variants like Delta and Omicron.

A study of the local dynamics of modified Patankar DeC and higher order modified Patankar–RK methods

Patankar schemes have attracted increasing interest in recent years because they preserve the positivity of the analytical solution of a production–destruction system (PDS) irrespective of the chosen time step size. Although they are now of great interest, for a long time it was not clear what stability properties such schemes have. Recently a new stability approach based on Lyapunov stability with an extension of the center manifold theorem has been proposed to study the stability properties of positivity-preserving time integrators. In this work, we study the stability properties of the classical modified Patankar–Runge–Kutta schemes (MPRK) and the modified Patankar Deferred Correction (MPDeC) approaches. We prove that most of the considered MPRK schemes are stable for any time step size and compute the stability function of MPDeC. We investigate its properties numerically revealing that also most MPDeC are stable irrespective of the chosen time step size. Finally, we verify our theoretical results with numerical simulations.

Fractional order prey–predator model incorporating immigration on prey: Complexity analysis and its control

In this paper, the Caputo fractional derivative is assumed to be the prey–predator model. In order to create Caputo fractional differential equations for the prey–predator model, a discretization process is first used. The fixed points of the model are categorized topologically. We identify requirements for the fixed points of the suggested prey–predator model’s local asymptotic stability. We demonstrate analytically that, under specific parametric conditions, a fractional order prey–predator model supports both a Neimark–Sacker (NS) bifurcation and a Flip bifurcation. We present evidence for NS and Flip bifurcations using central manifold and bifurcation theory. The parameter values and the initial conditions have been found to have a profound impact on the dynamical behavior of the fractional order prey–predator model. As the bifurcation parameter is increased, the system displays chaotic behavior. Numerical simulations are shown to demonstrate chaotic behaviors like bifurcations, phase portraits, invariant closed cycles, and attractive chaotic sets in addition to validating analytical conclusions. The suggested prey–predator dynamical system’s chaotic behavior will be controlled by the OGY and hybrid control methodology, which will also visualize the chaotic state for various biological parameters.

Effects of harvesting and planktivorous fish on bioeconomic phytoplankton-zooplankton models with ratio-dependent response functions and time delays

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.

Hopf bifurcation in a delayed prey–predator model with prey refuge involving fear effect

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.

Moving Modulating Pulse and Front Solutions of Permanent Form in a FPU Model with Nearest and Next-to-Nearest Neighbor Interaction

We consider a nonlinear chain of coupled oscillators, which is a direct generalization of the classical Fermi–Pasta–Ulam (FPU) lattice and exhibits, besides the usual nearest neighbor interaction, also next-to-nearest neighbor interaction. For the case of nearest neighbor attraction and next-to-nearest neighbor repulsion we prove that such a lattice admits, in contrast to the classical FPU model, moving modulating front solutions of permanent form, which have small converging tails at infinity and can be approximated by solitary wave solutions of the nonlinear Schrödinger equation. When the associated potentials are even, then the proof yields moving modulating pulse solutions of permanent form, whose profiles are spatially localized. Our analysis employs the spatial dynamics approach as developed by Iooss and Kirchgässner. The relevant solutions are constructed on a five-dimensional center manifold, and their persistence is guaranteed by reversibility arguments.

Bifurcation Analysis of a Wind Turbine Generator Drive System with Stochastic Excitation Under Both Displacement and Velocity Delayed Feedback

In this paper, a wind turbine generator drive system with stochastic excitation under both displacement and velocity delayed feedback is considered. Firstly, the center manifold method is used to approximate the delay term of the system, so that the Itô-stochastic differential equation can be obtained by random average method. Through the maximal Lyapunov exponential method, the local stochastic stability and random D-bifurcation conditions of the system are obtained. Secondly, it is verified that the increase of noise intensity and delay value induces the occurrence of random P-bifurcation of the system through Monte Carlo numerical simulations. In addition, the theoretical chaos threshold of the system is derived by the random Melnikov method. The results show that the chaos threshold decreases as the noise intensity increases, and the increase in time delay leads to a delay in the chaotic behavior of the system. Finally, the correctness and effectiveness of the chaos-theoretic analysis are verified based on the one-parameter bifurcation diagrams and the two-parameter bifurcation diagrams.