Direction Of Bifurcation Research Articles

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.

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.

Spatiotemporal dynamics in a diffusive predator-prey model with multiple Allee effect and herd behavior

In this paper, we investigated a diffusive predator-prey model with multiple Allee effect, herd behavior. Moreover, we consider the quadratic mortality on predator species. We found that the dynamics of system near the positive equilibria is quite rich. We are more concerned with the spatial dynamics near the positive equilibrium. The necessary conditions for Turing instability occurring are obtained. And the stability and direction of Hopf and steady state bifurcations are explored by using the normal form method. Furthermore, some numerical simulations are presented to support our theoretical analysis. We found that Allee effect does not alter the local stability of the boundary equilibrium and the transcritical bifurcation occurs at the highest intensity of Allee effect. The biomass conversion rate does affect the stability of the system and the occurrence of Turing instability. This indicates that the biomass conversion rate is essentially significant for the predator-prey system, and we can control the biological conversion rate to achieve the coexistence of the predator and prey. Finally, we summarize our findings in the conclusion.

Hopf Bifurcation and Self-Organization Pattern of a Modified Brusselator Model

This paper reports the Hopf bifurcation and self-organization pattern of a modified Brusselator model. The model is a non-standard Brusselator model, it involves the nonlinear restraint term. For the non-diffusive model, we give the types of unique positive equilibrium. It is found that the unique positive equilibrium may be focus, node, or center and we establish their stability, respectively. Especially, there exists the spatial homogeneous Hopf bifurcation when the equilibrium is a center. The first Lyapunov number technique is applied to perform the direction of the spatial homogeneous Hopf bifurcation. In the sequel, the occurrence conditions of the Turing instability and the spatial inhomogeneous Hopf bifurcation are given for the diffusive model. Moreover, by using the normal form theory, we show that the Hopf bifurcation is supercritical or subcritical. Finally, the self-organization patterns induced by the Turing instability and periodic solutions resulting from the Hopf bifurcation are displayed by employing numerical simulations. Our theoretical predictions and numerical results reveal that the modified Brusselator model enjoys the temporal period oscillation and spatial oscillation due to the Hopf bifurcation and Turing instability, respectively. These results may help us to figure out the spatio-temporal dynamics of such modified Brusselator model.

The influence of time delay and Gaussian white noise on the dynamics of tobacco smoking model

This article analyses the influence of psychological and social addictions such as drug and tobacco consumption problems on humans by using an appropriate nonlinear computational formulation. We propose a mathematical model to simulate the multiple stages and behaviors of the smoking addiction process. Firstly, we present the dynamics of the local and global stability of the tobacco smoking model at existing equilibrium points and discuss the Hopf-Bifurcation analysis of the delayed model by regarding the time delay as a bifurcation parameter and derive specific formulas for the stability and direction of the Hopf bifurcation employing the normal form theory and centre manifold theorem. Then, we enhance our delayed model by considering the effects of the social environment. Precisely, we incorporate additive white noises into our delayed system to simulate the stochastic factors. Then, we analytically present some statistical properties of the perturbed system. Finally, we performed the sensitivity analysis and some numerical examples to discuss the feasibility of our theoretical findings.

Impacts of inclusion of time delay in efforts on the dynamics of forestry biomass, concentration of greenhouse gases and elevated temperature

To understand the impacts of efforts applied to increase the density of forestry biomass on the dynamics of forestry biomass, concentration of greenhouse gases and elevated temperature a nonlinear mathematical model is proposed and analysed. The mathematical model involves four dynamical variables namely: the density of forestry biomass, efforts applied to increase the density of forestry biomass, concentration of greenhouse gases and elevated environmental temperature. Since, there is always a time lag between implementation of efforts and its outcome, therefore we extend our model by introducing time delay in efforts. It is found that as delay in efforts crosses a critical value, the delay model loses its stability and undergoes Hopf bifurcation. The stability and direction of Hopf bifurcation are analysed using normal form method and central manifold theory. Numerical simulations are performed to verify and validate our analytical results. It is observed that if efforts are implemented for appropriate time, the density of forestry biomass can be conserved but implementation of efforts with longer time delay has destabilizing effect on the system. Increasing rate of atmospheric temperature due to greenhouse gases, decreases the density of forestry biomass but this decrease can also be maintained by implementation of efforts. Therefore, implementation of efforts for sufficient period of time, plays a very vital role in increasing the density of forestry biomass and decreasing the concentration of greenhouse gases and elevated temperature.

Dynamic Analysis of the Project Investment Model with Time Delay and Density Constraints

The purpose of research, invention, and production is to effectively supply demand, which requires not only financial support but also time to complete. First, the differential dynamic model among funds, demand, and research (including invention and production) with time delay is constructed by using the predator-prey theory. Second, the sufficiency of local asymptotic stability of the positive equilibrium point of the model is obtained by using the Hopf bifurcation theory and stability theory. Then, the formulas for determining the direction of Hopf bifurcation and the stability of the periodic solution are calculated by using the central manifold theory and normative theory. Finally, the evolution process of the differential dynamic model with controlled time delay is numerically simulated so as to verify the correctness of the relevant analytical conclusions.

Bifurcations in a Plant-Pollinator Model with Multiple Delays

The plant-pollinator model is a common model widely researched by scholars in population dynamics. In fact, its complex dynamical behaviors are universally and simply expressed as a class of delay differential-difference equations. In this paper, based on several early plant-pollinator models, we consider a plant-pollinator model with two combined delays to further describe the mutual constraints between the two populations under different time delays and qualitatively analyze its stability and Hopf bifurcation. Specifically, by selecting different combinations of two delays as branch parameters and analyzing in detail the distribution of roots of the corresponding characteristic transcendental equation, we investigate the local stability of the positive equilibrium point of equations, derive the sufficient conditions of asymptotic stability, and demonstrate the Hopf bifurcation for the system. Under the condition that two delays are not equal, some explicit formulas for determining the direction of Hopf bifurcation and some conditions for the stability of periodic solutions of bifurcation are obtained for delay differential equations by using the theory of norm form and the theorem of center manifold. In the end, some examples are presented and corresponding computer numerical simulations are taken to demonstrate and support effectiveness of our theoretical predictions.

Analysis and simulation of a delayed HIV model with reaction–diffusion and sliding control

Selection of an appropriate therapy threshold to restrain the virus load is still challenging on structured treatment interruptions (STIs) for HIV. In this paper, we ponder that how the sliding control and multistability to regulate the treatment effect through comprehensive dynamics of a virus-immune model. Firstly, based on piecewise therapy, we propose a delayed reaction–diffusion virus-immune model under the homogeneous Neumann boundary condition. Secondly, the existence and stabilities of five kinds of equilibria as well as the direction and stability of spatial Hopf bifurcation at regular equilibrium are investigated. Thirdly, the sliding domain and the boundary node bifurcations are addressed by theoretical analysis. Finally, we appraise the effects of therapy threshold, sliding domain and multistability on HIV therapy by simulations, and further seek out the appropriate therapy threshold for infected patients with given physiological parameters and present the corresponding principles. Our explorations will provide evidence for HIV and other disease therapies.

Dynamics of Delayed Neuroendocrine Systems and Their Reconstructions Using Sparse Identification and Reservoir Computing

Neuroendocrine system mainly consists of hypothalamus, anterior pituitary, and target organ. In this paper, a three-state-variable delayed Goodwin model with two Hill functions is considered, where the Hill functions with delays denote the hormonal feedback suppressions from target organ to hypothalamus and to anterior in the reproductive hormonal axis. The existence of Hopf bifurcation shows the circadian rhythms of neuroendocrine system. The direction and stability of Hopf bifurcation are also analyzed using the normal form theory and the center manifold theorem for functional differential equations. Furthermore, based on the sparse identification algorithm, it is verified that the transient time series generated from the delayed Goodwin model cannot be equivalently presented by ordinary differential equations from the viewpoint of data when considering that a library of candidates are at most cubic terms. The reason is because the solution space of delayed differential equations is of infinite dimensions. Finally, we report that reservoir computing can predict the periodic behaviors of the delayed Goodwin model accurately if the size of reservoir and the length of data used for training are large enough. The predicting performances are evaluated by the mean squared errors between the trajectories generated from the numerical simulations and the reservoir computing.

A predator–prey system with prey social behavior and generalized Holling III functional response: Role of predator‐taxis on spatial patterns

In this paper, we investigate the intrinsic impact of the predator‐taxis coefficient on the formation of spatial patterns in a predator–prey system with prey social behavior subject to Neumann boundary conditions. By treating the predator‐taxis coefficient as a potential critical parameter for Turing bifurcation, we observe that the Turing pattern is fully captured by three distinct critical thresholds. Meanwhile, utilizing weakly nonlinear analysis and the amplitude equation, we establish the direction of Turing bifurcation. Our mathematical analysis reveals that the inclusion of the predator‐taxis coefficient in the predator–prey system may lead to the emergence of either subcritical or supercritical Turing bifurcation. Our numerical experiments confirm the theoretical findings and exhibit various spatial patterns with different values of predator‐taxis coefficients.

Dynamics of a Coccinellids-Aphids Model with Stage Structure in Predator Including Maturation and Gestation Delays

This work studies a three-dimensional predator–prey model with gestation delay and stage structure between aphidophagous coccinellids and aphid pests, where the interaction between mature coccinellids and aphids is inscribed by Crowley–Martin functional response function, and immature coccinellids and aphids act in the form of Holling-I type. We prove the positivity and boundedness of the solution of the nondelayed system and analyze its equilibrium point, local asymptotic stability, and global stability. In addition to the delays, the critical values of Hopf bifurcation occurring for different parameters are also found from the numerical simulation. The stability of the delayed system and Hopf bifurcation with different delays as parameters are also discussed. Our model analysis shows that the time delay essentially governs the system’s dynamics, and the stability of the system switches as delays increase. We also investigate the direction and stability of the Hopf bifurcation using the normal form theory and center manifold theorem. Finally, we perform computer simulations and depict diagrams to support our theoretical results.

Tipping point prediction and mechanism analysis of malware spreading in cyber–physical systems

Malware spreading, characterized by differential equations, in networked systems has become a common topic of study in the scientific literature. In this paper, we focus primarily on the propagation dynamics of malware in cyber–physical systems (CPSs). Although some work prior has been done to understand how the malware dynamics evolves in complex networks, these studies have restricted their attention to stability, bifurcation and oscillation based on ordinary differential equation models. Here, we address the question “how does one detect tipping phenomena which result in a total collapse of CPSs if the spread of malware is related to both time and space?” To this end, we use a reaction–diffusion equation to model spatio-temporal dynamic evolutions of malware in CPSs and accurately predict the emergence of tipping points due to the Turing instability and Hopf bifurcation for the first time. In order to further reveal the mechanism of tipping, the specific conditions of the Turing instability and Hopf bifurcation are given. It is confirmed that the introduction of space factor and transmission delay causes CPSs to change from stable to unstable, and then the tipping occurs. Moreover, the delay-dependent stability is discussed and the stability and direction of Hopf bifurcation are explored. These studies provide a strategic guidance for the prediction and control of malicious virus in CPSs. Finally, some numerical simulation examples are provided to substantiate the theoretical results.

Chaotic dynamics of the fractional order Schnakenberg model and its control

The Schnakenberg model is thought to be the Caputo fractional derivative. In order to create caputo fractional differential equations for the Schnakenberg model, a discretization process is first used. The fixed points in the model are categorized topologically. Then, we show analytically that, under certain parametric conditions, a Neimark-Sacker (NS) bifurcation and a Flip-bifurcation are supported by a fractional order Schnakenberg model. Using central manifold and bifurcation theory, we demonstrate the presence and direction of NS and Flip bifurcations. The parameter values and the initial conditions have been found to have a profound impact on the dynamical behavior of the fractional order Schnakenberg model. Numerical simulations are shown to demonstrate chaotic behaviors like bifurcations, phase portraits, period 2, 4, 7, 8, 10, 16, 20 and 40 orbits, invariant closed cycles, and attractive chaotic sets in addition to validating analytical conclusions. In order to support the system’s chaotic characteristics, we also compute the maximal Lyapunov exponents and fractal dimensions quantitatively. Finally, the chaotic trajectory of the system is stopped using the OGY approach, hybrid control method, and state feedback method.

Spatiotemporal Tipping Induced by Turing Instability and Hopf Bifurcation in a Population Ecosystem Model with the Fear Factor

Population ecosystems can display the tipping points at which extinctions of species happens. To predict the appearance of tipping points and to understand their evolution mechanism are of uttermost importance for ecological balance. Using techniques from bifurcation theory, we can predict the emergence of tipping points based on a spatiotemporal predator-prey system having a fear effect before an instability is encountered. In the case of no time delay, the tipping induced by Turing instability is studied. The conditions for Turing instability and local asymptotic stability of coexisting equilibrium points are given. It is ascertained that the introduction of diffusion causes the ecosystem to change from stable to unstable, and then the tipping occurs. Then, we investigate another tipping due to Hopf bifurcation. The delay-dependent stability criterion and Hopf bifurcation condition are derived, and the onset of Hopf bifurcations (tipping point) is also determined. In order to further probe into the mechanism of tipping evolution, explicit formulae are derived to ascertain the stability of bifurcated oscillations and the direction of bifurcation via the center manifold reduction. It is revealed that many tipping points may exist in ecological competition systems, and the tipping occurs many times as the fear delay increases. Finally, several simulation examples are provided to substantiate the analytical results.

Stability and bifurcation of a reaction-diffusion-advection model with nonlinear boundary condition

The dynamics of the reaction-diffusion-advection population models with linear boundary condition has been widely studied. This paper is devoted to the dynamics of a reaction-diffusion-advection population model with nonlinear boundary condition. Firstly, the stability of the trivial steady state is investigated by studying the corresponding eigenvalue problem. Secondly, the existence and stability of nontrivial steady states are proved by applying the Crandall-Rabinowitz bifurcation Theorem, the Lyapunov-Schmidt reduction method and perturbation method, in which bifurcation from simple eigenvalue and that from degenerate simple eigenvalue are both possible. The general results are applied to a parabolic equation with monostable nonlinear boundary condition, and to a parabolic equation with sublinear growth and superlinear boundary condition. Our theoretical results show that the nonlinear boundary condition can lead to the occurrence of various steady state bifurcations. Meanwhile, compared with the linear boundary condition, the nonlinear boundary condition can induce the multiplicity and growing-up property of positive steady-state solutions for the model with logistic interior growth. Finally, the numerical results show that the advection can change the bifurcation direction of some bifurcation, and affect the density distribution of the species.

Dynamics of a Delayed Predator–Prey Model with Prey Refuge, Allee Effect and Fear Effect

In this paper, we consider a Holling type II predator–prey system with prey refuge, Allee effect, fear effect and time delay. The existence and stability of the equilibria of the system are investigated. Under the variation of the delay as a parameter, the system experiences a Hopf bifurcation at the positive equilibrium when the delay crosses some critical values. We also analyze the direction of Hopf bifurcation and the stability of bifurcating periodic solution by the center manifold theorem and normal form theory. We show that the influence of fear effect and Allee effect is negative, while the impact of the prey refuge is positive. In particular, the birth rate plays an important role in the stability of the equilibria. Examples with associated numerical simulations are provided to prove our main results.

Pattern dynamics and bifurcation in delayed SIR network with diffusion network

The spread of infectious diseases often presents the emergent properties, which leads to more difficulties in prevention and treatment. In this paper, the SIR model with both delay and network is investigated to show the emergent properties of the infectious diseases’ spread. The stability of the SIR model with a delay and two delay is analyzed to illustrate the effect of delay on the periodic outbreak of the epidemic. Then the stability conditions of Hopf bifurcation are derived by using central manifold to obtain the direction of bifurcation, which is vital for the generation of emergent behavior. Also, numerical simulation shows that the connection probability can affect the types of the spatio-temporal patterns, further induces the emergent properties. Finally, the emergent properties of COVID-19 are explained by the above results.

Bifurcation analysis in an epidemic model on adaptive networks.

In this paper, we study a delayed adaptive network epidemic model in which the local spatial connections of susceptible and susceptible individuals have time-delay effects on the rate of demographic change of local spatial connections of susceptible and susceptible individuals. We prove that the Hopf bifurcation occurs at the critical value τ0 with delay τ as the bifurcation parameter. Then, by using the normal form method and the central manifold theory, the criteria for the bifurcation direction and stability are derived. Finally, numerical simulations are presented to show the feasibility of our results.

A class of natural pinus koraiensis population system with time delay and diffusion term

In this paper, we consider the long-term sustainability of the northeast Korean pine. We propose a class of natural Korean pine population system with time delay and diffusion term. First, by analyzing the roots distribution of the characteristic equation, we study the stability of the model system with diffusion terms and prove the occurrence of Hopf bifurcation. Second, we introduce lactation time delay into a population model with a diffusion term, based on stability theory of ordinary differential equation, norm form methods and center manifold theorem, the stability of bifurcating periodic solutions and the relevant formula for the direction of Hopf bifurcation are given. Finally, some numerical simulations are given.