Equilibrium Bifurcation Research Articles

Stability and bifurcation analysis of a size-stage-structured cooperation model

In this paper, we propose a size-stage-structured cooperation model which has two distinct life stages in facultative cooperator. The primary feature of this model is to consider size structure, stage structure and obligate and facultative symbiosis at the same time in a cooperation system. We use the method of characteristic to show that this new model can be reduced to a threshold delay equations (TDEs) model, which can be further transformed into a functional differential equations (FDEs) model by a simple change of variables. Such simplification allows us to apply the classical theory of FDEs and establish a set of sufficient conditions to investigate the qualitative analysis of solutions of the FDEs model, including the global existence and uniqueness, positivity and boundedness. What’s more, we use the geometric criteria to get the conclusions about stability and Hopf bifurcation of positive equilibrium because the coefficients of the characteristic equation depend on the bifurcation parameter. Finally, numerical simulations are carried out as supporting evidences of our analytical results. Our results show that the presence of size structure and stage structure plays an important role in the dynamic behavior of the model.

Non-axisymmetric magnetohydrodynamic equilibrium and stability in an axisymmetric toroidal device

This paper is dedicated to studying the radial broadening of helical states embedded in three-dimensional nonlinear magnetohydrodynamic (MHD) equilibria in an axisymmetric toroidal device. The stability of these helical equilibria is estimated by analyzing global MHD stability. An equilibrium bifurcation can develop a non-axisymmetric configuration with a helical core encompassed by an axis-symmetric plasma edge when the amplitude of an innermost magnetic helicity exceeds that of the primary axisymmetric magnetic component. It is found that internal negative magnetic shear drives the increase of the helicity and the decrease of the axisymmetric component via triple-mode nonlinear interactions, which induces the helical core to broaden radially. The helical structure mainly rests within the domain where the helicity is larger than the axisymmetric mode. Meanwhile, chaotic fields are generated outside the region predicted by the multi-region relaxed MHD model. Furthermore, the global stability analysis reveals that the helical equilibria are stabilized with flat or weak negative magnetic shear at the plasma core. As the core negative magnetic shear becomes significant, the helical equilibria become destabilized by kink modes even though the helical structure extends radially further out. This work is of importance and interest for reversed-field pinch plasma to reach steady quasi-single-helicity states.

Sliding dynamics and bifurcations of a human influenza system under logistic source and broken line control strategy.

This paper proposes a non-smooth human influenza model with logistic source to describe the impact on media coverage and quarantine of susceptible populations of the human influenza transmission process. First, we choose two thresholds $ I_{T} $ and $ S_{T} $ as a broken line control strategy: Once the number of infected people exceeds $ I_{T} $, the media influence comes into play, and when the number of susceptible individuals is greater than $ S_{T} $, the control by quarantine of susceptible individuals is open. Furthermore, by choosing different thresholds $ I_{T} $ and $ S_{T} $ and using Filippov theory, we study the dynamic behavior of the Filippov model with respect to all possible equilibria. It is shown that the Filippov system tends to the pseudo-equilibrium on sliding mode domain or one endemic equilibrium or bistability endemic equilibria under some conditions. The regular/virtulal equilibrium bifurcations are also given. Lastly, numerical simulation results show that choosing appropriate threshold values can prevent the outbreak of influenza, which implies media coverage and quarantine of susceptible individuals can effectively restrain the transmission of influenza. The non-smooth system with logistic source can provide some new insights for the prevention and control of human influenza.

Stability and bifurcation analysis of a multi-delay model for mosaic disease transmission

<abstract><p>A mathematical model is developed for analysis of the spread of mosaic disease in plants, which account for incubation period and latency that are represented by time delays. Feasibility and stability of different equilibria are studied analytically and numerically. Conditions that determine the type of behavior exhibited by the system are found in terms of various parameters. We have derived the basic reproduction number and identify the conditions resulting in eradication of the disease, as well as those that lead to the emergence of stable oscillations in the population of infected plants, as a result of Hopf bifurcation of the endemic equilibrium. Numerical simulations are performed to verify the analytical results and also to illustrate different dynamical regimes that can be observed in the system. In this research, the stabilizing role of both the time delay has been established i.e. when delay time is large, disease will persist if the infection rate is higher. The results obtained here are useful for plant disease management.</p></abstract>

HOPF BIFURCATION AND CHAOS OF COMBINATIONAL IMMUNE ANTI-TUMOR MODEL WITH DOUBLE DELAYS

In order to investigate the relations between tumor species growth and T cell activation assisted by dendritic cells, we establish a combinational immune anti-tumor model with double delays. Taking the activation rate of T cell and two time delays of tumor species growth and dendritic cell activation as parameters, we investigate the dynamical properties of the double delayed model, including the stability switches and the Hopf bifurcations of tumor-escape equilibrium and tumor-present equilibrium. With Hopf bifurcation curves, the center manifold theory and the normal form method, we find bi-stability states, the coexistence of two periodic solutions with different stabilities, two double Hopf bifurcation points, and use numerical simulations to show rich dynamic behaviors around the double Hopf bifurcation points, including the phase portraits and the corresponding Poincaré maps of chaotic attractors, as well as the progress transmission of unstable-oscillation-stable-oscillation. The theoretical and numerical results reveal the new methods of controlling tumor cells.

Contribution of a Ca<sup>2+</sup>-activated K<sup>+</sup> channel to neuronal bursting activities in the Chay model

<abstract><p>The central nervous system extensively expresses Ca<sup>2+</sup>-stimulated K<sup>+</sup> channels, which serve to use Ca<sup>2+</sup> to control their opening and closing. In this study, we explore the numerical computation of Hopf bifurcation in the Chay model based on the equilibrium point's stability and the center manifold theorem to illustrate the emergence of complicated neuronal bursting induced by variation of the conductance of the Ca<sup>2+</sup>-sensitive K<sup>+</sup> channel. The results show that the formation and removal of various firing activities in this model are due to two subcritical Hopf bifurcations of equilibrium based on theoretical computation. Furthermore, the computational simulations are shown to support the validity of the conceptual approach. Consequently, the conclusion could be helpful to improve and deepen our understanding of the contribution of the Ca<sup>2+</sup>-sensitive K<sup>+</sup> channel.</p></abstract>

Threshold control strategy for a Filippov model with group defense of pests and a constant-rate release of natural enemies.

In this paper, we establish an integrated pest management Filippov model with group defense of pests and a constant rate release of natural enemies. First, the dynamics of the subsystems in the Filippov system are analyzed. Second, the dynamics of the sliding mode system and the types of equilibria of the Filippov system are discussed. Then the complex dynamics of the Filippov system are investigated by using numerical analysis when there is a globally asymptotically stable limit cycle and a globally asymptotically stable equilibrium in two subsystems, respectively. Furthermore, we analyze the existence region of a sliding mode and pseudo equilibrium, as well as the complex dynamics of the Filippov system, such as boundary equilibrium bifurcation, the grazing bifurcation, the buckling bifurcation and the crossing bifurcation. These complex sliding bifurcations reveal that the selection of key parameters can control the population density no more than the economic threshold, so as to prevent the outbreak of pests.

Relaxation oscillations of a piecewise-smooth slow-fast Bazykin's model with Holling type Ⅰ functional response.

In this paper, we consider the dynamics of a slow-fast Bazykin's model with piecewise-smooth Holling type Ⅰ functional response. We show that the model has Saddle-node bifurcation and Boundary equilibrium bifurcation. Furthermore, it is also proven that the model has a homoclinic cycle, a heteroclinic cycle or two relaxation oscillation cycles for different parameters conditions. These results imply the dynamical behavior of the model is sensitive to the predator competition rate and the initial densities of prey and predators. In order to support the theoretical analysis, we present some phase portraits corresponding to different values of parameters by numerical simulation. These phase portraits include two relaxation oscillation cycles, an unstable relaxation oscillation cycle surrounded by a stable homoclinic cycle; the coexistence of a heteroclinic cycle and an unstable relaxation oscillation cycle. These results reveal far richer and much more complex dynamics compared to the model without different time scale or with smooth Holling type Ⅰ functional response.

Stability and Hopf Bifurcation Analyses of a Delayed Rumor Spreading Model and Its Discontinuous Control

In this paper, a delayed rumor propagation model with a discontinuous control mechanism is investigated. First, we investigate the model’s existence and boundedness. Then, conditions for the existence of the rumor-endemic equilibrium are derived utilizing differential inclusions. We examine the local asymptotic stability and Hopf bifurcation of the rumor-endemic equilibrium when the discontinuous control function is differential at the rumor-endemic equilibrium, considering the delay as the bifurcating parameter. When the discontinuous control function is not differential at the rumor-endemic equilibrium, we investigate the global asymptotically stable rumor-free and rumor-endemic equilibria using the Lyapunov functional technique. We conclude by providing two examples to validate our theoretical predictions. It is demonstrated that delay is a critical system parameter that can result in both Hopf bifurcation and chaos.

Complex dynamic analysis of a reaction-diffusion network information propagation model with non-smooth control

Rumors do the social’s perception seal, leaving people untouched with the true thing. Since the harmfulness of rumors is well known, in order to cut off the net of rumors and make the society run in order, it is necessary to have a deeper understanding of the spread of rumors. In this paper, the non-smooth reaction–diffusion system with considering the encouraging effect of secondary propagation of Internet platform on rumor propagation is used to study the dynamic system of rumor propagation. Firstly, the existence of the non-negative solution is proved by using the theory of upper and lower solutions for mixed monotone system. Secondly, the value of the basic reproduction number is calculated and the existence of positive equilibrium are discussed. Thirdly, the backward bifurcation and the saddle-node bifurcation are investigated. Fourthly, the stability of rumor propagation equilibrium and Hopf bifurcation are analyzed theoretically. Finally, some numerical simulations which proves the validity of the above theory are provided.

Unstable periodic orbits analysis in the Qi system

We use the variational method to extract the short periodic orbits of the Qi system within a certain topological length. The chaotic dynamical behaviors of the Qi system with five equilibria are analyzed by the means of phase portraits, Lyapunov exponents, and Poincaré maps. Based on several periodic orbits with different sizes and shapes, they are encoded systematically with two letters or four letters for two different sets of parameters. The periodic orbits outside the attractor with complex topology are discovered by accident. In addition, the bifurcations of cycles and the bifurcations of equilibria in the Qi system are explored by different methods respectively. In this process, the rule of orbital period changing with parameters is also investigated. The calculation and classification method of periodic orbits in this study can be widely used in other similar low-dimensional dissipative systems.

Memristor-coupled asymmetric neural networks: Bionic modeling, chaotic dynamics analysis and encryption application

With the rapid development of artificial intelligence, it has important theoretical and practical significance to construct neural network models and study their dynamical behaviors. This article mainly focuses on the bionic model and chaotic dynamics of the asymmetric neural network as well as its engineering application. We first construct a memristor-coupled asymmetric neural network (MANN) utilizing two asymmetrical sub-neural networks and a coupled multi-piecewise memristor synapse. Then, the chaotic dynamics of the proposed MANN is studied and analyzed by using basic dynamics methods like equilibrium stability, bifurcation diagrams, Lyapunov exponents, and Poincare mappings. Research results show that the proposed MANN exhibits multiple complex dynamical characteristics including infinitely wide hyperchaos with amplitude control, hyperchaotic initial-boosted behavior, and arbitrary number of hyperchaotic multi-structure attractors. More importantly, the phenomena of the infinitely wide hyperchaos and the hyperchaotic multi-structure attractors are observed in neural networks for the first time. Meanwhile, applying the hyperchaotic multi-structure attractors, a color image encryption scheme is designed based on the proposed MANN. Performance analyses show that the designed encryption scheme has some merits in correlation, information entropy, and key sensitivity. Finally, a physical circuit of the MANN is implemented and various typical dynamical behaviors are verified by hardware experiments.

Endemic Equilibrium and Forward Bifurcation in the Mathematical Model for Using Wolbachia to Control Spread of Zika Virus Disease

This paper focuses on the use of wolbachia to control the spread of zika virus disease. Zika virus disease is an arboviral disease that spreads through bites of female mosquitoes in the aedes family especially, aedes aegypti. Experimental studies have indicated that wolbachia could be used to prevent the spread of zika virus disease by infecting aedes aegypti with wolbachia in a laboratory and releasing them in the wild to mate with the wild aedes aegypti.
 A system of nonlinear ordinary differential equations is used to model the use of wolbachia to stop the spread of zika virus disease in the human and mosquito populations. as well as the population of wolbachia-infected aedes aegypti used as control. It is shown through bifurcation analysis that the model exhibits forward bifurcation, which confirms that a unique endemic equilibrium exists in the model when the control reproduction number, $ \mathcal{R}_c>1$. The existence of forward bifurcation in the model means that $ \mathcal{R}_c<1$ is enough to guarantee eradication of zika virus disease using wolbachia as a biocontrol. Hence, the spread of zika virus disease can be controlled irrespective of the initial sizes of infected human and mosquito populations

Five-dimensional memristive Hopfield neural network dynamics analysis and its application in secure communication

PurposeThis study aims to improve the complexity of chaotic systems and the security accuracy of information encrypted transmission. Applying five-dimensional memristive Hopfield neural network (5D-HNN) to secure communication will greatly improve the confidentiality of signal transmission and greatly enhance the anticracking ability of the system.Design/methodology/approachChaos masking: Chaos masking is the process of superimposing a message signal directly into a chaotic signal and masking the signal using the randomness of the chaotic output. Synchronous coupling: The coupled synchronization method first replicates the drive system to get the response system, and then adds the appropriate coupling term between the drive The synchronization error and the coupling term of the system will eventually converge to zero with time. The synchronization error and coupling term of the system will eventually converge to zero over time.FindingsA 5D memristive neural network is obtained based on the original four-dimensional memristive neural network through the feedback control method. The system has five equations and contains infinite balance points. Compared with other systems, the 5D-HNN has rich dynamic behaviors, and the most unique feature is that it has multistable characteristics. First, its dissipation property, equilibrium point stability, bifurcation graph and Lyapunov exponent spectrum are analyzed to verify its chaotic state, and the system characteristics are more complex. Different dynamic characteristics can be obtained by adjusting the parameter k.Originality/valueA new 5D memristive HNN is proposed and used in the secure communication

Equilibrium points, linear stability, and bifurcation analysis on the dynamics of a quantum dot light emitting diode system

Abstract In this paper a quantum dot light emitting diode (QDLED) system is modeled as a three dimensionless differential system. We identified three vital parameters Γ the rate of the output coupling rate of photons in the optical mode to the nonradiative decay rate of the number of carriers in the wetting layer (WL), Γ 1 the rate of the Einstein coefficient to the nonradiative decay rate of the number of carriers in the WL and δ o such that I/We is the divide the injection current on Einstein coefficient multiplying elementary charge. Using these parameters a novel way to stability of the dynamics based on the number of interior equilibrium solutions admitted by the system has been obtained. We investigate the considered model is reach in dynamics and identified various bifurcations that are experienced by the considered system from the parameter space. These are saddle-node bifurcation, trans-critical bifurcation, and pitchfork bifurcation. In this study we offer mathematical proof for the incidence of these bifurcations that take place in the considered dynamical system as the parameters move between the regions presented in the parameter space and accordingly, this method was enhanced with numerical results.

Dynamical analysis in a piecewise smooth predator–prey model with predator harvesting

The aim of this paper is to study the dynamical behaviors of a piecewise smooth predator–prey model with predator harvesting. We consider a harvesting strategy that allows constant catches if the population size is above a certain threshold value (to obtain predictable yield) and no catches if the population size is below the threshold (to protect the population). It is shown that boundary equilibrium bifurcation and sliding–grazing bifurcation can happen as the threshold value varies. We provide analytical analysis to prove the existence of sliding limit cycles and sliding homoclinic cycles, the coexistence of them with standard limit cycles. Some numerical simulations are given to demonstrate our results.

Allee effect in a Ricker type predator-prey model

The stability of the predator-prey model subject to the Allee effect is an interesting topic in recent times. The impact of a weak Allee effect on the stability of a discrete-time predator-prey model is investigated in this paper. Equilibrium analysis, stability analysis, and bifurcation theory are used to examine the mathematical properties of the proposed model. By using the Allee parameter as the bifurcation parameter, we provide sufficient conditions for the flip bifurcation. Numerical simulations are used to demonstrate our analytical conclusions.

Global studies on a continuous planar piecewise linear differential system with three zones

This paper is concerned with the global dynamics of a continuous planar piecewise linear differential system with three zones, where the dynamic of the one of the exterior linear zones is saddle and the remaining one is anti-saddle. We give all global phase portraits in the Poincaré disc and the complete bifurcation diagram including boundary equilibrium bifurcation curves, degenerate boundary equilibrium bifurcation curves, homoclinic bifurcation curves and double limit cycle bifurcation curves. Its application in a second-order memristor oscillator is shown. Finally, some numerical phase portraits are demonstrated to illustrate our theoretical results.

The large key space image encryption algorithm based on modulus synchronization between real and complex fractional-order dynamical systems.

This paper constructs and analyzes the dynamical properties of a new fractional-order real hyper-chaotic system and its corresponding complex variable system. A thorough analysis was done by employing stability of equilibrium points, phase plots, Lyapunov spectrum, and bifurcation analysis for the consequences of varying fractional-order derivative and parameter values on the system. For the first time, a modulus synchronization scheme is proposed to synchronize real and complex fractional-order dynamical systems. Based on Lyapunov stability theory, non-linear controllers are designed to achieve the proposed modulus synchronization scheme. A new modulus synchronization encryption algorithm with a large key space size for digital images is introduced for the application. The experimental results and analysis validate the desired algorithm. Also, we compare our result of the new encryption algorithm with the previously published literature and verify the efficacy of the considered scheme. Numerical simulations are given to validate the theoretical analysis

Dynamics of a delayed rumor spreading model with discontinuous threshold control

In this paper, we studied a delayed rumor spreading model with discontinuous threshold control. First, we studied the existence of equilibria of the subsystem. Regarding the delay as bifurcating parameter, the local asymptotic stability and Hopf bifurcation of the positive equilibrium are discussed by analyzing the corresponding characteristic equations of linearized systems. Then, we studied the existence of the sliding mode and analyzed the existence of the tangent equilibria, boundary equilibria, regular equilibria, and the stability of the pseudo-equilibrium. Finally, we provide some numerical simulations to verify the theoretical results.