Search

Under the background of double carbon, it is important to study forest pests and diseases to improve forest carbon sink. In this paper, we establish a delayed model associated with the larvae and adults of Saperda populnea, susceptible poplars, and infected poplars. First, we analyze the existence and stability of the equilibrium of the model. Second, we study the existence of Hopf bifurcation near the equilibrium and obtain the normal form of Hopf bifurcation by the multiple time scales method. Then, we analyze the direction and stability of Hopf bifurcating periodic solutions. Third, we analyze and conjecture some parameter values based on official data, and carry out numerical simulations to verify our results. Finally, we give some suggestions on the prevention and control of Saperda populnea. [ABSTRACT FROM AUTHOR]

This paper introduces diffusion into an immunosuppressive infection model with virus stimulation delay and Beddington-DeAngelis functional response. First, we study the stability of positive constant steady state solution and show that the Hopf bifurcation will exist under certain conditions. Second, we derive the normal form of the Hopf bifurcation for the model reduced on the center manifold by using the multiple time scales (MTS) method. Moreover, the direction and stability of the bifurcating periodic solution are investigated. Finally, we present numerical simulations to verify the results of theoretical analysis and provide corresponding biological explanations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a nonlinear discrete prey-predator system with fear effects. The existence, local stability and boundedness of positive equilibrium point are discussed. Using the center manifold theorem and bifurcation theory, the conditions for the existence of flip bifurcation and Neimark-Sacker bifurcation in the interior of R²+ are established. Furthermore, the numerical simulations not only show complex dynamical behaviors, but also verify our analysis results. A feedback control strategy is employed to control bifurcation and chaos in the system. [ABSTRACT FROM AUTHOR]

This work investigates the dynamic complications of the Ricker type predator-prey model in the presence of the additive type Allee effect in the prey population. In the modeling of discretetime models, Euler forward approximations and piecewise constant arguments are the most frequently used schemes. In Euler forward approximations, the model may undergo period-doubled orbits and invariant circle orbits, even while varying the step size. In this way, differential equations with piecewise constant arguments (Ricker-type models) are a better choice for the discretization of a continuous-time model because they do not involve any step size. First, the interaction between prey and predator in the form of the Holling-II type is considered. The essential mathematical features are discussed in terms of local stability and the bifurcation phenomenon as well. Next, we apply the center manifold theorem and normal form theory to achieve the existence and directions of flip bifurcation and Neimark-Sacker bifurcation. Moreover, this paper demonstrates that the outbreak of chaos can stabilize in the considered model with a higher value of the Allee parameter. The existence of chaotic orbits is verified with the help of a one-parameter bifurcation diagram and the largest Lyapunov exponents, respectively. Furthermore, different control methods are applied to control the bifurcation and fluctuating phenomena, i.e., state feedback, the Ott-Grebogi-Yorke, and hybrid control methods. Finally, to ensure our analytical results, numerical simulations have been carried out using MATLAB software. [ABSTRACT FROM AUTHOR]

In this paper, a six-neuron incommensurate fractional order BAM neural network model with multi-delays is considered. We demonstrate that the equilibrium point of the system loses its stability and Hopf bifurcation emerges when the delay passes through a critical value. And the relationship between the critical delay of Hopf bifurcation and size of fractional orders is found. Finally, some numerical simulations are given to verify the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics of a general nonlocal delayed reaction–diffusion–advection vector-disease model with spatial heterogeneity. The existence of the nonconstant positive steady state is identified by means of the Rabinowitz's bifurcation theory. Further, the stability of the positive steady state and the Hopf bifurcation are analyzed in terms of the eigenvalue distribution associated with the steady state. Moreover, the direction of the Hopf bifurcation and the stability of the bifurcated periodic solution are obtained by the center manifold reduction and normal form theory with the weighted inner product. It is found that in spatial heterogeneity, the large diffusion can change the stability of the infected host population due to the combined effects of the nonlocal dispersal and time delay. [ABSTRACT FROM AUTHOR]

This paper proposes a diffusive predator–prey model with Allee effect, time delay and anti-predator behavior. First, the existence and stability of all equilibria are analyzed and the conditions for the appearance of the Hopf bifurcation are studied. Using the normal form and center manifold theory, the formulas which can determine the direction, period and stability of Hopf bifurcation are obtained. Numerical simulations show that the Allee effect can determine the survival abundance of the prey and predator populations, and anti-predator behavior can greatly improve the stability of the coexisting equilibrium. [ABSTRACT FROM AUTHOR]

Aiming at the cause of limit cycles and the pathway of chaotic oscillation by doubly fed induction generator (DFIG) as its internal parameters change, this paper proposed a method to identify limit cycles and a numerical method to identify chaotic pathways. First, a simplified four-dimensional nonlinear dynamic model of the DFIG is developed. A simple criterion for the Hopf bifurcation parameter of the DFIG is given: the Routh–Hurwitz stability criterion. In addition, the relationship between the system at critical damping and the Hopf bifurcation is obtained. Second, the stability interval of the DFIG linearized system is verified using a time-domain approach when the system has recovered from a slight disturbance. Finally, periodic, quasi-periodic, and chaotic oscillations are predicted to occur based on the largest Lyapunov exponent and power spectrum to further determine the system's parameters. The path and threshold of the tuning parameter inducing the DFIG into chaos are obtained to provide a theoretical basis for revealing the mechanism inducing chaotic oscillations. The numerical simulation results demonstrate the effectiveness and feasibility of the proposed method. [ABSTRACT FROM AUTHOR]

Purpose: In order to reduce the hazards caused by the nonlinear torsional vibration of the shaft system of the direct-drive permanent magnet wind turbine, the dynamic characteristics of the controlled shaft system torsional vibration were analyzed. Methods: First, the shaft system of the direct-drive permanent magnet wind turbine is equivalent to a two-mass model. Based on considering the electromechanical coupling and the nonlinear change of the main shaft damping, the torsional vibration dynamic model of the shaft system is established by the Lagrange–Maxwell equation. Second, with the main shaft damping as the bifurcation parameter, the Routh–Hurwitz criterion is used to calculate the Hopf bifurcation critical point and stability range of the shafting system. Finally, a nonlinear state feedback controller is designed, and the controlled shafting system paradigm is obtained by the direct method. Results: The torsional vibration characteristics of shafting and the influence of supercritical Hopf bifurcation on shafting are discussed. In order to suppress the unstable torsional vibration of the shafting system and analyze the influence of the controller's linear gain and nonlinear gain on the torsional vibration of the shafting system, the numerical simulation verifies the correctness of the theoretical analysis. Conclusions: This paper can provide a theoretical basis for the stable operation of the direct-drive permanent magnet wind turbine. [ABSTRACT FROM AUTHOR]

In this paper, a simplified discrete-time SIR model with nonlinear incidence and recovery rates is discussed. Here, using the integral step size and the intervention level as control parameters, we mainly discuss three types of codimension-two bifurcations (fold-flip bifurcation, 1:3 resonance, and 1:4 resonance) of the simplified discrete-time SIR model in detail by bifurcation theory and numerical continuation techniques. Parameter conditions for the occurrence of codimension-two bifurcations are obtained by constructing the corresponding approximate normal form with translation and transformation of several parameters and variables. To further confirm the accuracy of our theoretical analysis, numerical simulations such as phase portraits, bifurcation diagrams, and maximum Lyapunov exponents diagrams are provided. In particular, the coexistence of bistability states is observed by giving local attraction basins diagrams of different fixed points under different integral step sizes. It is possible to more clearly illustrate the model's complex dynamic behavior by combining theoretical analysis and numerical simulation. [ABSTRACT FROM AUTHOR]

A phytoplankton-zooplankton prey-predator model has been investigated and the role of nutrient concentrations and toxin producing phyloplankton for controlling the planktonic bloom has been discussed. It is assumed that the nutrient uptake function for toxin producing phytoplankton (TPP) is taken as Holling type-II functional responce. We also assume two different types of predational forms: simple law of mass action and Holling type-II response term. It is found that constant nutrient input and dilution rate of nutrient and toxin influence the plankton ecosystem model and maintain stability around the coexistent equilibrium. Therefore stability analysis of all the feasible equilibria are studied. Our observations indicate that if the toxin liberation rate crosses a certain critical value, the systein enters into Hopf bifurcation so the existence of Hopf-bi furcation for the interior equilibrium of the system is explored. The local analysis shows that oscillatory regimes and their combinations. In this present paper it reveals the fact that nutrient as well as toxin producing phytoplankton control platiktonic bloom i.e. the rate of toxin production by toxin producing phytoplankton plays an important role for controlling oscillations in the plankton system. We also notice that mon tality functions of zooplankton has an important role in controlling oscillations, coexistence, survival or extinction of the zooplankton population. Due to a high rate of toxin production of phytoplankton species can change the steady-state behaviour. Computer simulations illustrate the results. [ABSTRACT FROM AUTHOR]

From the point of view of nonlinear dynamical systems, some neurological disorders can be indicated by bifurcations because bifurcations change the firing patterns of neurons; therefore, it is essential to control the bifurcations. We can avoid undesirable dynamical behaviors such as the behaviors of the Rulkov map model by controlling bifurcation which, then, can assist in modeling neuronal diseases. In this paper, we investigate the existence of Hopf bifurcation and analytically identify the type of bifurcation for the Rulkov map model; then, we apply a dynamic feedback controller using a washout filter to control the onset of Hopf bifurcation. Also, we can control the behaviors of the neurons, such as spiking or spiking-bursting behavior of neurons, and create the Hopf bifurcation for some parameters. The results analytically obtained in this paper can be applied to control some epileptic seizures. [ABSTRACT FROM AUTHOR]

The current paper is a review of some transformation techniques of partial differential equations (PDEs) using similarity techniques which have the ability to reduce the number of independent variables to be a single variable resulting in a simpler ordinary differential equation (ODEs). These techniques have the benefit to obtain some completely new solutions. The paper introduces a summary and literature review for method of characteristic lines, Bäcklund transformations, Cole-Hopf Transformation, Lie infinitesimals, Non-classical symmetry method, Potential Symmetry, direct methods and characteristic function method. [ABSTRACT FROM AUTHOR]

This paper carries out an analysis of the global properties of solutions of an in-host model of hepatitis C for general values of its parameters. A previously unknown stable steady state on the boundary of the positive orthant is exhibited. It is proved that the model exhibits Hopf bifurcations and hence periodic solutions. A general parametrization of positive steady states is given and it is determined when the number of steady states is odd or even, according to the value of a certain basic reproductive ratio. This implies, in particular, that when this reproductive ratio is greater than one there always exists at least one positive steady state. A positive steady state which bifurcates from an infection-free state when the reproductive ratio passes through one is always stable, i.e. no backward bifurcation occurs in this model. The results obtained are compared with those known for related models of viral infections. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of tumor–immune systems perturbed by the environmental noise and focus on the longtime behaviors. The existence and uniqueness of the globally positive solution to the tumor–immune system are proved using stochastic Lyapunov analysis and Itô's formula. We study the boundedness of moments for tumor cells and effector cells. By considering the dynamics on the boundary, applying the comparison theorem and the strong ergodic theorem, we obtain a threshold λ which is used to characterize the stochastic permanence in the sense that there is a unique invariant measure and extinction of the stochastic tumor–immune system. We also give biological interpretations about our analytical results of stochastic system. In addition, we present numerical examples and discussions to illustrate our analysis results. We find that the small noises preserve Hopf bifurcation of the deterministic system in stochastic setting and study numerically how the stochastic Hopf bifurcation with parameters occurs. [ABSTRACT FROM AUTHOR]

In this paper, we study a tumor-immune system interaction model with dendritic cell therapy and immune response delay. First, it is shown that the ODE version of the model has a Bogdanov--Takens (BT) singularity or a weak focus with multiplicity at most 1 for different parameter values. As the parameters vary, the ODE model undergoes supercritical Hopf bifurcation and supercritical BT bifurcation. Our analysis indicates that there exists a threshold value of the activation rate of T cells, below which tumor immune escape occurs, above or at which T cells and tumor cells coexist in the form of a stable periodic oscillation or steady state. Second, we study how the immune response delay affects the dynamics of the model. Our results reveal that the delay can destabilize the stable positive equilibrium through Hopf bifurcation. Furthermore, the direction and stability of Hopf bifurcation are derived. When there is a cusp, we show that it is a BT singularity for any delay and the delay model also undergoes BT bifurcation. Finally, numerical simulations are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

The dynamics of a general reaction–diffusion–advection two species model with nonlocal delay effect and Dirichlet boundary condition is investigated in this paper. The existence and stability of the positive spatially nonhomogeneous steady state solution are studied. Then by regarding the time delay τ as the bifurcation parameter, we show that Hopf bifurcation occurs near the steady state solution at the critical values τ n (n = 0 , 1 , 2 , ...) . Moreover, the Hopf bifurcation is forward and the bifurcated periodic solutions are stable on the center manifold. The general results are applied to a Lotka–Volterra competition–diffusion–advection model with nonlocal delay. [ABSTRACT FROM AUTHOR]

In this paper, we formulate and study a fractional-order network model with four neurons, bidirectional ring structure and self-delay feedback. For the scenario of non-identical neurons, we develop a new algebraic technique to deal with the characteristic equation with e − 4 s τ (τ is the self-feedback delay) term and thus establish the easy-to-check criteria to determine the Hopf bifurcation point of self-feedback delay by fixing communication delay in its stable interval. For the scenario of identical neurons, we apply the crossing curves method to the fractional functional equations and thus procure the Hopf bifurcation curve. The obtained results accommodate the fact that the model cannot preserve its stability behavior when the self-feedback delay crosses the Hopf bifurcation point in the positive direction. Finally, we deliberate on the correctness of our methodology through two demonstration examples. [ABSTRACT FROM AUTHOR]

In this paper, we provide a short proof for the Hopf bifurcation theorem in the Gurtin-MacCamy's population dynamics model. Here we use the Crandall and Rabinowitz's approach, based on the implicit function theorem. Compared with previous methods, here we require the age-specific birth rate to be slightly smoother (roughly of bounded variation), but we have a huge gain for the length of the proof. [ABSTRACT FROM AUTHOR]

The dynamics of the classical diffusive Nicholson's blowfly model with zero Dirichlet boundary condition is less understood. A challenging question is that how the stability of the unique and implicit positive steady state changes in parameters. Numerics suggests that it could be stable or unstable, and a time periodic solution may appear in the latter case. In this paper, we make an effort toward this question by considering a discretized version of the model. To overcome the difficulty caused by the implicit steady state, we first establish some monotonicity properties, which are then used to find the parameter regions such that the positive steady state is locally stable, or unstable due to the occurrence of Hopf bifurcations. A set of sufficient conditions for the global convergence of the positive equilibrium are also obtained. Finally, the global existence of periodic solution is studied by using the global Hopf bifurcation theorem and high-dimensional Bendixson's criterion. [ABSTRACT FROM AUTHOR]

Recently, Christian Cortés García proposed and studied a continuous modified Leslie–Gower model with harvesting and alternative food for predator and Holling-II functional response, and proved that the model undergoes transcritical bifurcation, saddle-node bifurcation and Hopf bifurcation. In this paper, we dedicate ourselves to investigating the bifurcation problems of the discrete version of the model by using the Center Manifold Theorem and bifurcation theory, and obtain sufficient conditions for the occurrences of the transcritical bifurcation and Neimark–Sacker bifurcation, and the stability of the closed orbits bifurcated. Our numerical simulations not only illustrate corresponding theoretical results, but also reveal new dynamic chaos occurring, which is an essential difference between the continuous system and its corresponding discrete version. [ABSTRACT FROM AUTHOR]

In this paper, we take physical and health integration as the entry point. Firstly, based on the transformation mechanism of adolescent idiopathic scoliosis we construct a time delay differential model. Moreover, using the theory of characteristic equation we discuss the stability of a positive equilibrium under the delays of τ = 0 and τ ≠ 0 . Furthermore, through numerical simulation, it has been verified the delay, τ , exceeds a critical value, the positive equilibrium loses its stability and Hopf bifurcation occurs. Lastly, we determine that sports have a positive effect on adolescent idiopathic scoliosis, directly reducing the number of people with adolescent idiopathic scoliosis. [ABSTRACT FROM AUTHOR]

A three-layer prey–predator model with two time-delays and one delay-dependent parameter is established in this paper. To begin, the paper calculates the conditions for each population in the model to maintain the quantity stable and Hopf bifurcation when τ 1 = τ 2 = 0 , τ 1 = 0 , τ 2 ≠ 0 , τ 2 is in the stable interval, and τ 1 > 0. The crossing curves, their type, and the direction of the crossing curves are then obtained using the crossing curve method, which is composed of the threshold values of the dynamic behavior change on the two time-delays plane when τ 1 , τ 2 > 0. The real data from the forage grass–Ochotona curzoniae–Buteo hemilasius food chain is used to conduct an empirical study of the model. When τ 1 , τ 2 > 0 , the feasible region of the crossing curves is open-ended, and the model's crossing curves on the (τ 2 , τ 1) plane are truncated. This indicates that the model's threshold distribution of dynamic behavior change is a regular curve made of several curves. The simulation using the time-delay value on the crossing curves shows that the model produces different dynamic behaviors such as stability, bifurcation, and chaos depending on the time-delay value on both sides of the curves. The critical values of dynamic behavior change are time-delay values on the crossing curves. The empirical study shows that increasing Ochotona curzoniae's environmental capacity can easily cause Hopf bifurcation of the system. At this time, the number of each population in the Plateau ecosystem constantly fluctuates, and Ochotona curzoniae is vulnerable to extinction. [ABSTRACT FROM AUTHOR]

In this paper, a size-structured population model is studied. Choosing the coefficient of birth function as the varying parameter, we prove the existence of Hopf bifurcation and discuss the bifurcation properties, such as the direction of Hopf bifurcation and the stability of bifurcated periodic solutions. The methods used in this paper include Hopf bifurcation theorem, center manifold theorem and normal form theory for the abstract Cauchy problems in a nondense domain. Numerical simulations are finally carried out to show the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a model called the k-nearest neighbor queue with the possibility of having delayed queue length feedback. We prove fluid limits for the stochastic queueing model and show that the fluid limit converges to a system of delay differential equations. Using the properties of circulant matrices, we derive a closed form expression for the value of the critical delay, which governs whether the delayed information will induce oscillations or a Hopf bifurcation in our queueing system. [ABSTRACT FROM AUTHOR]

In this paper, Myc/E2F/miR-17-92 network under Gaussian white noise is studied. Taking the time delay as the parameter, the Hopf bifurcation of the system is obtained, which causes the protein concentration to oscillate periodically. Under the influence of time delay and noise, the stochastic D-bifurcation of the system is obtained. It is worth noting that the occurrence of stochastic P-bifurcation is successfully captured. Thus a pattern of coexistence of high and low protein concentrations is founded in the network. The specific research methods of this paper are as follows: firstly, the system is reduced to a finite dimensional system by using stochastic center manifold and normal form theory. Then, using the stochastic averaging method, the Fokker–Planck–Kolmogorov equation of the system is constructed in which the statistical response in the stationary state is the probability density. Finally, the stochastic bifurcation analysis and numerical simulation are carried out. The agreements between the analytical method and those obtained numerically validate the effectiveness of the analytical investigations. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the classical SIR epidemic model by replacing the simple bilinear transmission rate by a nonlinear one. Our results show that in the presence of environmental fluctuations represented by Brownian motion and that mainly act on the transmission rate, the generalized non-concave force of infection adopted here, greatly affects the long-time behavior of the epidemic. Employing the Markov semigroup theory, we prove that the model solutions do not admit a unique stationary distribution but converge to 0 in p th moment for any p > 0. Furthermore, we prove that the disease extinguishes asymptotically exponentially with probability 1 without any restriction on the model parameters and we also determine the rate of convergence. This is an unexpected qualitative behavior in comparison with the existing literature where the studied epidemic models have a threshold dynamics behavior. It is also a very surprising behavior regarding the deterministic counterpart that can exhibit a rich qualitative dynamical behaviors such as backward bifurcation and Hopf bifurcation. On the other hand, we show by several numerical simulations that as the intensity of environmental noises becomes sufficiently small, the epidemic tends to persist for a very long time before dying out from the host population. To solve this problem and to be able to manage the pre-extinction period, we construct a new process in terms of the number of infected and recovered individuals which admits a unique invariant stationary distribution. Finally, we discuss the obtained analytical results through a series of numerical simulations. [ABSTRACT FROM AUTHOR]

Memristor-based neuromorphic systems have a neuro-bionic function, which is critical for possibly overcoming Moore's law limitation and the von Neumann bottleneck problem. To explore neural behaviors and complexity mechanisms in memristive circuits, this paper proposes an N-type locally active memristor, based on which a third-order memristive circuit is constructed. Theoretical analysis shows that the memristive circuit can exhibit not only various action potentials but also self-sustained oscillation and chaos. Based on Chua's theory of local activity, this paper finds that the neural behaviors and chaos emerge near the edge of chaos through subcritical Hopf bifurcation, in which the small unstable limit cycle is depicted by the dividing line between the attraction basin of the large stable limit cycle and the attraction basin of the stable equilibrium point. Furthermore, an analog circuit is designed to imitate the action potentials and chaos, and the simulation results are in agreement with the theoretical analysis. [ABSTRACT FROM AUTHOR]

In comparison with numerous available results with regard to bifurcations on account of fractional-order neural networks (FONNs) with the retarded delays, this paper is concerned with the stability and bifurcation of FONN with neutral delays. Delay-induced bifurcation results are firstly, procured and the bifurcation diagrams with respect to different delays are nicely plotted. Then it reveals that both time delay and fractional order have important effects in affecting the stability and bifurcation of the developed FONN. It further detects that FONN involving neutral delays can postpone the onset of bifurcations of the developed FONN compared with FONN with the retarded delays. It hints that neutral delays have a vital impact on the dynamics of FONNs in comparison with FONN with the retarded delays, which has outstanding performance in maintaining stability of FONNs. Eventually, numerical examples are addressed to check the availability of the developed results. The obtained criterion can be evaluated as an important contribution to the stability issue of the considered neural networks (NNs). [ABSTRACT FROM AUTHOR]

This paper reports on the dynamical analysis, field programmable gate array (FPGA) implementation of autonomous Josephson junction (JJ) jerk oscillator with cosine interference term (AJJJOCIT) and investigation of its collective behavior in a network. The AJJJOCIT derived from a resistive capacitive shunted JJ model with cosine interference term has two or no equilibrium points as a function of direct current (DC). One of the equilibrium points is unconditionally unstable and the other equilibrium point has a Hopf bifurcation where its expression depends on DC and coherence parameters. One-scroll self-excited chaotic attractor, one-scroll chaotic hidden attractor, steady state attractors, bistable periodic attractors, limit cycle and coexistence between periodic and one-scroll chaotic self-excited (or hidden) attractors are revealed in the AJJJOCIT during the numerical analysis. Moreover, the FPGA of AJJJOCIT is implemented and the FPGA results are qualitatively the same as those obtained during the numerical analysis. Finally, the collective dynamics of the AJJJOCIT are studied using a single-layer matrix of the AJJJOCIT. It is demonstrated that chimera states exist in the system and when increasing coupling strength, a completely synchronized network is revealed. [ABSTRACT FROM AUTHOR]

This paper aims to report complex bursting patterns in a parametrically and externally excited Jerk circuit system. Typically, the common compound bursting with two clusters in each period can be observed in this system. By the fast-slow analysis, we find that the active states related to this type of bursting are created by supercritical Hopf bifurcations. Interestingly, with the increase of parametric excitation amplitude, the common bursting pattern evolves into the ones that display complex structures. We show that the parametric excitation amplitude plays a vital role in influencing the dynamical behaviors, which is apparent from the two-parameters analysis. Then, a rich variety of dynamical phenomena including period-doubling and inverse period-doubling cascades in orbits of period 2, 4, and 8 can be observed with the increase of parametric excitation amplitude, thereby leading to complex bursting patterns with multiple-periodically active states. Besides, as a result of the period-doubling cascades, chaotic attractors are eventually discovered, which can be corroborated by the maximal Lyapunov exponent. In particular, periodic windows with period-doubling and inverse period-doubling cascades also emerge in the chaotic region when the parametric excitation amplitude increases to higher values. These give rise to the complex bursting patterns with chaotically active states. Based on this, two types of bursting patterns with complex structures, i.e., bursting patterns with multiple-periodically active states and bursting patterns with chaotically active states are revealed. Our results enrich the routes to complex bursting and deepen the understanding of bursting patterns with complex structures. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamical behaviors of a basic delayed physical model for aerosol–cloud–precipitation system, which was proposed by Koren and Feingold. The rain production delay τ is crucial for this system. The main work of the present paper is to study the effect of cloud and rain interaction on climate by the mathematical methods, such as double Hopf bifurcation analysis. First, we prove the uniqueness of the equilibrium of interest, i.e., the positive equilibrium, which has never been reported. Next, we use DDE-BIFTOOL to plot the bifurcation diagrams with respect to two bifurcation parameters, i.e., a and τ , and then find its double Hopf bifurcation points. Then, we study their unfolding and classification by the methods of multiple scales and normal form. Finally, we verify the results by numerical simulations. We find some complicated dynamic behaviors of the system via analytical method, such as stable equilibrium, stable periodic, and quasi-periodic solutions. For the stable equilibrium, the numerical simulations agree well with the analytical results. The stable equilibrium implies a balance between cloud formation and depletion (drizzle), and stable periodic solution is related to cycles of formation of thicker clouds that are later consumed by stronger rain (the moderately drizzle). These complex dynamical phenomena can be very helpful for the researchers to understand the mechanism of aerosol–cloud–precipitation system. And these results can also be used to predict rainfall and climate change, thereby reducing the impact of extreme weather, and its subsequent natural disasters and accidents on economic and social development. [ABSTRACT FROM AUTHOR]

HIV infects active uninfected CD4 + + T cells, and the active CD4 + + T cells are transformed from quiescent state in response to antigenic activation. Activation effect of the CD4 + + T cells may play an important role in HIV infection. In this paper, we formulate a mathematical model to investigate the activation effect of CD4 + + T cells on HIV dynamics. In the model, the uninfected CD4 + + T cells are divided into two pools: quiescent and active, and the stimuli rate of quiescent cells by HIV is described by saturated form function. We derive the basic reproduction number R 0 R 0 and analyze the existence and the stability of equilibria. Numerical simulations confirm that the system may have backward bifurcation and Hopf bifurcation. The results imply that R 0 R 0 cannot completely determine the dynamics of the system and the system may have complex dynamics, which are quite different from the models without the activation effect of CD4 + + T cells. Some numerical results are further presented to assess the activation parameters on HIV dynamics. The simulation results show that the changes of the activation parameters can cause the system periodic oscillation, and activation rate by HIV may induce the supercritical Hopf bifurcation and subcritical Hopf bifurcation. Finally, we proceed to investigate the effect of activation on steady-state viral loads during antiretroviral therapy. The results indicate that, viral load may exist and remain high level even if antiretroviral therapy is effective to reduce the basic reproduction number below 1. [ABSTRACT FROM AUTHOR]

Smoking poses an enormous threat to public health in the globe and continues to be one of the main cause of health problems. In this paper, a delayed smoking model with reversion class is proposed. Local stability of the smoking equilibrium and Hopf bifurcation is analyzed by taking the time delay as the bifurcation parameter. In particular, global exponential stability is proved by using the linear matrix inequality method. The theoretical results obtained are validated numerically and graphically. [ABSTRACT FROM AUTHOR]

In this paper, we present a memory-based diffusion predator-prey model that incorporates spatial heterogeneity and is subject to homogeneous Dirichlet boundary conditions. Prey species lack memory or cognitive abilities, exhibiting only random diffusion. In contrast, predators utilize memory-based self-diffusion. For the proposed model, we establish the existence and explicit expression of a spatially non-constant positive steady-state. Furthermore, we demonstrate that memory-based diffusion and the averaged memory period can lead to richer dynamics. Specifically, when the memory-based diffusion coefficient is not dominant, the averaged memory period has no impact on the non-constant steady-state. However, when the memory-based diffusion coefficient takes precedence, the averaged memory period can destabilize the non-constant steady-state, resulting in Hopf bifurcation and the emergence of spatially non-homogeneous periodic solutions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the normal form a class of partial functional differential equations, which takes account of delays in the diffusion terms as well as a wider scope of nonlinear terms. We first study the associated linear theory, mainly including the spectral properties of infinitesimal generator, formal adjoint and decomposition of phase space. Based on these results, the normal form theory for nonlinear equation is established, which can be used to study the local dynamics near the steady state for such equations. As an application, we consider the Hopf bifurcation problem of a scalar diffusive equation with delay not only involved in the diffusive term but also in reaction terms. The normal form, depending on the original coefficients, up to the third order term is calculated, which allows us to determine the direction of Hopf bifurcation and stability of bifurcated periodic solutions. The results are then applied to study the scalar population model with memory-based diffusion for modeling the movement of highly-developed animals. [ABSTRACT FROM AUTHOR]

Motivated by the study of multiple time scale system dynamics, we analyze the Murali–Lakshmanan–Chua (MLC) dissipative circuit exhibiting periodic bursting with different types of oscillations in this paper. Such patterns can be analyzed by treating the slow variable as a parameter of the nonsmooth fast subsystem. The discontinuous bifurcation structure of the subsystem can be described by the generalized differential of Clarke and the Jacobian matrix of the system. Bursting patterns as well as the oscillation mechanisms can be clarified by the discontinuous bifurcations, time series and phase portraits. For the chosen circuit parameter values, this designed circuit admits both MLC type bursting attractors and Duffing–van der Pol circuit type bursting attractors. It is found that not only the features combined with the circuit parameters of the system, but also the two switching boundaries may have an important impact on the origin of the bursting behaviors. In particular, the explicit analytic solutions of the proposed circuit are investigated in detail. We also show that the bursting can arise from either discontinuous saddle-node bifurcation or slow passage through a discontinuous Hopf bifurcation. Finally, the validity of the theoretical analysis has been well verified by the numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, an susceptible–infected–susceptible (SIS) epidemic patch model with media delay is proposed at first. Then the basic reproduction number R0$\mathcal {R}_0$ is defined, and the threshold dynamics are studied. It is shown that the disease‐free equilibrium is globally asymptotically stable if R0<1$\mathcal {R}_0<1$ and the disease is uniformly persistent if R0>1$\mathcal {R}_0>1$. When the dispersal rates of susceptible and infected populations are identical and less than a critical value, it is proved that the limiting model has a unique positive equilibrium. Furthermore, the stability of the positive equilibrium and the existence of local and global Hopf bifurcations are obtained. Finally, some numerical simulations are performed. [ABSTRACT FROM AUTHOR]

In this paper, we rigorously analyze an HIV-1 infection model with CTL immune response and three time delays which represent the latent period, virus production period and immune response delay, respectively. We begin this model with proving the positivity and bound-edness of the solution. For this model, the basic reproduction number R0 and the immune reproduction number R1 are identified. Moreover, we have shown that the model has three e-quilibria, namely the infection-free equilibrium E0, the infectious equilibrium without immune response E1 and the infectious equilibrium with immune response E2. By applying uctuation lemma and Lyapunov functionals, we have demonstrated that the global stability of E0 and E1 are only related to R0 and R1. The local stability of the third equilibrium is obtained under four situations. Further, we give the conditions for the existence of Hopf bifurcation. Finally, some numerical simulations are carried out for illustrating the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a quadratic chaotic system modeling self-excited and hidden attractors which is described by a system of three nonlinear ordinary differential equations with three real parameters. The primary goal is to establish the existence of two limit cycles that bifurcate based on the system's nature as an electronic circuits model, specifically via Hopf bifurcation. Notably, the application of the first and second Lyapunov coefficients is utilized to demonstrate the bifurcation of two limit cycles from an equilibrium point near a Hopf critical point. Furthermore, employing the first-order averaging theory enables us to confirm the presence of unstable periodic orbits originating from the zero-Hopf equilibrium. [ABSTRACT FROM AUTHOR]

This paper presents the stability and bifurcation of the Hindmarsh–Rose neuronal model with time delay. First, we discuss the existence and stability of the equilibria, revealing that the system can have a maximum of three equilibria which exhibit always stable, always unstable, and stability switching dynamics, depending on the stability of the equilibria when there is no time delay. Furthermore, an explicit formula for the time delay at which Hopf bifurcation occurs is derived, and the direction and stability of the Hopf bifurcation are also given. Finally, numerical simulations are carried out to support our theoretical results. We find that due to the existence of time delay, our system can exhibit many interesting dynamical behaviors depending on the number of equilibria and the increase in time delay. For instance, with increasing time delay, bistability transitions to the coexistence of a periodic solution and a stable equilibrium, followed by the emergence of quasi-periodic motion alongside stable equilibria, and eventually leading to chaotic motion coexisting with a stable equilibrium. [ABSTRACT FROM AUTHOR]

A two-degree-of-freedom nonlinear high-speed railway wheelset model with two time delays in the lateral and yaw dampers is studied. The aim is to investigate the effect of time delays on stability and Hopf bifurcation characteristics of the wheelset model. The local stability of the trivial equilibrium under different time delay conditions is qualitatively analyzed. Analytical studies reveal that the wheelset model undergoes stability switches with the variation of the time delays. The stability switches correspond to Hopf bifurcations that occur when the time delays cross critical values. Furthermore, properties of Hopf bifurcation including direction and stability of bifurcating limit cycles are studied by using the normal form theory and the center manifold theorem. Our findings indicate that time delays in both lateral dampers and yaw dampers influence the stability and direction of Hopf bifurcation. Additionally, the numerical results show that time delays in the lateral and yaw dampers not only affect the amplitude of the hunting motion of the wheelset but also the periodic and chaotic motions. If the time delays gradually increase, the wheelset will vibrate irregularly with large lateral displacements. The analytical results presented in this paper offer a theoretical reference for the stability design of wheelsets. [ABSTRACT FROM AUTHOR]

In this paper, a diffusive zooplankton–phytoplankton model with time delay and hunting cooperation is established. First, the existence of all positive equilibria and their local stability are proved when the system does not include time delay and diffusion. Then, the existence of Hopf bifurcation at the positive equilibrium is proved by taking time delay as the bifurcation parameter, and the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are investigated by using the center manifold theorem and the normal form theory in partial differential equations. Next, according to the theory of Turing bifurcation, the conditions for the occurrence of Turing bifurcation are obtained by taking the intraspecific competition rate of the prey as the bifurcation parameter. Furthermore, the corresponding amplitude equations are discussed by using the standard multi-scale analysis method. Finally, some numerical simulations are given to verify the theoretical results. [ABSTRACT FROM AUTHOR]

The slow–fast Leslie–Gower model with piecewise-smooth Holling type-I functional response and weak Allee effect is studied in this paper. It is shown that the model undergoes singular Hopf bifurcation and nonsmooth Hopf bifurcation as the parameters vary. The theoretical analysis implies that the predator's food quality and Allee effect play an important role and lead to richer dynamical phenomena such as the globally stable equilibria, canard explosion phenomenon, a hyperbolically stable relaxation oscillation cycle enclosing almost two canard cycles with different stabilities and so on. Moreover, the predator and prey will coexist as multiple steady states or periodic oscillations for different positive initial populations and positive parameter values. Finally, we present some numerical simulations to illustrate the theoretical analysis such as the existence of one, two or three limit cycles. [ABSTRACT FROM AUTHOR]

In this paper, a predator–prey system with multiple anti-predator behaviors is developed and studied, where not only the prey may spread between patches but also the fear effect and counter-attack behavior of the prey are taken into account. First, the stability and existence of coexistence equilibria are presented. The unique positive equilibrium may be a saddle-node or a cusp of codimension 2. Then, various transversality conditions of bifurcations such as saddle-node bifurcation, transcritical bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation are obtained. Moreover, compared with a single strategy, the multiple anti-predator strategies are more beneficial to the persistence and the population density of prey. [ABSTRACT FROM AUTHOR]

In this paper, we designed a population model that shows how a prey species defends itself against a generalist predator by exhibiting group defence. A non-monotonic functional response is used to represent the group defence functionality. We have demonstrated the model's local stability in the vicinity of the coexisting equilibrium solution employing a local Lyapunov function. Condition for existence of Hopf bifurcation is obtained along with its normal form. The suggested model has been validated by numerical simulations, which have also been used to verify the acquired analytical results. The parameters are subjected to sensitivity analysis by utilizing partial rank correlation coefficient (PRCC) and Latin hypercube sampling (LHS). The Z-type dynamic method is used to prevent population blow-up. [ABSTRACT FROM AUTHOR]

Recently, many scholars have discovered that fractional calculus possess infinite memory and can better reflect the memory characteristics of neurons. Therefore, this paper studies the Hopf bifurcation of a fractional‐order network with short‐cut connections structure and self‐delay feedback. Firstly, we use the Laplace transform to obtain the characteristic equation of the model, which is the transcendental equation containing four times transcendental item. Secondly, by selecting the communication delay as the bifurcation parameter and the other delay as the constant in its stability interval, the conditions for the occurrence of Hopf bifurcation are established; the bifurcation diagrams are provided to ensure that the derived bifurcation findings are accurate. Thirdly, in the case of identical neurons, the crossing curves method is exploited to the fractional‐order functional function equation to extract the Hopf bifurcation curve. Finally, two numerical examples are employed to confirm the efficiency of the developed theoretical outcomes. [ABSTRACT FROM AUTHOR]

In this paper, a diffusive predator-prey model with nonlocal competition and prey refuge is considered. The influence of parameters on the existence, multiplicity and stability of nonhomogeneous steady-state solutions is studied. It is obtained that an unstable positive nonconstant steady state exists in the neighborhood of the positive constant steady state. Compared with the model without nonlocal competition, the model with nonlocal competition can generate Hopf-Hopf bifurcation under some conditions. Through the qualitative analysis, the normal form at the Hopf-Hopf bifurcation singularity is calculated to analyze the different dynamic properties exhibited by the model in different parameter regions. In order to illustrate the feasibility of the obtained results and the dependence of the dynamic behavior on the nonlocal competition, numerical simulations are carried out. Through the numerical simulations, it is further shown that under certain conditions, the nonlocal competition will lead to the stablly spatially non-homogeneous periodic solutions and stablly spatially non-homogeneous quasi-periodic solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study a delayed HIV infection model with nonmonotonic immune response and perform stability and bifurcation analysis. Our results show that the delayed HIV infection system with nonmonotonic immune response has bistability and stable periodic solution appear. We find that both the uninfected and immune-free equilibria are globally asymptotically stable under certain conditions which are not affected by time delay. However, the time delay makes one immune equilibrium always unstable for τ ≥ 0 and also makes another immune equilibrium appear stability switches; meanwhile, the system will exhibit local Hopf bifurcation, global Hopf bifurcation, and saddle-node-Hopf bifurcation. Numerical simulations are carried out to verify our results. [ABSTRACT FROM AUTHOR]