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This paper proposes a novel fractional-order delayed Ross-Macdonald model for malaria transmission. This paper aims to systematically investigate the effect of both the incubation periods of Plasmodium and the order on the dynamic behavior of diseases. Utilizing inequality techniques, contraction mapping theory, fractional linear stability theorem, and bifurcation theory, several sufficient conditions for the existence and uniqueness of solutions, the local stability of the positive equilibrium point, and the existence of fractional-order Hopf bifurcation are obtained under different time delays cases. The results show that time delay can change the stability of system. System becomes unstable and generates a Hopf bifurcation when the delay increases to a certain value. Besides, the value of order influences the stability interval size. Thus, incubation periods and the order have a major effect on the dynamic behavior of the model. The effectiveness of the theoretical results is shown through numerical simulations.

The ecological theory of species interactions rests largely on the competition, interference, and predator–prey models. In this paper, we propose and investigate a three-species predator–prey model to inspect the mutual interference between predators. We analyze boundedness and Kolmogorov conditions for the non-spatial model. The dynamical behavior of the system is analyzed by stability and Hopf bifurcation analysis. The Turing instability criteria for the Spatio-temporal system is estimated. In the numerical simulation, phase portrait with time evolution diagrams shows periodic and chaotic oscillations. Bifurcation diagrams show the very rich and complex dynamical behavior of the non-spatial model. We calculate the Lyapunov exponent to justify the dynamics of the non-spatial model. A variety of patterns like interference, spot, and stripe are observed with special emphasis on Beddington–DeAngelis function response. These complex patterns explore the beauty of the spatio-temporal model and it can be easily related to real-world biological systems.

Chronic wasting disease (CWD) is a contagious prion disease among the deer family that has the potential to disrupt the ecosystems where deer occur in abundance. To understand the dynamics of this emerging infectious disease, we consider a simple eco-epidemic model where the host population is infected by CWD. Boundedness of the system is established. The structure of equilibria and their linearized stability are investigated. The persistence condition is discussed. By constructing a suitable Lyapunov function, we discuss the global stability of the endemic equilibrium. Local bifurcation (transcritical) around the boundary equilibria is developed. Sufficient conditions for the existence of Hopf-bifurcation are derived. Further, we have also introduced white type of noise into the system to investigate stochastic stability. This suggests that the deterministic model is robust with respect to stochastic perturbation. Some numerical simulations are performed to validate our results.

A three-dimensional stage-structured predator–prey model is proposed and analyzed to study the effect of predation and cannibalism of the organisms at the highest trophic level with non-constant harvesting. Time lag in maturation of the predator is introduced in the system and conditions for local asymptotic stability of steady states are derived. The length of the delay preserving the stability is also estimated. Moreover, it is shown that the system undergoes a supercritical Hopf bifurcation when the maturation time lag crosses a certain critical value. Computer simulations have been carried out to illustrate various analytical results.

This paper shows that the dynamics of the Lucas (J Monet Econ, 22:3–42, 1988) endogenous growth model with flow externalities may give rise to a 2-torus, a compact three-dimensional manifold enclosed by a two-dimensional surface. The implications of this result are relevant for many fields of economic theory. It is first of all clear that if we choose to initialize the dynamics in the basin of attraction of this trapping region, a continuum of perfect foresight solutions may be observed. A simple econometric exercise, linking the physical-to-human capital ratio (state variable) to the 5-years forward variance of the growth rate of an unbalanced sample of 183 countries, seems to provide empirical backing for the phenomenon. Other important consequences, relevant from the point of view of endogenous cycles theory, are also scrutinized in the paper.

The phenomenon produced by the Hopf bifurcations is of notable importance. In fact, a Hopf bifurcation—guaranteeing the existence of an unsteady periodic solution of the linearized problem at stake—is also an optimum limit cycle candidate of the nonlinear associated problem and, if non linearly globally attractive, is an absorbing set and an effective limit cycle. The present paper deals with the onset of Hopf bifurcations in thermal magnetohydrodynamics (MHD). Precisely, it is devoted to characterization—via a simple formula—of the Hopf bifurcations threshold in horizontal plasma layers between rigid planes, heated from below and embedded in a constant transverse magnetic field. This problem, remarked clearly and notably by the Nobel Laureate Chandrasekhar (Nature 175:417–419, 1955), constitutes a difficulty met by him and—for plasma layers between rigid planes electricity perfectly conducting—is, as far as we know, still not removed. Let $$m_0$$ m 0 be the thermal conduction rest state and let $$P_r, P_m, R, Q$$ P r , P m , R , Q , be the Prandtl, the Prandtl magnetic, the Rayleigh and the Chandrasekhar number, respectively. Recognized (according to Chandrasekhar) that the instability of $$m_0$$ m 0 via Hopf bifurcation can occur only in a plasma with $$P_m>P_r$$ P m > P r , in this paper it is shown that the Hopf bifurcations occur if and only if $$\begin{aligned} Q>Q_c=\displaystyle \frac{4\pi ^2[1+P_r(\mu /2\pi )^4]}{P_m-P_r}, \end{aligned}$$ Q > Q c = 4 π 2 [ 1 + P r ( μ / 2 π ) 4 ] P m - P r , with $$ \mu =7.8532$$ μ = 7.8532 . Moreover, the critical value of R at which the Hopf bifurcation occurs is characterized via the smallest zero of the second invariant of the spectrum equation governing the most destabilizing perturbation. The critical value of Q, in the free-rigid and rigid-free cases is shown to be $$\displaystyle \frac{1}{4}$$ 1 4 of the previous value.

Within this paper, we are focusing on the time delay effects on the improved positive position feedback (PPF) controller for the oscillations of a 2DOF nonlinear dynamical system which represents a thin-walled pre-twisted rotating blade. Herein, we are improving the performance of the traditional positive position feedback controller (PPF) that suffers from the existence of two high amplitude peaks on both sides of its minimum amplitude point. The improvement involves coupling double nonlinear saturation controllers to the main system to notch down the high peaks. As an active control process, the time delay is inherent in the process which produces a locus of Hopf bifurcation points separating between the stable and unstable regions of operation. Variation of different parameters is studied to relate their effects with time delay on the system operation. To ensure the validity of our proposed work, a comparison between the approached multiple scales analytical solutions and the computed Rung-Kutta numerical solutions is done at the end of this paper.

This paper deals with a general SEIR model for the coronavirus disease 2019 (COVID-19) with the effect of time delay proposed. We get the stability theorems for the disease-free equilibrium and provide adequate situations of the COVID-19 transmission dynamics equilibrium of present and absent cases. A Hopf bifurcation parameter τ concerns the effects of time delay and we demonstrate that the locally asymptotic stability holds for the present equilibrium. The reproduction number is brief in less than or greater than one, and it effectively is controlling the COVID-19 infection outbreak and subsequently reveals insight into understanding the patterns of the flare-up. We have included eight parameters and the least square method allows us to estimate the initial values for the Indian COVID-19 pandemic from real-life data. It is one of India’s current pandemic models that have been studied for the time being. This Covid19 SEIR model can apply with or without delay to all country’s current pandemic region, after estimating parameter values from their data. The sensitivity of seven parameters has also been explored. The paper also examines the impact of immune response time delay and the importance of determining essential parameters such as the transmission rate using sensitivity indices analysis. The numerical experiment is calculated to illustrate the theoretical results.

In this paper, we investigate the degenerate Hopf bifurcation of a Leslie–Gower predator–prey system with predator harvest. The known work discussed the Hopf bifurcation of this system when the first Lyapunov number does not vanish and gave an example of a stable weak focus with order 2. However, the thorough discussion of center-type equilibrium for all possible parameters has not been completed. In this paper, by computing the first two focal values, we decompose the variety with resultant and prove that the center-type equilibrium is a weak focus with order at most 2 for all the possible parameter values. Moreover, numerical simulations are employed to show the appearance of two limit cycles from degenerate Hopf bifurcation. Our results finish the study of Hopf bifurcation in this system.

In the present paper, we have studied a model for three species system with time lag considering two competing species and a predator species which is partially coupled with an alternative prey. It is also considered in the model that one of the competing population (commensal species) is supported by the predator species (host species). All the equilibrium points of the model are derived which are studied using stability and bifurcation theory. It is shown that the equilibrium levels of all the species increase or decrease with respect to alternative food source constant $$A$$ . It is also shown in the paper that introduction of the delay in the system exhibits Hopf bifurcation. Numerical Simulation of the model is carried out using hypothetical parametric values to support the analytical results. From the numerical simulation, it is shown that the length of delay is inversely proportional to the parameter $$A$$ which measures the level of interaction between prey and predator.

This paper is interested in gaining insights of approximate expressions of the bifurcating periodic solutions in a neuron model. This model shares the property of involving delay-dependent parameters. The presence of such dependence requires the use of suitable criteria which usually makes the analytical work so harder. Most existing methods for studying the nonlinear dynamics fail when applied to such a class of delay models. Although Xu et al. (Phys Lett A 354:126–136, 2006) studied stability switches, Hopf bifurcation and chaos of the neuron model with delay-dependent parameters, the dynamics of this model are still largely undetermined. In this paper, a detailed analysis on approximation to the bifurcating periodic solutions is given by means of the perturbation approach. Moreover, some examples are provided for comparing approximations with numerical solutions of the bifurcating periodic solutions. It shows that the dynamics of the neuron model with delay-dependent parameters is quite different from that of systems with delay-independent parameters only.

In this paper, we study the Turing–Hopf bifurcation in the predator–prey model with cross-diffusion considering the individual behaviour and herd behaviour transition of prey population subject to homogeneous Neumann boundary condition. Firstly, we study the non-negativity and boundedness of solutions corresponding to the temporal model, spatiotemporal model and the existence and priori boundedness of solutions corresponding to the spatiotemporal model without cross-diffusion. Then by analysing the eigenvalues of characteristic equation associated with the linearized system at the positive constant equilibrium point, we investigate the stability and instability of the corresponding spatiotemporal model. Moreover, by calculating and analysing the normal form on the centre manifold associated with the Turing–Hopf bifurcation, we investigate the dynamical classification near the Turing–Hopf bifurcation point in detail. At last, some numerical simulations results are given to support our analytic results.

In this paper, we study the global dynamics of a nonsmooth Rayleigh–Duffing equation $$\ddot{x}+a{\dot{x}}+b{\dot{x}}|{\dot{x}}|+cx+{\mathrm {d}}x^3=0$$ for the case $$d>0$$ , i.e., the focus case. The global dynamics of this nonsmooth Rayleigh–Duffing oscillator for the case $$d

In this paper, we show the existence of a Hopf bifurcation in a delayed single population model with patch structure. The effect of the dispersal rate on the Hopf bifurcation is considered. Especially, if each patch is favorable for the species, we show that when the dispersal rate tends to zero, the limit of the Hopf bifurcation value is the minimum of the “local” Hopf bifurcation values over all patches. On the other hand, when the dispersal rate tends to infinity, the Hopf bifurcation value tends to that of the “average” model.

This paper investigates the effect of model uncertainty on the nonlinear dynamics of a generic aeroelastic system. Among the most dangerous phenomena to which these systems are prone, Limit Cycle Oscillations are periodic isolated responses triggered by the nonlinear interactions among elastic deformations, inertial forces, and aerodynamic actions. In a dynamical systems setting, these responses typically emanate from Hopf bifurcation points, and thus a recently proposed framework, which address the problem of robustness from a nonlinear dynamics viewpoint, is employed. Briefly, the notion of robust bifurcation margin extends the concept of μ analysis technique from the robust control theory. The main contribution of this article is a systematic investigation of the numerous scenarios arising in the study of nonlinear flutter when uncertainties in the model are accounted for in the analyses. The advantages of adopting this framework include the possibility to: quantify relevant information for the determination of the nonlinear stability envelope; gain a more in-depth understanding of the physical mechanisms triggering subcritical and supercritical Hopf bifurcations; and reveal properties of the nominal system by identifying isolated branches not straightforward to detect with conventional numerical approaches., Nonlinear Dynamics, 103 (2), ISSN:0924-090X, ISSN:1573-269X

This paper proposes a coupled-circuit system composed of two Chua’s circuits with lossless transmission lines. By applying the Kirchhoff’s voltage and current laws, the equations that describe the coupled-circuit system are reduced to two coupled neutral-type differential equations with a time delay. Subsequently, the conditions for global stability are established using the inequality technology, and those for local stability and Hopf bifurcation are obtained by selecting the length of the transmission line as the bifurcation parameter. By using the normal-form theory and central manifold theorem, the formulas for the Hopf bifurcation direction and bifurcation periodic solution are obtained. Finally, the numerical simulations not only verify the theoretical analysis but also show that chaos exists near the Hopf bifurcation point.

Many service systems use technology to notify customers about their expected waiting times or queue lengths via delay announcements. However, in many cases, either the information might be delayed or customers might require time to travel to the queue of their choice, thus causing a lag in information. In this paper, we construct a neutral delay differential equation (NDDE) model for the queue length process and explore the use of velocity information in our delay announcement. Our results illustrate that using velocity information can have either a beneficial or detrimental impact on the system. Thus, it is important to understand how much velocity information a manager should use. In some parameter settings, we show that velocity information can eliminate oscillations created by delays in information. We derive a fixed point equation for determining the optimal amount of velocity information that should be used and find closed form upper and lower bounds on its value. When the oscillations cannot be eliminated altogether, we identify the amount of velocity information that minimizes the amplitude of the oscillations. However, we also find that using too much velocity information can create oscillations in the queue lengths that would otherwise be stable.

This paper is focused on local and global stability of a fractional-order predator–prey model with habitat complexity constructed in the Caputo sense and the corresponding discrete fractional-order system. Mathematical results like positivity and boundedness of the solutions of fractional-order predator–prey model is presented. Conditions for local and global stability of different equilibrium points are proved. It is shown that there may exist fractional-order-dependent instability through Hopf bifurcation. We have determined an extra stability region in the lower range of habitat complexity where all populations coexist in stable state for some fractional-order values but unstable for integer-order value. Dynamics of the discrete fractional-order model is shown to be more complex and depends on both the step-size and fractional-order. It shows Hopf bifurcation, flip bifurcation and chaos with respect to the step-size. Several examples are presented to substantiate the analytical results.

In this paper, the dynamical behaviors for multiple delayed latent virus model with virus-to-cell infection and cell-to-cell transmissions and humoral immunity are investigated. The virus-to-cell and cell-to-cell incidence rates are modeled by general nonlinear functions. The basic reproduction number $R_{0}$ R 0 and the humoral immune response number $R_{1}$ R 1 are calculated and proved to be threshold conditions determining the local and global properties of the virus model. The existence of Hopf bifurcation with immune delay as a bifurcation parameter is presented, and the effects of some key parameters on viral dynamics are revealed by numerical simulations.

In this paper, we develop an SEIS model with Holling type II function representing the innate immunity as well as the saturated treatment. We obtain the existence and stability criteria for the equilibrium points. We observe that when the reproduction number is less than unity, the disease-free equilibrium always exists and is locally asymptotically stable. The multiple endemic equilibrium points can exist independent of the basic reproduction number, and the system may experience bistability. We find that the system can encounter backward or forward bifurcation at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_0=1$$\end{document}R0=1, where the contact rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =\beta _0$$\end{document}β=β0 is the bifurcation parameter. Therefore, the disease-free equilibrium may not be globally stable. We deduce the criteria for the presence of Hopf bifurcation where the parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =\gamma ^*$$\end{document}γ=γ∗ acts as the bifurcation parameter and the system is a neutrally stable center. We also observe with the aid of a numerical example that a slight perturbation disrupts the neutral stability and the trajectories become either converging or diverging from the equilibrium point. Numerical simulation is performed with the help of MATLAB to justify the findings. We study the effect of nonlinearity of immunity function and the treatment rate on the dynamics of the disease spread. We find that when both are linear, the reproduction number is the same, but the system has a unique endemic equilibrium point that exists for reproduction number greater than unity. We find that there is neither backward bifurcation nor Hopf bifurcation. We also observe that the saturation in treatment enlarges the domain of backward bifurcation making disease eradication an extremely difficult task. The endemic equilibria in the case of saturated treatment may exist far more to the left of the bifurcation parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta = \beta _0$$\end{document}β=β0. Hence, the nonlinearity of immunity function and treatment function affects the dynamics of an SEIS model highly; therefore, one must be precautious to choose an appropriate function for both while modeling.

The main purpose of this paper is to investigate the complicated dynamical behaviours as well as the mechanism of a Filippov-type system. By introducing a non-smooth term and a periodic external excitation to a four-dimensional laser system, a new Filippov-type system with two scales can be obtained. When an order gap exists between the exciting frequency and the natural one, the whole excitation term can be considered as a slow-varying parameter. For smooth subsystems, the equilibrium branches along with bifurcations, such as fold bifurcation (FB) and Hopf bifurcation (HB) are derived. Boundary equilibrium bifurcation and non-smooth bifurcations, according to differential inclusion theory, are also explored by introducing an auxiliary parameter to the system. Several types of sliding movements, such as grazing-sliding and multisliding, can be observed with slow variation. In addition, due to the coexistence of stable attractors in the vector field, the symmetry breaking phenomenon of bursting attractors appears, the mechanism of which can be revealed by overlapping the transformed phase portrait and bifurcation curves.

In this paper, it is shown that when a flow undergoes a steady bifurcation breaking one reflection symmetry, the mean flow obtained by averaging the two possible asymmetric flow fields resulting from the instability remains marginally stable in the postcritical regime. This property is demonstrated rigorously through an asymptotic analysis which closely follows that proposed in Sipp and Lebedev (J Fluid Mech 792:620–657, 2007) for a Hopf bifurcation with focus on wakes. In the case of wakes, the marginal stability of the mean flow is well known and had several consequences documented in the literature. To the authors’ knowledge, the marginal stability of mean flows after a symmetry-breaking pitchfork bifurcation is demonstrated here for the first time. As an example of possible consequences of marginal stability, the self-consistent model proposed for wakes in Mantic-Lugo et al. (Phys Rev Lett 113:084501, 2014) and relying on marginal stability is also applied here to the symmetry-breaking instability of the flow in a channel with a sudden expansion. For this specific case, the marginal stability of the mean flow is first demonstrated by dedicated direct numerical simulations; successively, it is shown that the resulting self-consistent model predicts the nonlinear saturation of the instability with remarkable accuracy.

The ecological invasion problem in which a weaker exotic species invades an ecosystem inhabited by two strongly competing native species is modelled by a three-species competition-diffusion system. It is known that for a certain range of parameter values competitor-mediated coexistence occurs and complex spatio-temporal patterns are observed in two spatial dimensions. In this paper we uncover the mechanism which generates such patterns. Under some assumptions on the parameters the three-species competition-diffusion system admits two planarly stable travelling waves. Their interaction in one spatial dimension may result in either reflection or merging into a single homoclinic wave, depending on the strength of the invading species. This transition can be understood by studying the bifurcation structure of the homoclinic wave. In particular, a time-periodic homoclinic wave (breathing wave) is born from a Hopf bifurcation and its unstable branch acts as a separator between the reflection and merging regimes. The same transition occurs in two spatial dimensions: the stable regular spiral associated to the homoclinic wave destabilizes, giving rise first to an oscillating breathing spiral and then breaking up producing a dynamic pattern characterized by many spiral cores. We find that these complex patterns are generated by the interaction of two planarly stable travelling waves, in contrast with many other well known cases of pattern formation where planar instability plays a central role.

Reaction delays play an important role in determining the qualitative dynamical properties of a platoon of vehicles traversing a straight road. In this paper, we investigate the impact of delayed feedback on the dynamics of the Classical Car-Following Model (CCFM). Specifically, we analyze the CCFM in no delay, small delay and arbitrary delay regimes. First, we derive a sufficient condition for local stability of the CCFM in no-delay and small-delay regimes using. Next, we derive the necessary and sufficient condition for local stability of the CCFM for an arbitrary delay. We then demonstrate that the transition of traffic flow from the locally stable to the unstable regime occurs via a Hopf bifurcation, thus resulting in limit cycles in system dynamics. Physically, these limit cycles manifest as back-propagating congestion waves on highways. In the context of human-driven vehicles, our work provides phenomenological insight into the impact of reaction delays on the emergence and evolution of traffic congestion. In the context of self-driven vehicles, our work has the potential to provide design guidelines for control algorithms running in self-driven cars to avoid undesirable phenomena. Specifically, designing control algorithms that avoid jerky vehicular movements is essential. Hence, we derive the necessary and sufficient condition for non-oscillatory convergence of the CCFM. Next, we characterize the rate of convergence of the CCFM, and bring forth the interplay between local stability, non-oscillatory convergence and the rate of convergence of the CCFM. Further, to better understand the oscillations in the system dynamics, we characterize the type of the Hopf bifurcation and the asymptotic orbital stability of the limit cycles using Poincare normal forms and the center manifold theory. The analysis is complemented with stability charts, bifurcation diagrams and MATLAB simulations., Extended the existing results to the general model (the CCFM)

Double Hopf bifurcation analysis can be used to reveal some complicated dynamical behavior in a dynamical system, such as the existence or coexistence of periodic orbits, quasi-periodic orbits, or even chaos. In this paper, an algorithm for deriving the normal form near a codimension-two double Hopf bifurcation of a reaction–diffusion system with time delay and Neumann boundary condition is rigorously established, by employing the center manifold reduction technique and the normal form method. The dynamical behavior near bifurcation points are proved to be governed by twelve distinct unfolding systems. Two examples are performed to illustrate our results: for a stage-structured epidemic model, we find that double Hopf bifurcation appears when varying the diffusion rate and time delay, and two stable spatially inhomogeneous periodic oscillations are proved to coexist near the bifurcation point; in a diffusive Predator–Prey system, we theoretically proved that quasi-periodic orbits exist on two- or three-torus near a double Hopf bifurcation point, which will break down after slight perturbation, leaving the system a strange attractor.

In this paper, we report the existence of bursting oscillations in systems governed by a second-order differential equation with only one signum nonlinearity term and time-delay feedback, driven by the slowly varying external force. Depending on the sign of the strength of signum nonlinearity, two cases are studied: two-well and single-well potentials. The external force acts as the control parameter, the stability of equilibrium points is first discussed and the condition for Hopf bifurcation is established. Secondly, we present the effect of time delay on bursting oscillations when periodic forcing changes slowly. Our results show that the time delay is responsible for the appearance or disappearance of the bursting phenomenon. It is also found that the amplitude, the number of peaks and period of bursting oscillation depend on the value of the time delay. The bursting shapes obtained theoretically are exhibited experimentally using the real microcontroller simulation.

In this paper, the dynamics of a general differential equation with neutral type are investigated. Under certain assumptions, the stability of positive equilibrium and the existence of Hopf bifurcation are obtained by analyzing the distribution of eigenvalues. And global existence of positive periodic solutions is established by using the global Hopf bifurcation result of Krawcewicz et al. Finally, by taking neutral Nicholson’s blowflies model and neutral Mackey–Glass model as two examples, some numerical simulations are carried out to illustrate the analytical results.

A singularly perturbed system for doubly diffusive convection equations, called the artificial compressible system, is considered on a two-dimensional infinite layer for a parameters range where the Hopf bifurcation occurs in the corresponding incompressible system. The spectrum of the linearized operator in a time periodic function space is investigated in detail near the bifurcation point when the singular perturbation parameter is small. The results of this paper are the basis of the study of the nonlinear Hopf bifurcation problem and the singular limit of the time periodic bifurcating solutions.

Due to the fluctuation of membrane potential of neuron, there are complex time-varying electromagnetic fields in nervous systems, and the exciting electromagnetic field will further regulate the discharge activities of neurons. In this paper, the coupling of magnetic flux variables to the membrane potential is realised by using a magnetron memristor, and then a 5D extended Hindmarsh–Rose (e-HR) neuron model is established. With the help of Matcont software, the distribution and bifurcation properties of the equilibrium point in the e-HR model is analysed. It is found that there are subcritical Hopf bifurcation, coexisting oscillation modes and hidden limit cycle attractors with period 1 and period 2. In addition, by applying the washout controller, the subcritical Hopf bifurcation point can be transformed into the supercritical Hopf bifurcation point. Thus, the hidden oscillation behaviour of the model can be effectively eliminated. In order to analyse the influence of various parameters on the bifurcation behaviour, numerical simulation of two-parameter bifurcation, single-parameter bifurcation, maximum Lyapunov exponential and time response are given. It is found that the e-HR neuron has a complex bifurcation structure, i.e., the bifurcation structure with period-doubling bifurcations, inverse period-doubling bifurcations, period-adding bifurcations with and without chaos. At the same time, the study also finds that the coexistence behaviour of the periodic cluster discharge and the mixed-mode oscillations (MMOs) can be observed from the bifurcation structure with unique ‘periodic dislocation layer’ on the two-parameter plane. Interestingly, the bursting mode of the system is converted into MMOs when the system parameters are randomly perturbed. The results of this study provide useful insights into the complex discharge patterns and hidden discharge behaviours of neurons under electromagnetic induction.

A prey–predator model with constant-effort harvesting on the prey and predators is investigated in this paper. First, we discuss the number and type of the equilibria by analyzing the equations of equilibria and the distribution of eigenvalues. Second, with the rescaled harvesting efforts as bifurcation parameters, a subcritical Hopf bifurcation is exhibited near the multiple focus and a Bogdanov–Takens bifurcation is also displayed near the BT singularity by analyzing the versal unfolding of the model. With the variation of bifurcation parameters, the system shows multi-stable structure, and the attractive domains for different attractors are constituted by the stable and unstable manifolds of saddles and the limit cycles bifurcated from Hopf and Bogdanov–Takens bifurcations. Finally, a cusp point and two generalized Hopf points are found on the saddle-node bifurcation curve and the Hopf bifurcation curves, respectively. Several phase diagrams for parameters near one of the generalized Hopf points are exhibited through the generalized Hopf bifurcation.

In this paper, the Hopf bifurcation and Turing instability for a mussel–algae model are investigated. Through analysis of the corresponding kinetic system, the existence and stability conditions of the equilibrium and the type of Hopf bifurcation are studied. Via the center manifold and Hopf bifurcation theorem, sufficient conditions for Turing instability in equilibrium and limit cycles are obtained, respectively. In addition, we find that the strip patterns are mainly induced by Turing instability in equilibrium and spot patterns are mainly induced by Turing instability in limit cycles by numerical simulations. These provide a comprehension on the complex pattern formation of a mussel–algae system.

Dynamical analysis of a delayed tri-trophic food chain consisting of prey, an intermediate, and a top predator is investigated in this paper. The additive Allee effect is introduced in the prey population, and it is assumed that there is a time lag due to the gestation effect in the intermediate predator. The interference among the prey and the intermediate predator is according to Holling type II, while the interaction between the intermediate and top predators follows the Crowley–Martin functional response. The local stability and bifurcation analysis of the proposed model at the interior equilibrium point are studied. Numerical simulations are provided to ensure the mathematical results.

In this paper, we consider a diffusive predator–prey model with a time delay and prey toxicity. The effect of time delay on the stability of the positive equilibrium is studied by analyzing the eigenvalue spectrum. Delay-induced Hopf bifurcation is also investigated. By utilizing the normal form method and center manifold reduction for partial functional differential equations, the formulas for determining the property of Hopf bifurcation are given.

In this paper, an attempt has been made to understand the parametric excitation of a periodic orbit of nonlinear oscillator which can be a limit cycle, center or a slowly decaying center-type oscillation. For this a delay model is considered with nonlinear feedback oscillator defined in terms of Li\'enard oscillator description which can give rise to any one of the periodic orbits stated above. We have characterized the resonance and antiresonance behaviour for arbitrary nonlinear system from their stability and bifurcation analyses in reference to the standard delayed van der Pol system. An approximate analytical solution using Krylov--Bogoliubov (K-B) averaging method is utilised to recognize the sub-harmonic resonance and antiresonance, and average energy consumption per cycle. Direction of Hopf bifurcation and stability of the periodic solution bifurcating from the trivial fixed point are carried out using normal form and center manifold theory. The parametric excitation is also thoroughly investigated via bifurcation analysis to find the role of the control parameters like time delay, damping and nonlinear terms., Comment: Accepted (International Journal of Applied and Computational Mathematics)

In this paper, we propose a new fractional-order financial model which is a generalized version of the financial model reported in the previous publications. By applying a suitable time-delayed feedback controller, we have control for the chaotic behavior of the fractional-order financial model. We investigate the stability and the existence of a Hopf bifurcation of the fractional-order financial model. A new sufficient condition that guarantees the stability and the existence of a Hopf bifurcation for a fractional-order delayed financial model is presented by regarding the delay as bifurcation parameter. The investigation shows that the delay and the fractional order have an important effect on the stability and Hopf bifurcation of involved model. Some simulations justifying the validity of the derived analytical results are given. The obtained results of this article are innovative and are of great significance in handling the financial issues.

In this paper, the dynamical behavior of a SEIR epidemic system that takes into account governmental action and individual reaction is investigated. The transmission rate takes into account the impact of governmental action modeled as a step function while the decreasing contacts among individuals responding to the severity of the pandemic is modeled as a decreasing exponential function. We show that the proposed model is capable of predicting Hopf bifurcation points for a wide range of physically realistic parameters for the COVID-19 disease. In this regard, the model predicts periodic behavior that emanates from one Hopf point. The model also predicts stable oscillations connecting two Hopf points. The effect of the different model parameters on the existence of such periodic behavior is numerically investigated. Useful diagrams are constructed that delineate the range of periodic behavior predicted by the model.

Alcoholism is a social phenomenon that affects all social classes and is a chronic disorder that causes the person to drink uncontrollably, which can bring a series of social problems. With this motivation, a delayed drinking model including five subclasses is proposed in this paper. By employing the method of characteristic eigenvalue and taking the temporary immunity delay for alcoholics under treatment as a bifurcation parameter, a threshold value of the time delay for the local stability of drinking-present equilibrium and the existence of Hopf bifurcation are found. Then the length of delay has been estimated to preserve stability using the Nyquist criterion. Moreover, optimal strategies to lower down the number of drinkers are proposed. Numerical simulations are presented to examine the correctness of the obtained results and the effects of some parameters on dynamics of the drinking model.

This paper’s goal is to delve into the fractional modeling and bifurcation control for a predator-prey model with prey dispersal and gestation delay. First, the bifurcation criteria for the uncontrolled system are obtained by viewing gestation delay as a bifurcation parameter. It is revealed that gestation delay can induce periodic oscillations. Then, an extended feedback controller is deeply conceived to suppress Hopf bifurcation for the underlying system. The results reflect that the stability behaviors of the uncontrolled system are saliently enhanced by adjusting feedback gain and feedback delay if other coefficients are fixed. To protrude the correctness and excellent feature of our works, two simulation examples are eventually carried out.

In this paper, a novel four-dimensional continuous chaotic system is constructed from the simplified Lorenz-like system. The novel system has three equilibria, strange attractors, coexisting attractors, and performs Hopf bifurcation with the variation of system parameters. The coexisting attractors, which are the most remarkable dynamic features of the system, are numerically studied. The coexisting attractors show that the system coexists as a pair of point, periodic, and chaotic attractors. Some basic dynamic behaviours are studied as well. The synchronisation control problem of the system is analysed. The theoretical and numerical analyses demonstrate that the system can easily achieve synchronisation by using the passive control technique.

In this paper, mixed-mode oscillations and bifurcation mechanism for a Filippov-type system including two time-scales in the frequency domain are demonstrated. According to classic Chua’s system, we investigate a non-smooth dynamical system including two time-scales. As there exists an order gap between the exciting frequency and the natural one, the whole external excitation term can be considered as a slow-changing parameter, which results in two smooth subsystems divided by the non-smooth boundary. In addition, the critical condition about fold bifurcation (FB) is studied, and by applying the Hopf bifurcation (HB) theorem, specific formulas for determining the existence of HBs are presented. By introducing an auxiliary parameter via differential inclusions theory, the non-smooth bifurcations on the boundary are discussed. Then, the equilibrium branches and the bifurcations are derived, and two typical cases associated with different bifurcations are considered. In light of the superposition between the bifurcation curve and the transformed phase portrait, the dynamical behaviours of the mixed-mode oscillations as well as sliding movement along the non-smooth boundary are obtained, which reveal the corresponding dynamical mechanism.

Ecological stoichiometry is the study of the balance of multiple elements in ecological interactions and processes (Sterner and Elser in Ecological Stoichiometry: The Biology of Elements from Molecules to the Biosphere, 2002). Modeling under this framework enables us to investigate the effect nutrient content on organisms whether the imbalance involves insufficient or excess nutrient content. This phenomenon is called the “stoichiometric knife-edge”. In this paper, a discrete-time predator–prey model that captures this phenomenon is established and qualitatively analyzed. We systematically expound the similarities and differences between our discrete model and the corresponding continuous analog. Theoretical and numerical analyses show that while the discrete and continuous models share many properties, differences also exist. Under certain parameter sets, the models exhibit qualitatively different dynamics. While the continuous model shows limit cycle, Hopf bifurcation, and saddle-node bifurcation, the discrete-time model exhibits richer dynamical behaviors, such as chaos. By comparing the dynamics of the continuous and discrete model, we can conclude that stoichiometric effects of low food quality on predators are robust to the discretization of time. This study can possibly serve as an example for pointing to the importance of time scale in ecological modeling.

In this paper, we are concerned with a delayed smoking model in which the population is divided into five classes. Sufficient conditions guaranteeing the local stability and existence of Hopf bifurcation for the model are established by taking the time delay as a bifurcation parameter and employing the Routh–Hurwitz criteria. Furthermore, direction and stability of the Hopf bifurcation are investigated by applying the center manifold theorem and normal form theory. Finally, computer simulations are implemented to support the analytic results and to analyze the effects of some parameters on the dynamical behavior of the model.

This paper shows the effects of microgrid (MG) integration, location, penetration and load levels on the power systems oscillating stability. The analysis work was carried out in the IEEE 14 bus test system which is widely used in stability studies. Stability studies were carried out with the help of eigenvalue analysis over linearized system models. HOPF bifurcation point detection was performed to show the effect of MGs on the system loadability margin. In the study results, it is seen that MGs affect system stability positively by increasing system loadability margin and has a damper effect on the critical modes of the system and the electromechanical local modes, but they reduce the damping amount of the electromechanical interarea modes.

In this paper, a mathematical model consisting of three populations with discrete time delays is considered. By analyzing the corresponding characteristic equations, the local stability of each of the feasible equilibria of the system is addressed and the existence of Hopf bifurcations at the coexistence equilibrium is established. The direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are analyzed using the theory of normal form and center manifold. Discussion with some numerical simulation examples is given to support the theoretical results.

The problem of determining the bounds of applicability of perturbation expansions in terms both of the system parameters and the state-space variable amplitude is a key point in the perturbation analysis of nonlinear systems. In the present paper an analysis in a finite neighborhood of a Hopf bifurcation is presented in order to analyze the conditions under which a Normal Form zero-divisors-based approach fails to describe the local dynamics and, therefore, a small divisor approach is required. The condition of "smallness" referred to the divisors is analyzed from both a qualitative and a quantitative point of view. Finally, a simple but effective analytical and numerical example is introduced to illustrate the theoretical issues along with an interpretation within a codimension-two framework.

In this paper, the authors consider a four-compartmental HIV epidemiological model, which describes the interaction between HIV virus and two target cells, CD4 T cells and macrophages in vivo. It is proved that the bilinear incidence can cause the backward bifurcation, where a locally asymptotically stable disease-free equilibrium co-exists with a locally asymptotically stable endemic equilibrium when the basic reproduction number $$(R_{0})$$ is less than unity. It is shown that a sequence of Hopf bifurcations occur at the endemic equilibrium by choosing one parameter of the model as the bifurcation parameter. Meanwhile, the global asymptotic stabilities of the equilibria are established by constructing suitable Lyapunov functions under some conditions. Furthermore, the authors develop an extended model by incorporating with the intracellular delays and derive global asymptotic stability of the delayed model by constructing Lyapunov functions. Some numerical simulations for justifying the theoretical analysis results are also given.

The delayed competitive web-site system with incommensurate fractional orders, based on the Lotka–Volterra competition model, is firstly proposed in this paper. It is demonstrated that there is a stability switch for time delay, Hopf bifurcation occurs when time delay crosses through a critical value and each order has important influence on the creation of bifurcation. Furthermore, a nonlinear delayed feedback control is successfully designed to postpone the onset of Hopf bifurcation, extend the stability domain, and then the system possesses the stability in a larger parameter range. Finally, numerical simulations are included to illustrate the efficiency of the obtained theoretical results.

In this paper, a chemostat model with general monotone response functions and two discrete time delays is proposed to describe the dynamical behavior about predator–prey system. First, by analyzing the characteristic equation associated with the model, we obtain the conditions of the existence and stability of extinction equilibrium and positive equilibrium. Choosing delays as bifurcation parameters, the existence of Hopf bifurcations is investigated in detail. Second, by virtue of the Poincare normal form method and center manifold theorem, explicit formulas are derived to determine the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions. Finally, some numerical simulations are carried out to illustrate the theoretical results and the biological significance.

In this paper, dynamics of a stage-structured epidemic model with delays and nonlinear incidence rate is analyzed. Local stability and existence of Hopf bifurcation is discussed by choosing possible combination of the delays as the bifurcation parameter. It is proved that the unique endemic equilibrium is locally asymptotically stable when the delay is suitably small and a bifurcating periodic solution will be caused once the delay passes through the corresponding critical value of the delay. We make use of the normal form theory and center manifold theorem to obtain the explicit formulas for determining the properties of the Hopf bifurcation. Numerical simulations supporting our obtained findings are carried out in the end.

Experimental investigations have shown that the pre-Bötzinger complex (pre-BötC) within the mammalian brainstem generates the inspiratory phase of respiratory rhythm. Based on a single-compartment model of a pre-BötC inspiratory neuron, we, in this paper, use semi-analytical, numerical as well as fast-slow dynamical methods to investigate the effects of sodium conductance (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{\text{Na}}$$\end{document}gNa) and potassium conductance (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{{\text{K}}}$$\end{document}gK) on the firing activities of pre-BötC and try to reveal the dynamical mechanisms behind them. We show how \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{{\text{Na}}}$$\end{document}gNa and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{\text{K}}$$\end{document}gK affect the bifurcations of the fast-subsystem and how the the firing patterns of pre-BötC transit according to the bifurcations.