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We analyze the replicator-mutator equations for the rock-paper-scissors game. Various graph-theoretic patterns of mutation are considered, ranging from a single unidirectional mutation pathway between two of the species, to global bidirectional mutation among all the species. Our main result is that the coexistence state, in which all three species exist in equilibrium, can be destabilized by arbitrarily small mutation rates. After it loses stability, the coexistence state gives birth to a stable limit cycle solution created in a supercritical Hopf bifurcation. This attracting periodic solution exists for all the mutation patterns considered, and persists arbitrarily close to the limit of zero mutation rate and a zero-sum game. [ABSTRACT FROM AUTHOR]

In this paper we construct two limit cycles with a heteroclinic polycycle for the three-dimensional continuous model of the rock–scissors–paper (RSP) game between bacteriocin producing bacteria. Our construction gives a partial answer to an open question posed by Neumann and Schuster (2007) concerning how many limit cycles can coexist for the RSP game. [ABSTRACT FROM AUTHOR]

In this paper, we consider the influence of a nonlinear contact rate caused by multiple contacts in classical SIR model. In this paper, we unversal unfolding a nilpotent cusp singularity in such systems through normal form theory, we reveal that the system undergoes a Bogdanov-Takens bifurcation with codimension 2. During the bifurcation process, numerous lower codimension bifurcations may emerge simultaneously, such as saddle-node and Hopf bifurcations with codimension 1. Finally, employing the Matcont and Phase Plane software, we construct bifurcation diagrams and topological phase portraits. Additionally, we emphasize the role of symmetry in our analysis. By considering the inherent symmetries in the system, we provide a more comprehensive understanding of the dynamical behavior. Our findings suggest that if this occurrence rate is applied to the SIR model, it would yield different dynamical phenomena compared to those obtained by reducing a 3-dimensional dynamical model to a planar system by neglecting the disease mortality rate, which results in a stable nilpotent cusp singularity with codimension 2. We found that in SIR models with the same occurrence rate, both stable and unstable Bogdanov-Takens bifurcations occur, meaning both stable and unstable limit cycles appear in this system. [ABSTRACT FROM AUTHOR]

In this paper, to investigate the dynamic interplay between supply and demand, with a focus on collectability, a novel mathematical model was introduced via conformable operator. This model considers the possibility that operating expenses or a lack of raw materials causes a manufacturing delay than the supply of goods instantly matching demand. This maturation (delay) is represented by the delay factor (τ) . Stability analysis revolves around the equilibrium point other than zero. Chaotic behavior emerges through Hopf bifurcation at the critical delay parameter value. If this delay parameter is even slightly perturbed, oscillatory limit cycles can be induced in the market dynamics, leading to equilibrium with brisk market expansion, frequent recessions, and sudden collapses. We conducted sensitivity and directional analysis on a number of factors while also examining the stability and duration of the Hopf bifurcation. Numerical findings were validated using MATLAB. Additionally, the Caputo operator was used to examine the fractional of demand and supply dynamics. Importantly, we assumed a pivotal role in advancing fair labor practices and fostering economic growth on a national scale. In this paper, to investigate the dynamic interplay between supply and demand, with a focus on collectability, a novel mathematical model was introduced via conformable operator. This model considers the possibility that operating expenses or a lack of raw materials causes a manufacturing delay than the supply of goods instantly matching demand. This maturation (delay) is represented by the delay factor . Stability analysis revolves around the equilibrium point other than zero. Chaotic behavior emerges through Hopf bifurcation at the critical delay parameter value. If this delay parameter is even slightly perturbed, oscillatory limit cycles can be induced in the market dynamics, leading to equilibrium with brisk market expansion, frequent recessions, and sudden collapses. We conducted sensitivity and directional analysis on a number of factors while also examining the stability and duration of the Hopf bifurcation. Numerical findings were validated using MATLAB. Additionally, the Caputo operator was used to examine the fractional of demand and supply dynamics. Importantly, we assumed a pivotal role in advancing fair labor practices and fostering economic growth on a national scale. [ABSTRACT FROM AUTHOR]

In this paper, the existence and number of non-contractible limit cycles of the Josephson equation $ \beta \frac{d^{2}\Phi}{dt^{2}}+(1+\gamma \cos \Phi)\frac{d\Phi}{dt}+\sin \Phi = \alpha $ are studied, where $ \phi\in \mathbb S^{1} $ and $ (\alpha,\beta,\gamma)\in \mathbb R^{3} $. Concretely, by using some appropriate transformations, we prove that such type of limit cycles are changed to limit cycles of some Abel equation. By developing the methods on limit cycles of Abel equation, we prove that there are at most two non-contractible limit cycles, and the upper bound is sharp. At last, combining with the results of the paper (Chen and Tang, J. Differential Equations, 2020), we show that the sum of the number of contractible and non-contractible limit cycles of the Josephson equation is also at most two, and give the possible configurations of limit cycles when two limit cycles appear. [ABSTRACT FROM AUTHOR]

In this paper, we use a semi-discretization method to consider the predator–prey model of Leslie type with ratio-dependent simplified Holling type IV functional response. First, we discuss the existence and stability of the positive fixed point in total parameter space. Subsequently, through using the central manifold theorem and bifurcation theory, we obtain sufficient conditions for the flip bifurcation and Neimark–Sacker bifurcation of this system to occur. Finally, the numerical simulations illustrate the existence of Neimark–Sacker bifurcation and obtain some new dynamical phenomena of the system—the existence of a limit cycle. Corresponding biological meanings are also formulated. [ABSTRACT FROM AUTHOR]

The paper's primary objective is to examine the dynamic behavior of an integer and noninteger predator–prey system with a Holling type IV functional response in the Caputo sense. Our focus is on understanding how harvesting influences the stability, equilibria, bifurcations, and limit cycles within this system. We employ qualitative and quantitative analysis methods rooted in bifurcation theory, dynamical theory, and numerical simulation. We also delve into studying the boundedness of solutions and investigating the stability and existence of equilibrium points within the system. Leveraging Sotomayor's theorem, we establish the presence of both the saddle-node and transcritical bifurcations. The analysis of the Hopf bifurcation is carried out using the normal form theorem. The model under consideration is extended to the fractional reaction–diffusion model which captures non-local and long-range effects more accurately than integer-order derivatives. This makes fractional reaction–diffusion systems suitable for modeling phenomena with anomalous diffusion or memory effects, improving the fidelity of simulations in turn. An adaptable numerical technique for solving this class of differential equations is also suggested. Through simulation results, we observe that one of the Lyapunov exponents has a negative value, indicating the potential for the emergence of a stable-limit cycle via bifurcation as well as chaotic and complex spatiotemporal distributions. We supplement our analytical investigations with numerical simulations to provide a comprehensive understanding of the system's behavior. It was discovered that both the prey and predator populations will continue to coexist and be permanent, regardless of the choice of fractional parameter. [ABSTRACT FROM AUTHOR]

A nonlinear phenomenon in nature is often modeled by a system of differential equations with parameters. The bifurcation occurs when a parameter varies in such systems, causing a qualitative change in its solution. In this paper, we study one of the most exciting bifurcations, which is Hopf bifurcation. We use tools from algebraic topology to analyze and reveal supercritical and subcritical Hopf bifurcations. [ABSTRACT FROM AUTHOR]

In this paper, we classify the phase portraits of an SIR epidemic dynamics model. Depending on the values of the parameters, this model can exhibit seven different phase portraits. In particular, from a biological point of view we prove that the unique attractors of this model are one or two equilibrium points depending on the values of the parameters and from the phase portraits follow the basins of attraction of these equilibria. [ABSTRACT FROM AUTHOR]

This paper explores a predator–prey system featuring fear and disease within the predator population,utilizing the Rosenzweig–MacArthur model with Holling type-II functional response. The primary focus lies in investigating the impact of a fear factor, wherein the prey's growth rate is hindered due to predator-induced fear. Additionally, the model accounts for the spread of disease among predators,leading to a division between susceptible and infected predator subpopulations. The inclusion of an Allee effect in the susceptible predator further enriches the model. The study involves a thorough examination, encompassing local and global stability analysis as well as Hopf bifurcation analysis around the interior equilibrium point. Numerical simulations underscore a noteworthy observation: an escalation in interaction force propels the system into chaotic dynamics,marked by stable focus, limit cycles and period-doubling phenomena. A noteworthy finding pertains to the influence of the Allee parameter (θ) on chaotic dynamics. As the Allee parameter values increase, the system tends to stable focus through a sequence of chaotic states, period-doubling and limit cycles. Subsequently, the paper introduces the role of another pivotal parameter, the fear factor, into the chaotic dynamics. Intriguingly, chaos transforms into stable focus through diverse nonlinear phenomena, including period-doubling and limit cycles. This nuanced exploration of parameters sheds light on the intricate dynamics governing the predator–prey system, offering a comprehensive understanding of the interplay between fear, disease and ecological factors. So our observation throughout this paper that how chaos behaves here after one by one injection of our new features: fear factor and Allee parameter? [ABSTRACT FROM AUTHOR]

Purpose: Van der Pol (VDP) oscillator is a typical self-excited system with observable limit cycle oscillations (LCOs). To suppress LCOs, a VDP oscillator coupled nonlinear energy sink (NES) is proposed in this paper. To achieve the optimal vibration reduction effect, the sensitive parameters of the VDP-NES system are investigated and its vibration reduction mechanism is revealed. Methods: First, the dynamic model of the coupled NES system is established by utilizing Newton's law. Then we reveal the vibration reduction mechanism based on stability and bifurcation theory, which may be a new angle of view to investigate the VDP oscillator with NES. Finally, the influence of different parameters on the vibration reduction effect is discussed based on numerical simulation. Results: The LCOs of VDP oscillator are caused by Hopf bifurcation, and its amplitude is closely related to the eigenvalue of the equilibrium point. When the NES is coupled, the unstable equilibrium point becomes stable, then the LCOs are completely suppressed. Although the stability of the equilibrium point of the system with NES keeps invariant, the real part of the eigenvalue decreases. As a result, the LCOs are partially suppressed, and the amplitude decreases by about 70%. In addition, the self-excited intensity and damping ratio are sensitive to the vibration reduction effect. Conclusion: Coupling NES can have a better vibration reduction effect on the LCOs, and vibration reduction is closely related to Hopf bifurcation and the equilibrium point of the system. The research shows that putting off the Hopf bifurcation to widen the parameter range of the stable equilibrium point can obtain the optimal vibration control effect. At the same time, decreasing the real part of the eigenvalue also can reduce the amplitude of LCOs. It may provide some ideas for vibration control strategies in practical engineering. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the celebrated Haken--Kelso--Bunz model, describing the dynamics of bimanual coordination, in the presence of delay. We study the linear dynamics, stability, nonlinear behavior, and bifurcations of this model by both theoretical and numerical analysis. We calculate in-phase and antiphase limit cycles as well as quasi-periodic solutions via double Hopf bifurcation analysis and center manifold reduction. Moreover, we uncover further details on the global dynamic behavior by numerical continuation, including the occurrence of limit cycles in phase quadrature and 1-1 locking of quasi-periodic solutions. [ABSTRACT FROM AUTHOR]

In this paper, we provide a rigorous description of the birth of canard limit cycles in slow-fast systems in R 3 through the folded saddle-node of type II and the singular Hopf bifurcation. In particular, we prove – in the analytic case only – that for all 0 < ϵ ≪ 1 there is a family of periodic orbits, born in the (singular) Hopf bifurcation and extending to O (1) cycles that follow the strong canard of the folded saddle-node. Our results can be seen as an extension of the canard explosion in R 2 , but in contrast to the planar case, the family of periodic orbits in R 3 is not explosive. For this reason, we have chosen to call the phenomena in R 3 , the "dud canard". The main difficulty of the proof lies in connecting the Hopf cycles with the canard cycles, since these are described in different scalings. As in R 2 , we use blowup to overcome this, but we also have to compensate for the lack of uniformity near the Hopf bifurcation, due to its singular nature; it is a zero-Hopf bifurcation in the limit ϵ = 0. In the present paper, we do so by imposing analyticity of the vector-field. This allows us to prove existence of an invariant slow manifold, that is not normally hyperbolic. [ABSTRACT FROM AUTHOR]

In this paper, we deal with a classical object, namely, a nonhyperbolic limit cycle in a system of smooth autonomous ordinary differential equations. While the existence of a center manifold near such a cycle was assumed in several studies on cycle bifurcations based on periodic normal forms, no proofs were available in the literature until recently. The main goal of this paper is to give an elementary proof of the existence of a periodic smooth locally invariant center manifold near a nonhyperbolic cycle in finite-dimensional ordinary differential equations by using the Lyapunov–Perron method. In addition, we provide several explicit examples of analytic vector fields admitting (non)-unique, (non)- C ∞ -smooth and (non)-analytic periodic center manifolds. [ABSTRACT FROM AUTHOR]

This paper is devoted to the high codimension bifurcations of a classical predator–prey system with Allee effects and generalized Holling type III functional response p (x) = m x 2 a x 2 + b x + 1 where b > - 2 a . We show that the maximal orders of nilpotent saddle, cusp singularity and weak focus are all three. The unfoldings of a cusp singularity and of a nilpotent saddle of order 3 with a fixed invariant line are developed. The dependence of codimension of degenerate Hopf bifurcation on b is thoroughly investigated. It is proven that there exist a homoclinic loop of order 2 and a heteroclinic loop of order 2 for - 2 a < b < 0 , and three limit cycles for b > 0 . Together with existing work for Holling type I, II and IV functional responses, our results complement the analysis of the classical predator–prey systems with Allee effects and four types of Holling functional responses. Furthermore, simple formulas are derived to characterize the order of nilpotent saddle, through which the existence and order of the heteroclinic loop can be easily obtained for a general class of predator–prey systems with any smooth functional response. [ABSTRACT FROM AUTHOR]

The tea green leafhopper Empoasca onukii is one kind of insect pest threatening the tea production, and the mite Anystis baccarum has been used as an agent for pest control. In this paper, we introduce a generalist predator-prey model to study the dynamics for informing biological control. There have been some bifurcation studies of the generalist predator-prey model in the last few years. Except for the bifurcations include saddle-node bifurcation of codimension 1 and 2, Hopf bifurcations, and Bogdanov-Takens bifurcation of codimension 2 and 3, we also present the bifurcations of nilpotent singularities of elliptic and focus type of codimension 3. We find that the nilpotent singularities are associated with a cubic Liénard system, and the nilpotent bifurcations are three-parameter bifurcations of a codimension 4 nilpotent focus. Furthermore, we show that the nilpotent focus serves as an organizing center to connect all the codimension 3 bifurcations in the system. We also present the bifurcation diagrams to unfold the nilpotent singularities of codimension 3. One interesting observation is that we show numerically the existence of three limit cycles in the system. [ABSTRACT FROM AUTHOR]

The Kolmogorov model is a class of significant ecological models and is initially introduced to describe the interaction between two species occupying the same ecological habitat. Limit cycle bifurcation problem is close to Hilbertis 16th problem. In this paper, we focus on investigating bifurcation of limit cycle for a class of quartic Kolmogorov model with two positive equilibrium points. Using the singular values method, we obtain the Lyapunov constants for each positive equilibrium point and investigate their limit cycle bifurcations behavior. Furthermore, based on the analysis of their Lyapunov constants' structure and Hopf bifurcation, we give the condition that each one positive equilibrium point of studied model can bifurcate 5 limit cycles, which include 3 stable limit cycles. [ABSTRACT FROM AUTHOR]

In this paper, we study a predator–prey system with the simplified Holling IV functional response and antipredator behavior such that the adult prey can attack vulnerable predators. The model has been investigated by Tang and Xiao, and the existence and stability of all possible equilibria are determined. In addition, they performed a bifurcation analysis and showed that the system undergoes a Codimension 2 Bogdanov–Takens bifurcation. In this paper, for the same model, we further show that the cusp‐type Bogdanov–Takens bifurcation can be of Codimension 3, which acts as an organizing center for the whole bifurcation set. In addition, we propose the existence of Hopf bifurcation of Codimension 2 and the coexistence of stable limit cycle and unstable limit cycle. In particular, we show that the antipredator behavior has great effect on the dynamics of the model, it may cause the predator population to extinct while the prey population will increase up to the carrying capacity. Numerical simulations including bifurcation diagrams and phase portraits are performed to illustrate and confirm the theoretical results. These results may enrich the dynamics of predator–prey systems. [ABSTRACT FROM AUTHOR]

Neuronal networks control motor output through multi-rhythmic electrical activities with different frequencies and amplitudes of different neurons. For example, the pre-Bötzinger complex related to respiratory function exhibits sigh (slow frequency and large amplitude) and eupnea (fast frequency and small amplitude) rhythms. In the present paper, rhythm transitions modulated by several physiological factors are simulated in a coupling model composed of sigh and eupnea compartments, and the roles of ionic currents and the underlying bifurcation mechanism of the rhythm transitions are acquired. The transition from sigh- and eupnea-rhythm to the eupnea rhythm induced by the blockage of calcium or hyperpolarization-activated channels is related to supercritical Hopf bifurcation of the sigh compartment, and to the sigh rhythm caused by the blockage of persistent sodium channel or change of extracellular potassium concentration is associated with subcritical Hopf/fold limit cycle bifurcation of the eupnea compartment. The parameter regions of ionic currents and dynamical mechanism of rhythm transitions in the pre-Bötzinger complex are helpful for the modulation to the respiratory rhythms. [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]

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]

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]

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]

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]

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, we introduce constant-yield prey harvesting into the Holling–Tanner model with generalist predator. We prove that the unique positive equilibrium is a cusp of codimension 4. As the parameter values change, the system exhibits degenerate Bogdanov–Takens bifurcation of codimension 4. Using the resultant elimination method, we show that the positive equilibrium is a weak focus of order 2, and the system undergoes degenerate Hopf bifurcation of codimension 2 and has two limit cycles. By numerical simulations, we demonstrate that the system exhibits homoclinic bifurcation and saddle–node bifurcation of limit cycles as the parameters are varied. The main results show that constant-yield prey harvesting and generalist predator can lead to complex dynamic behavior of the model. [ABSTRACT FROM AUTHOR]

In this paper, we study a predator--prey mite model of Leslie type with generalized Holling IV functional response. The model is shown to have very rich bifurcation dynamics, including subcritical and supercritical Hopf bifurcations, degenerate Hopf bifurcation, focustype and cusp-type degenerate Bogdanov--Takens bifurcations of codimension 3, originating from a nilpotent focus or cusp of codimension 3 that acts as the organizing center for the bifurcation set. Coexistence of multiple steady states, multiple limit cycles, and homoclinic cycles is also found. Interestingly, the coexistence of two limit cycles is guaranteed by investigating generalized Hopf bifurcation and degenerate homoclinic bifurcation, and we also find that two generalized Hopf bifurcation points are connected by a saddle-node bifurcation curve of limit cycles, which indicates the existence of global regime for two limit cycles. Our work extends some results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we propose a Leslie–Gower predator–prey model with strong Allee effect on prey and fear effect on predator. We discuss the existence and local stability of equilibria by making full use of qualitative analytical theory. It is shown that the above system exhibits at most two positive equilibria and it can undergo a series of bifurcation phenomena. We indicate that the dynamical behavior of the model is closely related to the fear effect on predator. In detail, when the fear effect parameter p = p ∗ , the system will undergo degenerate Hopf bifurcation. There exist two limit cycles (the inner is stable and the outer is unstable). However, when p = p ∗ , the system will undergo degenerate Bogdanov–Takens bifurcation. Also, by numerical simulation, we conclude that the stronger the fear effect, the bigger the density of prey species. The above shows that fear effect on predator is beneficial to the persistence of the prey species. Our results can be seen as a complement to previous works [González-Olivares et al., 2011; Pal & Mandal, 2014]. [ABSTRACT FROM AUTHOR]

It has been well established in the literature that the predator-induced fear has indirect impact on prey but can have comparable effects on prey population as direct killing, and that the crowding effect or self-limitation of population plays pivotal roles in determining the dynamics and interactions among populations. In this paper, we first propose and investigate a deterministic prey-predator model incorporating simultaneously fear effect in prey and crowding effect in predators. The model has rich dynamics, including one up to three positive equilibria, complex bifurcations (saddle-node, Hopf and Bogdanov-Takens bifurcations), and two types of bistability (between two interior equilibria or between an interior equilibrium and an interior limit cycle). Thus the model is easily affected by external environmental fluctuations. When environmental noises are involved, some new dynamics can be observed for the developed stochastic model. Especially, for the scenarios when the deterministic model exhibits bistability, we can observe noise-induced frequent transitions between two different interior attractors (two interior equilibria or an interior equilibrium and an interior limit cycle). The tipping points of noise intensities for the occurrence of such transitions are estimated by constructing the confidence ellipse/band for the equilibrium/limit cycle. These indicate that the predators and prey can coexist in two different modes and switch randomly between them. [ABSTRACT FROM AUTHOR]

In the brain, feedforward inhibition is a fundamental regulator of balancing neural excitation and inhibition, and its dysfunction may cause epilepsy. In this paper, we investigate the influence of feedforward inhibition on absence seizures and its dynamical mechanisms based on a neural mass model. On the one hand, pyramidal neurons (PY), fast kinetic interneurons ( I fast ) and slow kinetic interneurons ( I slow ) form two feedforward inhibition pathways, which are the PY– I fast – I slow pathway and the PY– I slow – I fast pathway, respectively. When changing the synaptic strength in different pathways, the system shows simple oscillation state, spike and wave discharge (SWD), and multi-spike and wave discharge (m-SWD). On the other hand, the one-parameter bifurcation analysis reveals that the system develops fold bifurcations, Hopf bifurcations, fold bifurcations on limit cycle, and period doubling bifurcations when state transitions occur. In particular, when the system is in the bistable region, the dynamic state of this region is closely related to the stable limit cycle and stable fixed point. Therefore, feedforward inhibition pathways are indeed involved in the regulation of absence seizures, excitatory connections in both feedforward inhibition pathways are more effective than the inhibitory connections in the control of absence seizures. More interestingly, the feedforward inhibition pathway PY– I slow – I fast has a stronger regulation of absence seizures than PY– I fast – I slow . These results provide a theoretical basis for a more detailed understanding of the underlying mechanisms of neurological disorders. [ABSTRACT FROM AUTHOR]

Subsynchronous oscillation events of grid‐connected voltage source converter (VSC) pose a potential risk to the security and stability of power systems. It is particularly challenging to reveal the subsynchronous oscillation mechanism precisely because the grid‐connected VSC consists of non‐linear components such as saturation. The describing function and the generalized Nyquist criterion are used in this paper to investigate the bifurcation types and dominant parameters of grid‐connected VSC. First, the condition of a supercritical Hopf bifurcation is investigated, which generates subsynchronous oscillations. Then, the dominant parameters of supercritical Hopf bifurcation are deduced, including the grid strength, the active power reference, or the proportional coefficient of phase‐locked loop. Finally, it is examined how the saturation's limit value impacts the limit cycle's amplitude and stability. The findings demonstrate that a system's stability depends on whether its closed‐loop frequency characteristic curve intersects the describing function's negative reciprocal curve. When the two curves intersect, the supercritical Hopf bifurcation appears, and changing the limit value can affect the limit cycle's amplitude. However, if the two curves do not intersect, altering the limit value cannot cause the supercritical Hopf bifurcation, and the subsynchronous oscillation will not happen. Variations in grid strength, active power reference, or proportional coefficient of phase‐locked loop will significantly affect the Nyquist curve, determining if supercritical Hopf bifurcation appears. [ABSTRACT FROM AUTHOR]

Hidden chaotic attractors is a fascinating subject of study in the field of nonlinear dynamics. Jerk systems with a stable equilibrium may produce hidden chaotic attractors. This paper seeks to enhance our understanding of hidden chaotic dynamics in jerk systems of three variables (x , y , z) with nonlinear terms from a predefined set: { − x y + d z 2 , − x y + d z tanh (z) } , where d is a real parameter. The behavior of the systems is analyzed using rigorous Hopf bifurcation analysis and numerical simulations, including phase portraits, bifurcation diagrams, Lyapunov spectra, and basins of attraction. For certain jerk systems with a subcritical Hopf bifurcation, adjusting the coefficient of a linear term can lead to hidden chaotic behavior. The adjustment modifies the subcritical Hopf equilibrium, transforming it from an unstable state to a stable one. One such jerk system, while maintaining its equilibrium stability, experiences a sudden transition from a point attractor to a stable limit cycle. The latter undergoes a period-doubling route to chaos, which may be followed by a reverse route. Therefore, by perturbing certain jerk systems with a subcritical Hopf equilibrium, we can gain insights into the formation of hidden chaotic attractors. Furthermore, adjusting the coefficient of the nonlinear term z tanh (z) in certain systems with a stable equilibrium can also lead to period-doubling routes or reverse period-doubling routes to hidden chaotic dynamics. Both findings are significant for our understanding of the hidden chaotic dynamics that can emerge from nonlinear systems with a stable equilibrium. [ABSTRACT FROM AUTHOR]

We continue to study the nonsmooth van der Pol-Duffing oscillator x ˙ = y x ˙ = y , y ˙ = a 1 x + a 2 x 3 + b 1 y + b 2 | x | y y ˙ = a 1 x + a 2 x 3 + b 1 y + b 2 | x | y , where a i , b i a i , b i are real and a 2 b 2 ≠ 0 a 2 b 2 ≠ 0 , i = 1 , 2 i = 1 , 2. Notice that the sum of indices of equilibria is − 1 − 1 for a 2 > 0 a 2 > 0 and 1 1 for a 2 < 0 a 2 < 0. When a 2 > 0 a 2 > 0 , the nonsmooth van der Pol-Duffing oscillator has been studied completely in the companion paper. Attention goes to the bifurcation diagram and all global phase portraits in the Poincaré disc of the nonsmooth van der Pol-Duffing oscillator for a 2 < 0 a 2 < 0 in this paper. The bifurcation diagram is more complex, which includes two Hopf bifurcation surfaces, one pitchfork bifurcation surface, one homoclinic bifurcation surface, one double limit cycle bifurcation surface and one bifurcation surface for equilibria at infinity. When b 2 > 0 b 2 > 0 is fixed, this nonsmooth van der Pol-Duffing oscillator cannot be changed into a near-Hamiltonian system for small a 1 , b 1 a 1 , b 1. Moreover, the global dynamics of the nonsmooth van der Pol-Duffing oscillator and the van der Pol-Duffing oscillator are different. [ABSTRACT FROM AUTHOR]

In this paper, we study the global existence of a bifurcating periodic solution for a two zooplankton-phytoplankton model with two delays. First, we demonstrate that the bifurcating periodic solution exists when one delay increases and the other delay remains unchanged. Second, we give simulation to describe the bifurcating periodic solution when one delay increases. Our work answers the question in Sect. 5 (Shi and Yu in Chaos Solitons Fractals 100:62–73, 2017). [ABSTRACT FROM AUTHOR]

In this paper, we study Poincaré, Hopf and homoclinic bifurcations of limit cycles for planar near-Hamiltonian systems. Our main results establish Hopf and homoclinic bifurcation theories by higher order Melnikov functions, obtaining conditions on upper bounds and lower bounds of the maximum number of limit cycles. As an application, we concern a cubic near-Hamiltonian system, and study Hopf and homoclinic bifurcations in detail, finding more limit cycles than [26]. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the complex dynamics of a predator-prey model, specifically the Leslie-Gower model, with additive Allee effect and simplified Holling Ⅲ functional response. The model has been analyzed for various bifurcations, including the nilpotent cusp singularity of codimension 3, Bogdanov-Takens bifurcation of codimension 3, and Hopf bifurcation of codimension 2. Additionally, a codimension 2 cusp of limit cycles and the coexistent acute angle region of three limit cycles have been identified. Notably, the isola bifurcation of limit cycles, which indicates a new mechanism of sustained oscillation, has been observed for the first time in a Leslie-Gower predator-prey model with additive Allee effect. One-parameter and two-parameter bifurcation diagrams and corresponding phase portraits have been presented to verify the theoretical results. From a biological perspective, the additive Allee effect may result in system collapse, leading to the extinction of the predator population and survival of the prey. The finding of the isola bifurcation of limit cycles is of significant interest, highlighting a novel mechanism of sustained oscillation in this complex system. [ABSTRACT FROM AUTHOR]

The lumbar disc has complex structure and material properties, and if it develops disease, it cannot heal itself. Therefore, prevention of lumbar disc herniation is very important. In this paper, the model of lumbar disc is built, and the nonlinear dynamic response of the system is researched. A modified Van der Pol model is imported to describe the stress–strain curves of the lumbar disc. The system's dynamic model is set up, and harmonic balance method is applied to revise the natural frequency of the system. The product of numerical simulation reveals that the lumbar disc has complex dynamic characteristics, including balance point, limit cycle bifurcation and stochastic Hopf bifurcation. By changing the parameters, we can avoid large-scale periodic movement of the lumbar disc which causes lumbar disc herniation. These results contribute to the prevention of lumbar disc herniation. [ABSTRACT FROM AUTHOR]

In the Antarctic, the whale population had been reduced dramatically due to the unregulated whaling. It was expected that Antarctic krill, the main prey of whales, would grow significantly as a consequence and exploratory krill fishing was practiced in some areas. However, it was found that there has been a substantial decline in abundance of krill since the end of whaling, which is the phenomenon of krill paradox. In this paper, to study the krill–whale interaction we revisit a harvested predator–prey model with Holling I functional response. We find that the model admits at most two positive equilibria. When the two positive equilibria are located in the region { (N , P) | 0 ≤ N < 2 N c , P ≥ 0 } , the model exhibits degenerate Bogdanov–Takens bifurcation with codimension up to 3 and Hopf bifurcation with codimension up to 2 by rigorous bifurcation analysis. When the two positive equilibria are located in the region { (N , P) | N > 2 N c , P ≥ 0 } , the model has no complex bifurcation phenomenon. When there is one positive equilibrium on each side of N = 2 N c , the model undergoes Hopf bifurcation with codimension up to 2. Moreover, numerical simulation reveals that the model not only can exhibit the krill paradox phenomenon but also has three limit cycles, with the outmost one crosses the line N = 2 N c under some specific parameter conditions. [ABSTRACT FROM AUTHOR]

In this paper, two prey–predator models with distributed delays are presented based on the growth and loss rates of the predator, which are much smaller than that of the prey, leading to a singular perturbation problem. It is obtained that Hopf bifurcation can occur, where the coexistence equilibrium becomes unstable leading to a stable limit cycle. Subsequently, considering the perturbation parameter 0 < ≪ 1 , the fact that the solution crossing the transcritical point converges to a stable equilibrium is discussed for the model with Holling type I using the linear chain criterion, center-manifold reduction, the geometric singular perturbation theory and entry–exit function. The existence and uniqueness of relaxation oscillation cycle for the model with Holling type II are obtained. In addition, numerical simulations are provided to verify the analytical results. [ABSTRACT FROM AUTHOR]

The influences of Allee effect and density‐dependent mortality on population growth are of great significance in ecology. In this paper, we first consider a Gause‐type predator–prey model with simplified Holling type IV functional response, strong Allee effect on prey, and density‐dependent mortality of predator. It is shown that the system exhibits rich and complex dynamics like bistability, local and global bifurcations such as transcritical bifurcation, saddle‐node bifurcation, Hopf bifurcation, cusp bifurcation, Bogdanov–Takens bifurcation, Bautin bifurcation, homoclinic bifurcation, saddle‐node bifurcation of limit cycle, and heteroclinic bifurcation. Next, we separately explore the influences of Allee effect and density‐dependent mortality on the dynamics of the same model. The results show that the strong Allee effect induces the occurrence of heteroclinic bifurcation and the reduction of the number of limit cycles, while the density‐dependent mortality stabilizes the system. Since the analytical expressions of the interior equilibria are difficult to derive, the results are verified numerically. [ABSTRACT FROM AUTHOR]

In this paper, we study the codimensions of Hopf bifurcation and Bogdanov–Takens bifurcation of a predator–prey model with alternative prey and prey refuges, which was proposed by Chen et al. [2023]. The results show that the predator–prey model can undergo a supercritical Hopf bifurcation or a Bogdanov–Takens bifurcation of codimension two under certain parameter conditions. It means that there are some predator–prey models with an alternative prey and prey refuges which have a limit cycle or a homoclinic loop. Moreover, it is also shown that the codimension of Hopf bifurcation is at most one and codimension of Bogdanov–Takens bifurcation is at most two. [ABSTRACT FROM AUTHOR]

In most HIV models, the emergence of backward bifurcation means that the control for basic reproduction number less than one is no longer effective for HIV treatment. In this paper, we study an HIV model with CTL response and cell‐to‐cell transmission by using the dynamical approach. The local and global stability of equilibria is investigated, the relations of subcritical Hopf bifurcation and supercritical bifurcation points are revealed, especially, the so‐called new type bifurcation is also found with two Hopf bifurcation curves meeting at the same Bogdanov–Takens bifurcation point. Forward and backward bifurcation, Hopf bifurcation, saddle‐node bifurcation, Bogdanov–Takens bifurcation are investigated analytically and numerically. Two limit cycles are also found numerically, which indicates that the complex behavior of HIV dynamics. Interestingly, the role of cell‐to‐cell interaction is fully uncovered, it may cause the oscillations to disappear and keep the so‐called new type bifurcation persist. Finally, some conclusions and discussions are also given. [ABSTRACT FROM AUTHOR]

This paper studies the existence and multiplicity of periodic orbits of a stage-structured predator-prey model with Beddington-DeAngelis functional response. We derive an existence criterion of periodic orbit in terms of inequalities of system's nine parameters and prove that the system admits at least two limit cycles or three limit cycles via subcritical Hopf bifurcation or generalized Hopf bifurcation theory and hypersurface theory. We also prove that each one-parameter system possesses at least one limit cycle when it is larger or smaller than its Hopf bifurcating value, or between its Hopf bifurcating values. Combining theoretic analysis and numerical simulations, we investigate global bifurcations of limit cycles for a varying parameter system, which provides a plenty of bifurcating properties and again suggests that the system has at least three limit cycles. Similar results are obtained for the system without mutual interference. [ABSTRACT FROM AUTHOR]

In this paper, the bursting oscillation phenomenon in coupled systems with two time scales is introduced. Firstly, several types of bifurcation are briefly introduced: fold bifurcation, Hopf bifurcation, fold limit cycle bifurcation, homoclinic bifurcation, etc. The bursting oscillations of the system with two excitation terms and the bifurcation delay in the bursting oscillations are considered. Secondly, some simple bursting oscillations are introduced, such as fold/fold bursting, fold/supHopf bursting, subHopf/subHopf bursting, fold/LPC bursting, Hopf/LPC bursting, fold/homoclinic bursting, Hopf/homoclinic bursting, etc. At the same time, the system also has some complex bursting oscillations, such as asymmetric bursting, delayed bursting, bursting with hysteresis loop, etc. Finally, the practical applications of bursting oscillations, such as dynamic vibration absorbers and nonlinear vibration energy harvesting technology, are introduced. [ABSTRACT FROM AUTHOR]

In this work. we consider an eco-epidemiological model with disease in prey. The role of alternative food is being investigated. Alternative food source for predator is another significant component in the interaction of ecological species and incorporation of this aspect in eco-epidemiological investigations may give some fascinating outcomes. The contribution of this paper is to consider different competition coefficients within the prey population. which leads to the emergent carrying capacity. The pernianence and uniform persistence are established. Moreover. we discuss the local and global stability analysis. For bifurcation analysis. the transmission rate has been taken as parameter. The Hopf bifurcation analysis around the endemic equilibrium point is discussed. Further. we also pay attention to the chaotic dynamics which is produced by the disease. At the end. extensive numerical simulations are performed to demonstrate the effectiveness of our hypothetical outcomes. The numerical simulations reveal that the three species eco-epidemiological system induced chaos when disease spreads iii low level. If we increase the force of infection. then the system shows stable focus through chaos to period-double. period-double to limit cycle. We conclude that chaotic dynamics can be coiitrolled by the alternative food as well as the competition coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we study the factor of the fear effect in a predator–prey model with prey refuge and a non-differentiable fractional functional response due to the group defense. Since the functional response is non-differentiable, the dynamics of this system are considerably different from the dynamics of a classical predator–prey system. The persistence, the stability and the existence of the steady states are investigated. We examine the Hopf bifurcation at the unique positive equilibrium. Direct Hopf bifurcation is studied via the central manifold theorem. When the value of the fear factor decreases and is less than a threshold κ H , the limit cycle appears, and it disappears through a loop of heteroclinic orbits when the value of the fear factor is equal to a value κ h e t . [ABSTRACT FROM AUTHOR]

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

The Leslie–Gower model, a kind of predator–prey model with weak Allee effect, is studied in this paper. The existence and stability of non-negative equilibria are first discussed. Then, we investigate several bifurcation phenomena undergoing positive equilibria, such as saddle-node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation, etc. Some possible dynamical behaviors of this model are illustrated by numerical simulation. The bifurcation diagrams for the cases of codimensions 2 and 3 are given respectively. The coexistence of a periodic cycle and a homoclinic cycle, and two limit cycles enclosing an unstable equilibrium are also proved. This appears to be the first study of the Leslie–Gower model including the influence of weak Allee effect on prey. [ABSTRACT FROM AUTHOR]