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• We introduce the effect of changing environmental carrying capacity on evolution of asymmetric rock-paper-scissors game. • According to assumptions on environmental gradients, the system can exhibit various survival states including multistability of single species survival. • Symmetry-breaking of competition rates and environmental carrying capacity can be significant factors to yield rich behavior of species survival in systems of cyclic competition. • Considering ecological factors is found to be an important issue on understanding mechanisms of evolution among cyclically competing species in the perspective of maintaining coexistence and promoting biodiversity. Asymmetrical rock-paper-scissors (RPS) competition has been perceived as a crucial factor in shaping species biodiversity, and understanding this ecological issue in a multi-species paradigm is rather difficult because community dynamics usually depend on distinct factors such as abiotic environments, biotic interactions and symmetry-breaking phenomenon. To address this problem, we employ a Lotka-Volterra competitive system consisting of both symmetrical, asymmetrical interactions and abiotic environment components. We discover that that asymmetrical RPS competition in heterogeneous environments can yield much richer dynamical behaviors, compared to the symmetrical and asymmetrical competition in homogeneous environments. While it is observed that species coexistence outcomes and/or oscillatory solutions are maintained as in the case of homogeneous environments, the nonuniformity in the environmental carrying capacities may lead to extra dynamics with regards to the appearance of survival states; for instance, coexistence of any two-species and single-species persistence states, which are not evident in the previous modelling studies. By means of bifurcation analysis, various salient features of the dynamical systems, including the emergence of certain attractors (e.g., different steady states, stable limit cycles and heteroclinic cycles) and co-dimension one bifurcations (e.g., transcritical and supercritical Hopf bifurcations) are realized in this ecological model. Overall, this modelling work provides a novel attempt to simultaneously encompass not only symmetry-breaking phenomenon through RPS competition, but also heterogeneity in the environments. This framework can provide additional insights to better understand various mechanisms underlying the effects of distinct ecological processes on multi-species communities. [ABSTRACT FROM AUTHOR]

• The rock-paper-scissors model is presented for two-person non-zero sum game in which the strategies of rock mutate to scissors and paper. • The evolutionary dynamics of the population of species rock, paper and scissors is studied when the patches are connected through random walk. • When the patches are coupled, the state of synchronization and stability is observed. • When mutation is allowed, the limit cycle converges to stable state and the transition from one phase to another phase is observed. • The replicator-mutator equations are solved analytically as well as numerically. Connectivity is the safety network for biodiversity conservation because connected habitats are more effective for saving the species and ecological functions. The nature of coupling for connectivity also plays an important role in the co-existence of species in cyclic-dominance. The rock-paper-scissors game is one of the paradigmatic mathematical model in evolutionary game theory to understand the mechanism of biodiversity in cyclic-dominance. In this paper, the metapopulation model for rock-paper-scissors with mutations is presented in which the total population is divided into patches and the patches form a network of complete graph. The migration among patches is allowed through simple random walk. The replicator-mutator equations are used with the migration term. When migration is allowed then the population of the patches will synchronized and attain stable state through Hopf bifurcation. Apart form this, two phases are observed when the strategies of one of the species mutate to other two species: co-existence of all the species phase and existence of one kind of species phase. The transition from one phase to another phase is taking place due to transcritical bifurcation. The dynamics of the population of species of rock, paper, scissors is studied in the environment of homogeneous and heterogeneous mutation. Numerical simulations have been performed when mutation is allowed in all the patches (homogeneous mutation) and some of the patches (heterogeneous mutation). It has been observed that when the number of patches is increased in the case of heterogeneous mutation then the population of any of the species will not extinct and all the species will co-exist. [ABSTRACT FROM AUTHOR]

• We introduce population flow and its effect on biodiversity in evolutionary dynamics of rock-paper-scissors game. • According to assumptions on population flows, the system can exhibit various survival states including persistent coexistence and multistability of single group survival. • Population flow can change the carrying simplex for evolutions of the system. • The basin structure for multistability may be spirally entangled and discontinuous. • The coexistence state can exhibit oscillatory dynamics according to the magnitude of population flow. Classic frameworks of rock-paper-scissors game have been assumed in a closed community that a density of each group is only affected by internal factors such as competition interplay among groups and reproduction itself. In real systems in ecological and social sciences, however, the survival and a change of a density of a group can be also affected by various external factors. One of common features in real population systems in ecological and social sciences is population flow that is characterized by population inflow and outflow in a group or a society, which has been usually overlooked in previous works on models of rock-paper-scissors game. In this paper, we suggest the rock-paper-scissors system by implementing population flow and investigate its effect on biodiversity. For two scenarios of either balanced or imbalanced population flow, we found that the population flow can strongly affect group diversity by exhibiting rich phenomena. In particular, while the balanced flow can only lead the persistent coexistence of all groups which accompanies a phase transition through supercritical Hopf bifurcation on different carrying simplices, the imbalanced flow strongly facilitates rich dynamics such as alternative stable survival states by exhibiting various group survival states and multistability of sole group survivals by showing not fully covered but spirally entangled basins of initial densities due to local stabilities of associated fixed points. In addition, we found that, the system can exhibit oscillatory dynamics for coexistence by relativistic interplay of population flows which can capture the robustness of the coexistence state. Applying population flow in the rock-paper-scissors system can ultimately change a community paradigm from closed to open one, and our foundation can eventually reveal that population flow can be also a significant factor on a group density which is independent to fundamental interactions among groups. [ABSTRACT FROM AUTHOR]

The movements of the plankton in the ocean are driven by random diffusion and cognitive judgement with herd-taxis. In this paper, we formulate a phytoplankton–zooplankton model with time delay in the herd-taxis effect diffusion and homogeneous Neumann boundary conditions. The conditions to guarantee the existence of the coexistence equilibrium of the model are given. By analyzing the distribution of the eigenvalues of the characteristic equation, the local asymptotic stability of the coexistence equilibrium is achieved under certain condition. When there is no time delay in the herd-taxis effect, the model can possess the Turing bifurcation when we consider the nonlinear diffusion term, which leads to instability. When taking the time delay into account, the Hopf bifurcation occurs instead as the time delay varies. Furthermore, we investigate the situation without the fact of time, that is the steady-state bifurcation and the stability of bifurcating solution. Finally, the stability of the coexistence equilibrium, the Turing bifurcation and the Hopf bifurcation of the system are modeled by numerical simulation. The simulations shown are coordinated with the theoretical results which we arrive at in the former part of the paper. The results illustrate that the time delay in the herd-taxis effect of the zooplankton influence the dynamics of the plankton system. [ABSTRACT FROM AUTHOR]

The paper focuses on studying the global dynamics of a simple prey–predator model. The model incorporates cooperative hunting behavior among predators and takes into account the effect of harvesting on the predator population. The interaction between the prey and predator follows the Crowley–Martin functional response. The study involves analyzing the equilibrium points of the system and investigating their stability properties through mathematical analysis. Various types of bifurcations, including Hopf bifurcation, saddle–node bifurcation, and transcritical bifurcation, are numerically demonstrated in the figures, highlighting the dynamic behavior of the model. One of the intriguing findings of the study is the occurrence of bistability and tristability in the model. To incorporate stochasticity into the model, white noise is added to the deterministic system. This allows for the examination of transitions between different steady states in the stochastic system. We have conducted an analysis of species persistence and extinction in relation to the presence of noise. The paper presents the stochastic sensitivity function (SSF) technique and the use of confidence ellipses to assess the likelihood of such transitions occurring in the system. Overall, the study provides insights into the complex dynamics of prey–predator interactions, considering factors such as cooperation, harvesting, and stochasticity. The results contribute to our understanding of population dynamics and the potential effects of environmental and human-induced perturbations on ecosystem stability. • Study on a cooperative prey-predator model with harvesting. • Bifurcations like saddle-node, Hopf, and transcritical, have been explored. • Bistability and tristability are observed in the proposed model system. • Critical radius of transition is determined by evaluating the confidence ellipses. • Noise in the stochastic system leads to transitions between different equilibria. [ABSTRACT FROM AUTHOR]

Active materials hold substantial potential for self-sustaining motion, however, the dependency on movable light or extensive area illumination in current structures often restricts the design and practical deployment of such active machines. In this study, we report a novel zero-energy-mode, self-rolling unicycle equipped with a liquid crystal elastomer rod engine, which fueled by a fixed, small-area light source. By employing an elaborate dynamic model for liquid crystal elastomer, we derive the lateral curvature of the liquid crystal elastomer rod and the driving moment necessary for the unicycle's self-rolling. The study results demonstrate that the self-rolling of the unicycle is driven by a moment induced by the shift in the center of gravity of the curved liquid crystal elastomer rod. Our numerical simulations indicate the presence of a supercritical Hopf bifurcation point delineating the transition between the self-rolling mode and the static mode within the unicycle system. Additionally, the rolling velocity of the unicycle relies on a handful of key system parameters, notably including light intensity, light penetration depth, length of the liquid crystal elastomer rod, rolling friction coefficient, total mass of the unicycle, and wheel radius of the unicycle. The experimental validation of a self-rolling unicycle with zero-energy-mode has been conducted. The self-rolling unicycle constructed in this paper has excellent characteristics of simple structure, zero-energy-mode, horizontal constant light source, and small area light source, providing valuable insights for the utilization of photoresponsive liquid crystal elastomer rods in soft robotics, medical devices, energy harvesting systems, and actuators. • A light-fueled self-rolling unicycle with a liquid crystal elastomer rod engine is proposed. • The experiment confirmed the functionality of the light-powered self-rolling unicycle with an LCE rod engine. • The steady state solution of a self-rolling unicycle with zero energy mode is obtained. • The system parameters can affect the motion patterns and self-rolling velocity. [ABSTRACT FROM AUTHOR]

This paper focuses on the species interaction within a tri-trophic food chain model involving species' mutual interference, characterized by Crowley–Martin functional response with strong Allee effect in prey species. First, the evaluation of the solutions' positivity and boundedness is conducted. Subsequently, the local stability of equilibrium points is analysed, and the Lyapunov function is employed to assess the global stability of interior equilibrium. We have also established adequate criteria for detecting local bifurcations. Sotomayor's theorem infers the presence of saddle–node and transcritical bifurcations. Utilizing the centre manifold theory and the Hopf bifurcation theorem, the emergence and stability of Hopf bifurcation are investigated. Following this, numerical simulations are performed so as to validate the theoretical findings and analyse the impacts of Allee effect in greater details. The analysis reveals that the system undergoes a range of local bifurcations in co-dimension one, including saddle–node, subcritical and supercritical Hopf bifurcations. Furthermore, the existence of global dynamics can be observed by the appearance of Generalized-Hopf bifurcation in the co-dimension two bifurcation diagram. The findings suggest that strong Allee effect could trigger instability and induce a bi-stability phenomenon in the system's dynamical behaviours. • A tri-trophic food chain model with a strong Allee effect is considered. • The occurrence of transcritical and saddle–node bifurcations is derived using Sotomayor's theorem. • The analysis of intricate dynamical features is conducted by means of co-dimension one and two bifurcations. • The model exhibits diverse behaviours and bi-stability phenomena at various Allee thresholds. • Prey with strong Allee effect have a higher chance of survival. [ABSTRACT FROM AUTHOR]

This paper explores the stability and bifurcation of a Caputo fractional-order BAM neural network with time delay and inertia terms. Subsequently, the limitation in bifurcation characteristics of Caputo fractional-order delayed inertial BAM neural network (CFODIBAMNN) is surpassed. By analyzing the stability of the system without time delay, the direct method is applied that involves solving the eigenvalues to determine its stability. Ulteriorly, analyzing the system in the presence of time delays and choosing the time delay as the bifurcation parameter to identify the Hopf bifurcation properties of the system. Eventually, during the verifications, the critical values of bifurcation are accurately calculated, the fluctuation and phase diagrams for the ranges of different fractional order are simulated and the impact of fractional orders on system stability is analyzed. [ABSTRACT FROM AUTHOR]

Animals have evolved fearful behavior throughout a lengthy evolutionary history, and such action should be the result of interactions between prey and predator. At present, the fear factor considered by many scholars in predator–prey models is reasonable to a certain extent, for example in some unicellular and bacterial species, but for most species, the fear factor should be density-dependent of both species due to sexual reproduction. Therefore, based on the existing works, this paper puts forward some more reasonable conditions that should be satisfied by fear factor, and makes theoretical analysis and numerical simulations for specific type of expression. The study shows that fear plays an important role in enriching model dynamics, and that the model dynamics are sensitive to moderate levels of fear. Lower levels of fear maintain the same dynamics as the model without fear, higher levels of fear can exclude periodic solutions (or limit cycles) to promote model stability, while moderate levels of fear may lead to the emergence of Hopf bifurcations. By selecting a different set of parameters, the Hopf bifurcation in our model can be both supercritical and subcritical, contrasting to the typical predator–prey model in which it frequently appears supercritical. • The general form of fear factors that depend on predator and prey density is proposed for the first time. • The specific form of fear factor is analyzed theoretically and simulated numerically. • Research has found that fear plays a key role in enriching ecosystem dynamics and promoting ecosystem stability. • It also found that fear may be an important factor in the extinction of species. [ABSTRACT FROM AUTHOR]

To demonstrate and elucidate bursting patterns and their bifurcation mechanisms, a two-neuron Hopfield neural network is proposed in this paper. The proposed non-autonomous model has a time-varying equilibrium point whose stability undergoes continuous evolution in response to changes in stimulation, and exhibits chaotic dynamics, especially the quasi-periodic and periodic bursting patterns. Over a full bursting cycle, the stability evolution of the time-varying equilibrium point triggers Hopf bifurcation and fold bifurcation, leading to the emergence of quasi-periodic or periodic bursting. To elucidate the bifurcation mechanisms, the transitions between the spiking state and the resting state are demonstrated, thereby identifying the Hopf/Hopf quasi-periodic bursting and fold/fold/Hopf periodic bursting. In addition, a simple analog electronic circuit is designed for the physical implementation of the non-autonomous model, and a printed-circuit board based hardware circuit is made to test the experimental results to verify the numerical results. • Non-autonomous model of two-neuron Hopfield neural network considering stimulation is proposed. • Chaotic dynamics, quasi-periodic spiking/bursting patterns and periodic bursting patterns are exhibited. • Bifurcation mechanisms to quasi-periodic bursting and periodic bursting are elucidated theoretically. • Simple analog electronic circuit is designed and hardware circuit is made to test the experimental results. [ABSTRACT FROM AUTHOR]

The benefits of self-excited motion include the capacity to draw energy directly from the external environment, as well as autonomy and portability, and the creation of additional self-oscillating systems holds the promise of finding applications in various fields, including medical devices, autonomous robotics, energy harvesting, sensors, and more. Inspired by the magnetic perpetual motion, this paper introduces a conceptual perpetual motion machine utilizing liquid crystal elastomer (LCE) and steady illumination. The system consists of a turntable and multiple motion ducts, each containing a mass block. A nonlinear dynamics model for the LCE-based perpetual motion machine operating under steady illumination is established, considering the dynamic LCE model. Theoretical analysis and numerical simulations, employing the classical fourth-order Runge-Kutta method, are employed to examine the system's dynamic behaviors. The system exhibits a supercritical Hopf bifurcation between static and self-rotation patterns. Notably, the external environment's energy input counteracts damping dissipation, preventing the system from halting due to damping and other factors. Theoretical condition for Hopf bifurcation in the perpetual motion machine's self-rotation is derived and validated via numerical computations. Extensive quantitative investigations are conducted to determine the influence of system parameters on self-rotation frequency. The results illustrate that through parameter adjustments, the motion pattern and frequency of the system can be controlled effectively. The proposed system offers continuous rotation in a zero-energy mode, presenting remarkable benefits, including energy efficiency, reduced noise, decreased wear, and robust stability. These attributes position it for significant future applications in power generation, energy harvesting, sensors, and various other fields. [Display omitted] • A light-powered liquid crystal elastomer-based perpetual motion machine was conceptualized. • The perpetual motion machine has two typical motion patterns: static and self-rotation pattern. • Theoretical Hopf bifurcation condition was derived and validated via numerical computations. • The self-rotation frequency can be designed by several parameters. [ABSTRACT FROM AUTHOR]

Highlights • Turing instability condition for a Gierer–Meinhardt system of discrete time and space form are obtained. • Coupled map lattice is applied to study the spatiotemporal dynamics. • Hopf, flip bifurcation to chaos, together with Turing bifurcation are coupled to account for the complex patterns. • Striking patterns such as circle, spiral, mosaic and chaos are shown by numerical simulation. Abstract This paper is concerned with the spatiotemporal behaviors of a Gierer–Meinhardt system in discrete time and space form. Through the linear stability analysis, the parametric conditions are gained to ensure the stability of the homogeneous steady state of the system. Based on the bifurcation theory, as well as center manifold theorem, we derive the critical parameter values of the flip, Neimark–Sacker and Turing bifurcation respectively. Besides, the specific parameter expression to form patterns are also determined. In order to identify chaos among regular behaviors, we calculate the Maximum Lyapunov exponents. The results obtained in this paper are illustrated by numerical simulations. From the simulations, we can see some complex dynamics, such as period doubling cascade, invariant cycles, periodic windows, chaotic behaviors, and some striking Turing patterns, e.g. circle, mosaic, spiral, spatiotemporal chaotic patterns and so on, which can be produced by flip-Turing instability, Neimark–Sacker–Turing instability and chaos. [ABSTRACT FROM AUTHOR]

Highlights • Predator-Prey model in which only the prey species is infected with some disease, is proposed and analyzed. • D. Mukherjee model is extended by incorporating the recovery rate in the infected prey species and the possibility of Hopf bifurcation around non zero equilibrium point is also discussed. • Role of recovery of infected prey species in eliminating the limit cycle oscillations and thus making the interior equilibrium point stable, is studied. • A monitory agency has been introduced which monitors the exploitation of resources by implementing certain tax for each unit biomass of the predator population. • The optimization of the total economical net profits from harvesting of predator species, taking taxation as control parameter, has been done. Abstract In this paper, we present and analyze a prey-predator system, in which prey species can be infected with some disease. The model presented in this paper is motivated from D. Mukherjee's model in which he has considered an SI model for the prey species. There are substantial evidences that infected individuals have the ability to recover from the disease if vaccinated/ treated properly. In this regard, Mukherjee's model is modified by considering SIS model for prey species. Theoretical and numerical simulations show that the recovery of infected prey species plays a crucial role in eliminating the limit cycle oscillations and thus making the interior equilibrium point stable. The possibility of Hopf bifurcation around non zero equilibrium point using the recovery rate as a bifurcation parameter, is discussed. Further, the model is extended by incorporating the harvesting of predator population. A monitory agency has been introduced which monitors the exploitation of resources by implementing certain taxes for each unit biomass of the predator population. The main purpose of the present research is to explore the effect of recovery rate of prey on the dynamics of the system and to optimize the total economical net profits from harvesting of predator species, taking taxation as control parameter. [ABSTRACT FROM AUTHOR]

This paper proposes a stochastic delay model for a mechanical centrifugal governor system with noise. The Hopf bifurcation of the system is obtained with the delay as a parameter. Considering the effect of delay, the stochastic bifurcation of the system is obtained, as well as the stochastic bifurcation diagrams of the model by numerical analysis. It is found that large delays can lead to the occurrence of stochastic bifurcations, which destabilize the system. Moreover, retaining the quadratic term of ϵ induces a new dynamical phenomenon - stochastic P-bifurcation. When spectral density is used as a bifurcation parameter, the periodicity of the system is influenced. The results of this paper are a reference value for preventing instability in mechanical systems. Meanwhile, the theoretical work and numerical simulations in this paper can be extended to other systems. [ABSTRACT FROM AUTHOR]

To study the current trend in the society of online gaming addiction. A two delayed social networking game addiction model with categorized contagion rate is presented in this paper. The influence of the two delays and sufficient prerequisites for the reality of Hopf bifurcation are explored by means of involving the feasible combos of the double delays as a bifurcation limitation. Furthermore, specific formulae for deciding direction of the Hopf bifurcation and permanence of the bifurcating periodic options are obtained by exhausting normal form theory and centre manifold theory. To end with, some numerical replications are conceded out with the assist of MAT LAB software program in directive to interpretation for the approve of theoretical results. • A two delayed social networking game addiction model. • Sufficient prerequisites for the reality of Hopf bifurcation are explored. • Specific formulae for deciding direction of the Hopf bifurcation and permanence of the bifurcating periodic options are obtained. • Numerical replications are conceded. [ABSTRACT FROM AUTHOR]

In the classical evolutionary game theory, mutation is usually considered as a constant, however strategy mutation is affected by strategies in the real game process. Therefore, the main purpose of this paper is to study the effects of mutation feedback and time delays on strategy dynamics, where mutation is a linear feedback related to strategy. First, we construct a co-evolutionary game model with two time delays induced by mutation feedback and analyze the existence and stability of the equilibria of the non-time delay system. The conditions for the coexistence of the two strategies and the mutation rate are obtained. Second, Sotomayor's theorem is used to explore the transcritical bifurcation of the system. Then, the existence of Hopf bifurcation is investigated by using feedback delay and payoff delay as bifurcation parameters in the time-delay system. Furthermore, we discuss the direction of the Hopf bifurcation, stability and periodic change of the periodic solution in detail. Finally, a series of numerical simulations are used to describe the theoretical analysis. The main results are as follows: (i) Mutation causes negative feedback to cooperative strategy. (ii) As the time delay increases, the stable equilibrium point becomes unstable, and which branches a stable limit cycle. When the time delay continues to increase sufficiently, the stable limit cycle becomes unstable and produces irregular oscillation and chaos. (iii) When the two time delays are large enough, the coexistence of the two strategies becomes that defective strategy is dominant, and the mutation rate also reaches the maximum. • A co-evolutionary game model with time delays and mutation feedback is proposed. • Mutation is not a constant but is affected by strategy. • Mutation causes negative feedback to cooperative strategy. • The time delay is large enough, strategies produce irregular oscillations and chaos. • Two delays are large enough, strategies' coexistence becomes the dominant defect. [ABSTRACT FROM AUTHOR]

The phenomenon of algal blooms caused by global warming frequently occurs in global water bodies, however, the mechanism by which temperature affects algal growth is still unclear. In this paper, we investigate the dynamics of a deterministic algae–zooplankton model with water temperature, as well as its corresponding stochastic version. Mathematical theoretical work mainly analyzes the stability, persistence and Hopf bifurcation of the deterministic model, and performs a survival analysis on the stochastic model and shows the existence of a unique ergodic stationary distribution, which in turn provides a theoretical basis for numerical simulations. Numerical analysis indicates that the variation of temperature can generate complex impacts on the fundamental mechanisms underlying the growth of algae, whether in deterministic or stochastic environment. One of the most interesting results indicates that the decrease of temperature can destabilize the algae–zooplankton interactions, but increase their permanence; the increase of temperature can stabilize the algae–zooplankton interactions, but reduce their permanence. [ABSTRACT FROM AUTHOR]

In this paper, a time-delayed coupled FitzHugh–Rinzel neuron system (two neurons) with electrical and chemical couplings is discussed. First, when choosing different strengths of electrical (or chemical) coupling, the number and position of equilibria of the neuron system without time delays affected by the variation of chemical (or electrical) coupling are investigated in detail. Next, based on the discussion about the equilibria of the coupled neuron system without time delays, the codimension one bifurcations of equilibria or limit cycles including static bifurcation, Hopf bifurcation and fold bifurcation of limit cycles are deduced when choosing the strength of electrical coupling or chemical coupling as the bifurcation parameter, respectively. Furthermore, the codimension two bifurcations of equilibria are discussed which exhibit more fascinating and complicated dynamical behaviors. Synchronization problems influenced by the strength of electrical coupling are also considered which contain the complete synchronous and asynchronous state. After the discussion of the neuron system without time delays, the impact of time delays on the stability of symmetric equilibria of coupled FitzHugh–Rinzel neuron system is explored. The phenomenon of stability switching by a Hopf bifurcation is well detected. Delay-dependent Hopf curves are found and the dynamical behaviors near the intersection point of Hopf bifurcation (Hopf–Hopf bifurcation point) in the parameter plane of two time delays are investigated where there exists the coexistence of two limit cycles with different frequencies. The numerical simulations are exhibited after the discussion of each part. • The dynamics of a time-delayed coupled FitzHugh-Rinzel neuron system with electrical and chemical couplings is discussed. • The co-existence of equilibria, codimension one and two bifurcations, synchronization problems are considered. • Delay-dependent Hopf curves and dynamical behaviors near the Hopf-Hopf bifurcation point are investigated. [ABSTRACT FROM AUTHOR]

Synthetic drug addiction represents an escalating worldwide concern. In this paper, we introduce a comprehensive synthetic drug addiction model that considers both psychosocial and environmental factors. Additionally, our model encompasses aspects related to relapse and the drug supply chain. Through a dynamic analysis of this model, we uncover a multitude of intricate dynamical phenomena, including saddle–node bifurcation, Hopf bifurcation. Meanwhile, we have also obtained an elliptic-type nilpotent singularity. Furthermore, we provide bifurcation diagrams and corresponding phase diagrams to offer epidemiological insights into these intricate dynamic phenomena. [ABSTRACT FROM AUTHOR]

The complex mixed-mode oscillation patterns are proposed and analyzed in a modified Rayleigh-Duffing oscillator based on the bifurcation theory in this paper. Four mixed-mode oscillations, namely "Homoclinic/Homoclinic-Homoclinic/Homoclinic" intermittent type, "fold/Homoclinic-Homoclinic/supHopf" intermittent type, "fold/Homoclinic-supHopf/supHopf" intermittent type and "fold/Homoclinic" type, are discussed in detail. Considering the low frequency excitations as slow-changing state variables, a modified autonomous system is obtained. The bifurcation characteristics of the fast subsystem are presented briefly by using the bifurcation theory. Then, we investigate the generation principle of the bifurcation delay phenomenon that performs a critical role in the production of two mixed-mode oscillations. This paper presents a fact that the dynamical behaviors are sensitive to the values of the system parameters and the parameters determine different forms of the repetitive spiking states that leads to different patterns of the mixed-mode oscillations. In addition, the theoretical analysis and numerical simulations are compared to illustrate the correctness of this paper. • A route to mixed-mode oscillations named "Hopf bifurcation delay" is proposed. • A modified autonomous system is obtained by considering the low frequency excitations as slow-varying state variables. • Two Hopf-bifurcation-delay-triggered mixed-mode oscillations and two pure bifurcation-structure-induced mixed-mode oscillations are investigated. • Dynamical evolutions among different bursting patterns are studied by the two-parameter bifurcation diagram. [ABSTRACT FROM AUTHOR]

• A diffusive plankton model with defensive and offensive effects is considered. • Hopf bifurcations of plankton model without and with diffusion are studied. • Turing instability of diffusive model is analyzed. • The impacts of defense and offense on the coexistence of plankton are discussed. This paper investigates the dynamic behavior of a diffusive plankton model with defensive and offensive effects in two cases. For the single compartment model, we first derive the sufficient conditions for the stability and Hopf bifurcation of coexisting equilibrium, which implies that the changes of defense and offense can cause oscillation of planktonic population. Then the properties of Hopf bifurcation are discussed by center manifold theorem. For the spatially extended model, we obtain the sufficient conditions for Turing instability and Hopf bifurcation. It is observed that spatial patterns put in place, under the interaction of diffusion, defense and offense. Finally, some numerical simulations are carried out to support the analytical results. [ABSTRACT FROM AUTHOR]

• Four mathematical models for derivation of CSA of surge chamber are constructed. • Stability of hydropower station with surge chamber is analyzed. • Formulas for CSA of surge chamber are derived. • Verification and comparison of different CSAs of surge chamber are conducted. • Correctness and rationality of formulas for CSA of surge chamber are explained. This paper aims to study the critical sectional area (CSA) of surge chamber considering the nonlinearity of head loss of diversion tunnel and steady output of turbine. Firstly, three basic equations for hydropower station with surge chamber are established. Four mathematical models for the derivation of CSA of surge chamber are constructed. Then, the stability of hydropower station with surge chamber is analyzed by Hopf bifurcation. Based on the critical stable state of hydropower station, the formulas for the CSA of surge chamber are derived. Finally, the verification and comparison of different CSAs are conducted. The correctness and rationality of obtained formulas are explained. The results indicate that, under load increase operation condition, both the nonlinearity of head loss of diversion tunnel and nonlinearity of steady output of turbine can reduce the value of CSA of surge chamber and are favorable for the stability of hydropower station. Under load decrease operation condition, the rules are opposite. Under both load increase and load decrease operation conditions, the effect of the nonlinearity of head loss of diversion tunnel on CSA of surge chamber is much more significant than that of the nonlinearity of steady output of turbine. The formula for CSA of surge chamber considering both the nonlinearity of head loss of diversion tunnel and nonlinearity of steady output of turbine can be expressed as an amplification coefficient times of the Thoma formula. That formula has a higher precision than Thoma formula. [ABSTRACT FROM AUTHOR]

• A delayed epidemic model with logistic growth and saturated treatment is studied. • Backward bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation may occur. • Effects of varying parameters are studied in numerical simulations. • Presence of logistic growth and high treatment rate can decrease infection levels. In this paper, we incorporate a nonlinear incidence rate and a logistic growth rate into a SIR epidemic model for a vector-borne disease with incubation time delay and Holling type II saturated treatment. We compute the basic reproduction number and show that it completely determines the local stability of the disease-free equilibrium. Sufficient conditions for the existence of backward bifurcation and Hopf bifurcation are also established. Furthermore, we determine the direction and stability of the Hopf bifurcation around the endemic equilibrium by means of the center manifold theory. Our study reveals that the model admits a Bogdanov–Takens bifurcation when the time delay and the maximal disease transmission rate are varied. Numerical simulations are presented to illustrate the dynamics of the model and to study the effects caused by varying the treatment rate and delay parameters. [ABSTRACT FROM AUTHOR]

The fear effect on prey populations due to the presence of a predator's sound or smell can be more impactful than direct predation. In this paper, we formulate a diffusive predator–prey model incorporating the nonlocal fear effect. The existence and boundedness of solutions are showed. We also study the stability of constant steady states by using the characteristic equation and Lyapunov functionals. Steady-state bifurcations are carried out in detail by the Lyapunov-Schmidt method. The analyses demonstrate that the fear effect in the system can change the stability of the constant steady state and there exist spatially nonhomogeneous steady states. Hopf bifurcations are also investigated and the results show that high levels of fear can stabilize the model by excluding the existence of periodic solutions. Finally, numerical simulations are sketched to illustrate our theoretical findings. This work can help us further understand the impact of fear effect on the stability of constant steady states, steady-state bifurcations and Hopf bifurcations of predator–prey models. • A diffusive predator–prey system incorporating nonlocal fear effect is formulated. • The existence and boundedness of solutions are discussed. • We investigate the local and global stability of constant steady states. • Steady-state bifurcations and Hopf bifurcations are also studied. • Numerical simulations are sketched to illustrate our theoretical findings. [ABSTRACT FROM AUTHOR]

This paper investigates the pattern dynamics of a harvested predator–prey model with no-flux boundary conditions. Firstly, we analyze the positive equilibrium types of the local temporal model. We find that they can be classified as nodes, foci, or centers depending on the harvesting coefficient within a certain parameter range. Furthermore, the direction of the Hopf bifurcation is determined by employing the first Lyapunov coefficient. In the subsequent analysis, we present the conditions for the existence of Turing instability and classify the different pattern selections using amplitude equations with the assistance of weakly nonlinear analysis by treating the harvesting coefficient as a critical parameter. Finally, the spot patterns and mixed patterns are respectively displayed in 2D space, on spherical and torus surfaces with various harvesting coefficient values. Especially, we can numerically demonstrate that the diffusion rate of the prey population will strongly affect the pattern structures of the model. These results can provide a reference for understanding the interaction dynamics of the model. [ABSTRACT FROM AUTHOR]

For the development of the theoretical framework of slow–fast dynamics, it is essential to study the mechanism of bursting oscillations with high co-dimensional bifurcations in the high-dimensional vector field. This paper is aimed to explore the bursting oscillations in the normal form up to the third order of a vector field with triple Hopf bifurcation at the origin. We establish an eight-dimension system with the slow–fast effect, in which two time scales exist. The fast subsystem is represented as a six-dimensional vector field with triple Hopf bifurcation at the origin, while the slow subsystem is adopted to adjust the variation of parameters in the fast subsystem, which may lead to abundant dynamic behaviors. To better understand the qualitatively different behaviors in the fast subsystem, the trivial solution, single-mode, two-mode, and three-mode movements are introduced based on the projections of trajectory on the three sub-planes. As a result, several types of bursting phenomena, such as Hopf/Hopf bursting oscillations with single-mode, Hopf/Hopf/fold bursting oscillations with single-mode, Hopf/Hopf bursting oscillations with two-mode have been revealed, the mechanism of which is obtained upon the superposition of equilibrium branches together with bifurcations in the fast subsystem as well as transformed phase portrait. In addition, for bursting oscillations on the whole phase space, the amplitude of spiking oscillations may change according to two different limit cycles, which leads to an abrupt change of the oscillating amplitude on the sub-plane, corresponding to different projections of Poincaré. It can be observed the state variables on the different sub-planes alternate between synchronized and non-synchronized states in the fast subsystem, which may be accounted for by the bifurcations. • Hopf bifurcation leads to transitions between different single modes or between single-mode and two-mode. • Synchronization and non-synchronization between state variables on different sub-planes occur. • Complexity mechanisms of the slow–fast dynamics are investigated. • Bursting oscillations behave in quasi-periodic types instead of the high dimensional torus. [ABSTRACT FROM AUTHOR]

In this paper, we consider a predator–prey model with fear response delay, gestation delay, fear effect, prey refuge, harvesting and Crowley–Martin type functional response. First, we discuss the model without delays and corroborate the positivity and boundedness of the solution. Then, we give some sufficient conditions for the existence and stability of three equilibriums. For the model with delays, we not only analyze the local stability of the positive equilibrium and the occurrence of Hopf bifurcation, but also obtain the crossing curves to study the stability switches of the positive equilibrium on the delays plane. Furthermore, we calculate the normal form of Hopf bifurcation and hence get the direction of Hopf bifurcation and the stability of bifurcated periodic solutions. At last, we support our findings by numerical simulations. • Considering the cost of fear into both reproduction and death term of prey. • Fear that influence birth and death rate of prey have similar impact on populations. • The stability crossing curve method is creatively applied to the model. • The model can exhibit stability switches on the delays plane. [ABSTRACT FROM AUTHOR]

Although there are many factors that influence population fluctuations, the impact of delay on population dynamics is one of the most significant ones. In this paper, we explore how the Allee effect in combination with a double delay (competition and cooperation) affects predator–prey population dynamics. One can determine the requirements for the equilibrium's asymptotic stability and the presence of Hopf bifurcations by investigating the roots of a characteristic equation. The rigorous mathematical proofs of the stability and direction of the Hopf bifurcating periodic solutions are derived with the help of the normal form theory and the center manifold theorem. Numerical simulation shows that different delay mechanisms can generate different behaviors, such as the competitive delay that produces regular and irregular oscillations, as well as the chaotic, while the cooperative delay results in the stability switch. These theoretical and numerical results are illustrated that depending on the chosen delayed mechanism, delay may not necessarily lead to instability, and also can help for understanding the biological implications corresponding to the interaction dynamics of predator and prey. [ABSTRACT FROM AUTHOR]

Our main goal in this paper is to study the number of small-amplitude isolated periodic orbits, so-called limit cycles, surrounding only one equilibrium point a class of polynomial Kolmogorov systems. We denote by M K (n) the maximum number of limit cycles bifurcating from the equilibrium point via a degenerate Hopf bifurcation for a polynomial Kolmogorov vector field of degree n. In this work, we obtain another example such that M K (3) ≥ 6. In addition, we obtain new lower bounds for M K (n) proving that M K (4) ≥ 13 and M K (5) ≥ 22. • Considering Kolmogorov systems of degree 2, we give the second example where 6 limit cycles bifurcating from the origin. • For Kolmogorov systems of degree 3, we show that we can obtain, at least, 13 limit cycles bifurcation from the origin. • For Kolmogorov systems of degree 4, and we offer an example where it is possible to obtain 22 limit cycles. • In addition, we show a study of lower bounds of limit cycles for families of Kolmogorov systems of degrees 3 and 4. [ABSTRACT FROM AUTHOR]

This paper proposes and studies the dynamics of a Holling-type II predator–prey interaction system that incorporates the following three components: (1) a prey refuge; (2) predation fear and its carry-over effects; and (3) environmental noise in both prey and predator populations. The impacts of those three components are studied through both rigorous analysis and numerical simulations. Analytical results show that the introduction of prey refuge, predation fear, and its carry-over effects can generate Hopf bifurcation. It is found that increasing prey refuge and predation fear effect in a reasonable region can stabilize the system, while excessive refuge strength would lead to the extinction of predators. The theoretical results of the corresponding system with environmental noise include (1) sufficient conditions for the existence of a unique ergodic stationary distribution of the SDE system by constructing appropriate stochastic Lyapunov functions; (2) the explicit probability density function of the distribution by solving the Fokker–Planck equation; and (3) the extinction conditions of prey and/or predator species at an exponential rate in the long run. Our work shows that the proposed model, incorporating prey refuge, predation fear, carry-over effect, and environmental noise, exhibits rich and complex dynamic behaviors. Moreover, our results indicate that small environmental noise can save the prey and predator from extinction, while large environmental noise can drive the species to extinction. These interesting findings provide more perspectives on the protection and control of species in complex communities. • A predation system comprises a refuge, fear and carry-over effects, and environmental noise. • Refuge and fear induce Hopf bifurcation, while excessive refuge strength can lead to extinction. • A stationary distribution, probability density function and extinction conditions are obtained. [ABSTRACT FROM AUTHOR]

• We study the bio-economic harvesting of an intraguild fishery model via independent harvesting strategies. • Bifurcation and numerical analyses are presented to show distinctive result that a Hopf bifurcation occurs with respect to harvesting parameter instead of resource enrichment parameter as in the literature. • Trivial and nontrivial bionomic equilibrium solutions are investigated with certain possibilities and restrictions. • Optimal harvesting policy is explored by utilizing the Pontryagin Maximum Principle. • Harvesting parameter is the key element for the stability. Latterly, variable carrying capacity in the predator–prey population models has been widely studied to provide more realistic understanding about population dynamics. In the present paper, we discuss the bio-economic harvesting of an intraguild fishery model where the logistic carrying capacities of both predator and prey fish populations are variable based on the shared biotic resource. Distinct from most fishery models, we incorporate the independent harvesting strategies on the fisheries with different harvesting efforts by assuming them to have different economic values. In our model, the prey fish population is assumed to be infected directly by an anthropogenic toxicant from the environment while the predator is infected indirectly when they feed on prey. We investigate the local stability of trivial equilibria with Routh–Hurwitz criterion and the global stability of non-trivial equilibrium by constructing a Lyapunov function. Bifurcation and numerical analyses are presented to show distinctive result that a Hopf bifurcation occurs with respect to harvesting parameter instead of resource enrichment parameter as in the literature. Bionomic equilibrium is explored with several possibilities and restrictions. The nontrivial bionomic equilibrium is found to depend critically on the resource density. Finally, the optimal harvesting policy of the three dimensional model is derived by utilizing Pontryagin Maximum Principle. From this study, harvesting parameter is a crucial factor for the stability of the intraguild system. The objective is to obtain the optimal sustainable yield that provides maximum monetary profit while conserving the marine ecosystem. [ABSTRACT FROM AUTHOR]

• We provide an epidemic SIR model with long-range temporal memory governed by delay differential equations with fractional-order. • The existence of steady states and the asymptotic stability of the steady states are discussed. • The occurrence of Hopf bifurcation is captured when the time-delay passes through a critical value. • Theoretical results are validated numerically. In this paper, we provide an epidemic SIR model with long-range temporal memory. The model is governed by delay differential equations with fractional-order. We assume that the susceptible is obeying the logistic form in which the incidence term is of saturated form with the susceptible. Several theoretical results related to the existence of steady states and the asymptotic stability of the steady states are discussed. We use a suitable Lyapunov functional to formulate a new set of sufficient conditions that guarantee the global stability of the steady states. The occurrence of Hopf bifurcation is captured when the time-delay τ passes through a critical value τ *. Theoretical results are validated numerically by solving the governing system, using the modified Adams-Bashforth-Moulton predictor-corrector scheme. Our findings show that the combination of fractional-order derivative and time-delay in the model improves the dynamics and increases complexity of the model. In some cases, the phase portrait gets stretched as the order of the derivative is reduced. [ABSTRACT FROM AUTHOR]

• We firstly introduce a fractional-order PD scheme to control the Hopf bifurcation caused embedded in congestion control system. • The condition of stability, the criterion of Hopf bifurcation and the range of periodic oscillations for the congestion control system are derived. • The initiation of the inherent Hopf bifurcation can be effectively delayed or advanced by setting appropriate proportional and differential gain parameters. In this paper, we consider the control of nonlinear dynamic in a congestion control system. A novel fractional-order proportional-derivative (PD) feedback law is firstly designed to control the Hopf bifurcation caused by the system. The proposed P D 1 n controller has the different order with the original congestion system. Meanwhile, the communication delay is elected as the bifurcation parameter to study the stability, bifurcations and periodic oscillations of the controlled congestion system. By the stability mechanism of fractional-order systems, we can obtain the criteria for satisfying the stability and Hopf bifurcation. It has been discovered that the Hopf bifurcation point can be delayed or advanced under the adjustment of appropriate control gain parameters and the order. Therefore, the congestion system becomes controllable and the desirable behaviors can be realized. Finally, numerical simulations are presented to demonstrate the validity of the main results and the efficiency of the control strategy in the designed fractional-order PD controller. [ABSTRACT FROM AUTHOR]

• The fractional Chen system is analyzed. • Stability properties with respect to free entry parameters are described. • A fractional Hopf bifurcation is investigated rigorously. • Other bifurcation phenomena are observed numerically. • The occurrence of a chaotic attractor for low derivative orders is discussed. The paper provides a theoretical analysis of some local bifurcations in the fractional Chen system. Contrary to the integer-order case, basic bifurcation properties of the fractional Chen system are shown to be qualitatively different from those described previously for the fractional Lorenz system. Further, the fractional Hopf bifurcation in the Chen system is expressed rigorously with respect to general entry parameters. Based on these observations, some particularities of the fractional dynamics of the Chen system are documented and its chaotic behavior for low derivative orders is discussed. [ABSTRACT FROM AUTHOR]

• Imposition of environmental tax plays a vital role in dynamics of greenhouse gases. • Corruption fades the effects of ecological taxation policy. • Delay in the imposition of environmental tax may unstabilize the system. • Stability of the system can be maintained by quantified applications of fair taxation policy. In this research paper, we plan to study the relation among concentration of greenhouse gases, human population and environmental tax. For this purpose, a deterministic nonlinear mathematical model is proposed and analyzed in regard to the boundedness and persistence of its solutions, equilibria and their stabilities. Whenever, a policy is implemented, an automated delay comes into existence because outcomes of application of any policy always take some time to become visible. To see the effects of delay, analyses for the stability and direction of Hopf bifurcation for delay system are done. Critical value of the delay parameter is calculated theoretically and numerically, and then verified graphically. For verifications and descriptions of analytical findings, numerical simulations are performed. Graphical comparisons for different values of various parameters provides us some realistic and interesting results. Overall, model analysis shows that the increasing level of greenhouse gases can be controlled by applying environmental tax and imposing some restrictions on corruption. [ABSTRACT FROM AUTHOR]

• The dynamics of predator-prey system with two time delays and Monod–Haldane response function are considered. • The local stability of the interior equilibrium is established and the conditions for existence of the Hopf bifurcation are obtained. • The direction of the Hopf bifurcation and the stability of the bifurcation period solutions are investigated. • The Hopf bifurcation of the original system is controlled by using parameter disturbance and state feedback control. • The numerical examples are given to carry out to support the obtained theoretical findings. In this paper, the dynamics of a stage-structured predator-prey system with two time delays and Monod–Haldane response function are considered. By taking the two time delays as the bifurcation parameter, the local stability of the interior equilibrium is established and the conditions for existence of the Hopf bifurcation are obtained. Furthermore, based on the normal form method and center manifold theorem, the direction of the Hopf bifurcation and the stability of the bifurcation period solutions are investigated. In addition, by using parameter disturbance and state feedback control to act on the system, we succeeded in controlling the Hopf bifurcation of the original system. Numerical examples are given to carry out to support the obtained theoretical findings. [ABSTRACT FROM AUTHOR]

• The model considering with different fractional derivative definitions give rise to different behaviours. • Hopf bifurcation and chaos exists for the model considering with conformable fractional order derivative. • Source rate of immune cells plays critical role in cancer Dynamics. In this paper, tumor-immune system interaction has been considered by two fractional order models. The first and the second model consist of system of fractional order differential equations with Caputo and conformable fractional derivative respectively. First of all, the stability of the equilibrium points of the first model is studied. Then, a discretization process is applied to obtain a discrete version of the second model where conformable fractional derivative is taken into account. In discrete model, we analyze the stability of the equilibrium points and prove the existence of Neimark-Sacker bifurcation depending on the parameter σ. Moreover, the dynamical behaviours of the models are compared with each other and we observe that the discrete version of conformable fractional order model exhibits chaotic behavior. Finally, numerical simulations are also presented to illustrate the analytical results. [ABSTRACT FROM AUTHOR]

• Dynamical behaviour of a Leslie–Gower model with Holling type-IV functional response and nonlinear age-selective prey harvesting has been studied. • Qualitative analysis, bifurcation theory, bionomic equilibrium, MSY and optimal harvesting policy has been analyzed. • The age-selective harvesting system under goes a Hopf bifurcation with respect to the maturation delay. • Conditions for the MSY and existence of bionomic equilibrium are analyzed and Pontryagin's maximum principle has been used to find the path of a singular control and Pontryagin's maximum principle has been used to find the path of a singular control. In this paper, we have studied the dynamical behaviour of a Leslie-Gower model with Holling type-IV functional response and nonlinear prey harvesting where target prey for harvesting are above a particular age which is called maturation delay. The model system has been studied using qualitative analysis, bifurcation theory, bionomic equilibrium, MSY and optimal harvesting policy. We explain that the interior equilibrium is delay dependent and locally asymptotically stable. The age-selective harvesting system under goes a Hopf-bifurcation with respect to the maturation delay. Conditions for the MSY and existence of bionomic equilibrium are analyzed and Pontryagin's maximum principle has been used to find the path of a singular control. Difficult analytical results are featured by numerical examples to better understand the system in a simple way. [ABSTRACT FROM AUTHOR]

Highlights • A diffusive predator-prey system with time-delay and competitive interference is considered. • Time delay induced instability and Hopf bifurcation at coexisting equilibrium are obtained. • The direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are determined. Abstract In this paper, we studied a diffusive predator-prey model with competitive interference and Crowley-Martin type functional response. The conditions for local stability of coexisting equilibrium are given by analyzing the eigenvalue spectrum. By using delay as bifurcation parameter, conditions for occurrence of Hopf bifurcation are also given. The property of bifurcating period solutions is investigated by calculating the normal form. Some numerical simulations are performed to support our theoretical result. Our conclusions show that diffusion and delay are two factors that should be considered in establishing the predator-prey model, since they can induce spatially bifurcating period solutions. [ABSTRACT FROM AUTHOR]

Highlights • We study the impact of herd shape on the the prey and predator equilibrium densities. • In the absence of the time delay we studied the local stability and bifurcation. • The effect of the time delay on the stability of the interior equilibrium where Hopf bifurcation is obtained. Abstract In this paper, a delayed predator-prey model with the presence of a social behavior for the prey population has been investigated. A new functional response is obtained. We studied the effect of the herd shape for the prey population on the prey and predator equilibrium densities. The analysis of the system has been also established where the boundedness, stability, Hopf bifurcation, stability of Hopf bifurcation are obtained. [ABSTRACT FROM AUTHOR]

Highlights • Three fractional-order discrete chaotic maps are proposed. • Important dynamics of three maps are investigated. • A stabilizing strategy is introduced for the proposed maps. • A combined synchronization scheme is presented. Abstract In this paper, we propose three systems, namely the fractional Lozi map, the fractional Lorenz map, and the fractional flow map. We study some of the important dynamics of these maps including the bifurcation graphs. In addition, we propose control laws aimed at stabilizing and synchronizing a combination of these three maps. The convergence of the stabilized states and the synchronization errors is established by means of the linearization method. Numerical simulations are also presented to confirm the main findings of the study. [ABSTRACT FROM AUTHOR]

Highlights • Influence of group defense and prey harvesting in predator-prey model is discussed. • Harvesting rate plays a crucial rule for coexistence of both the species. • Sufficient conditions for the existence and stability of equilibria are established. • Occurrence of several bifurcations is proved. • Numerical simulation shows chaotic and periodic oscillations of the system. Abstract We discuss the stability and bifurcation analysis of predator-prey model in the presence of group defense and non-linear harvesting in prey. Mathematically, we analyze the dynamics of the system such as boundedness of the solutions, existence and stability conditions of the equilibrium points. The model undergoes saddle-node, transcritical and Hopf-Andronov bifurcations. The direction of Hopf bifurcation by calculating the first Lyapunov number is examined. The effects of prey harvesting rate and death rate of predator on the model by considering them as bifurcation parameters are analyzed. In this paper, we dedicate ourselves to the investigation of the complex dynamics of the model to maintain the coexistence of the species which is important for ecological balance in the real environment. Several numerical simulations are performed to substantiate our analytical findings. [ABSTRACT FROM AUTHOR]

Abstract The stability and bifurcation behavior of a kind of active magnetic bearing rotor are investigated in this paper. The analysis is carried out both analytically and numerically. It is found that a Hopf bifurcation occurs in the system by using center manifold and normal form. Numerical simulation is conducted to validate the theoretical predictions. More precisely, the dynamic characteristics of this system in two-dimensional parameter space are analyzed, the phase diagrams assist us to identify multi-attractor coexisting that makes the dynamical behaviors of the system become more enrich and complex. These results we represent can be useful in designing and selection of suitable operating parameters. As a result, the system can avoid the undesirable behavior. [ABSTRACT FROM AUTHOR]

Highlights • A network with wireless access links with a delay differential equation (DDE) model and input delay is considered. • Integral sliding mode control is assumed as the nonlinear robust active queue management in the network. • Hopf bifurcation analysis is investigated with the transmission delay considered as the Hopf bifurcation parameter. • The direction and the stability of the bifurcation periodic solutions are determined. • Simulation results are provided to illustrate the system behavior with different parameters. Abstract Transmission control protocol (TCP), which is usually implemented in the transport layer of a communication network, can detect the congestion after its occurrence. But congestion control can be more complicated in networks with wireless access links due to time-varying fading and packet error rate (PER). Thus, it is essential to use robust active queue management (RAQM) to prevent congestion with certain tolerance against serious disturbance of the wireless environment. As the stability of a TCP/RAQM is tremendously related to the transmission delay, this paper focuses on Hopf bifurcation analysis of a network governed by nonlinear delayed differential equations and an integral sliding mode control as a nonlinear RAQM. The transmission delay is assumed as the bifurcation parameter. It is shown that if the delay passes through a critical value, the closed loop system would not remain stable and the Hopf bifurcation occurs. Moreover, based on the normal form theory and the center manifold theorem, the direction and the stability of the bifurcation periodic solutions are determined. Finally, simulation results are given to confirm the analytical achievements and to show the effectiveness of the procedure compared to some recent protocols. [ABSTRACT FROM AUTHOR]

Abstract Stability and Hopf bifurcation of a class of delayed complex-valued neural networks are investigated in this paper. First, using proper translations and coordinate transformations, we decompose the activation functions and connection weights into their real and imaginary parts, so as to construct an equivalent real-valued system. Then, the sufficient conditions for Hopf bifurcation and its directions are provided through normal form theory and central manifold theorem. In the end, some numerical simulations are given to illustrate the correctness of the results. [ABSTRACT FROM AUTHOR]

Highlights • Local and global stability analysis. • Numerical technique for fractional-order reaction–diffusion equation. • Modelling with the Caputo and Atangana–Baleanu fractional derivatives. • Numerical simulation results in one and two dimensions for different values of fractional power. Abstract In this paper, we numerically study a nonlinear time-fractional reaction-diffusion equation involving the Caputo and Atangana–Baleanu fractional derivatives of order α ∈ (0, 1). A novel algorithm known as the Laplace Adams–Bashforth method is formulated for the approximation of these derivatives. In the simulation framework, a tri-tropic food chain system is considered in which the classical time-derivatives are replaced with non-integer order derivatives. Mathematical analysis of the main system is examined for both stability and Hopf-bifurcations to occur. Numerical simulation results show the existence of chaotic behaviours and spatiotemporal oscillations as well as the emergence of some Turing patterns (such as, spots and stripes) in two-dimensional space. [ABSTRACT FROM AUTHOR]

Highlights • A two-step family of Adams–Bashforth method is derived for the approximation of the Atangana–Baleanu fractional derivative. • Mathematical analysis and numerical simulation of time-fractional multicomponent systems • Application of the Atangana–Baleanu derivative based on nonlocal and nonsingular kernels. • Simulation results showing chaotic behaviors for different values of alpha. Abstract In this paper, we consider the mathematical analysis and numerical simulation of time-fractional multicomponent systems. Here, the classical time derivatives in such systems are replace with the Atangana–Baleanu fractional derivative in the sense of Caputo. This derivative is found useful in the sense that it combines both the non-local and nonsingular kernels in its formulation. A two-step family of Adams–Bashforth method is derived for the approximation of the Atangana–Baleanu derivative. Numerical experiments presented for different instances of α , 0 < α ≤ 1 correspond to our theoretical findings. [ABSTRACT FROM AUTHOR]

• We considered an opinion model of threshold-type delay differential equations. • We proved that the model with either a transcritical forward or a backward bifurcation, can exhibit Hopf bifurcation. • We showed that the obtained Hopf bifurcation appears only on upper bifurcated branches of the equilibria. • We performed numerical simulations to illustrate and support our theoretical results. In a recent paper [R. Qesmi, Dynamics of an opinion model with threshold-type delay, Chaos, Solitons & Fractals 98 (2020). (https://doi.org/10.1016/j.chaos.2020.110379)], we proposed a mathematical model of threshold-type delay differential equations describing the relationship between two subpopulations with opposite opinions and the opinion spread dynamics. The study there showed the possibility of a transcritical forward and backward bifurcations of positive equilibria. In the present paper, we show that the opinion model undergoes a Hopf bifurcation through which one of the bifurcation branches loses the stability and periodic solutions appear. One of the important consequences of the obtained dynamics is that the consensus of the both opinions could be lost by maintaining the balance between the time taken for an individual to become convinced of the outsider opinion, which need be short, and the number of individuals converted to the local opinion which need be low. Finally, we provide numerical simulations to illustrate and support our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a cross-diffusion predator–prey system with Holling type functional response. We study the local stability, Turing instability, spatial pattern formation, Hopf and Turing–Hopf bifurcation of the equilibrium. Numerical simulation with zero-flux boundary conditions discloses that the system under consideration experiences the occurrence of cross-diffusion-driven instability. The dynamical system in Turing space emerges spots, stripe-spot mixtures and labyrinthine patterns, which reveals that the interaction of both self- and cross-diffusions play a significant role on the pattern formation of the present system in a way to enrich the pattern. We obtain the normal form of the Turing–Hopf bifurcation and observe that the system has stably spatially homogeneous periodic solutions, stable constant and nonconstant steady-state solutions, which indicates that the intrinsic growth rate coefficient and the environmental carrying capacity coefficient are two important factors for predator–prey system, and affect the stability of predator–prey system. • A cross-diffusion predator–prey system with habitat complexity and Holling type functional response is considered. • The Turing pattern is studied by the amplitude equation. • The Turing–Hopf bifurcation is investigated by using the method of the normal form and the center manifold theory. [ABSTRACT FROM AUTHOR]

The aim of this paper is to reveal the dynamical mechanism of bursting oscillations in non-smooth dynamical systems, and the interest is focused on the effects of period-doubling bifurcation and chaotic attractor on the bursting oscillations. By modifying a fourth-order Chua's circuit, a dynamical system possessing non-smooth switching manifold and multiple scale variables, is established. By taking the slow-varying variable as a bifurcation parameter, Subcritical non-smooth Hopf bifurcation, C-bifurcation, and period-doubling bifurcation are obtained in the fast subsystem. Further more, chaotic attractors, which are generated from a series of period-doubling bifurcations, are also observed. Based on the numerical simulations and bifurcation analysis of the fast subsystem, eight typical bursting patterns are obtained, and their dynamical mechanism is revealed. Study shows that period-2 limit cycle in the fast subsystem may cause the slow–fast system to oscillates in alternating small and large amplitudes; A complex spiking attractors made up of limit cycles and chaotic attractors may lead to spiking states changing from chaotic oscillations to periodic ones, and the trajectory may jump directly from a chaotic attractor to other stable one without undergoing any bifurcation of the fast subsystem. [ABSTRACT FROM AUTHOR]