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We are concerned in this paper with dynamic behaviors and an optimal control problem of a new delayed epidemic model. There are three major ingredients. The first one is the dynamic behaviors of the state system. The locally asymptotic stability of the disease-free equilibrium and the endemic equilibrium are investigated and the effect of time delay on stability is also discussed. It is also found that the Hopf bifurcation appears at a specific time delay. The second, which is the main new ingredient of this paper, is an optimal control problem. Applying vaccine strategy in the system, an optimal control problem is proposed to minimize the total number of infected individuals as much as possible, maximize the total number of the uninfected individuals, and reduce the total control cost. In view of Pontryagin's maximum principle, the specific characteristics of the optimal control policy are given. The third ingredient is the numerical simulations of the theoretical results. • The model considers time delay, competition and spread of disease. • It studies dynamics and an optimal control problem of a delayed model. • Examples and numerical simulations are given to verify the theoretical results. [ABSTRACT FROM AUTHOR]

This paper serves as a continuation and expansion of the previous achievement by B. Han and D. Jiang in their article (Han and Jiang, 2024). Our initial step involves transforming the more complex stochastic Holling-II model featuring a stronger kernel into a simplified yet degenerate stochastic system composed of five interconnected equations. For the deterministic component of this model, we delve into an extensive examination of the local asymptotic stability of its positive equilibrium state. Subsequently, for the stochastic model, we derive critical conditions that determine the thresholds for exponential extinction and persistence of both predator and prey populations. Importantly, our findings not only encompass scenarios where there are no stochastic disturbances but also shed light on how environmental noise impacts the population dynamics within the predator–prey system. Through the exploration of the homologous Fokker–Planck equations, we present approximate representations characterizing the probability density function of the stochastic predator–prey model. To substantiate these theoretical advancements, several illustrative examples are provided, offering numerical elucidations of our proposed results and emphasizing the profound effects of stochastic noises and some important parameters on the behaviors of the stochastic model. • A stochastic Holling II predator–prey model with general distributed delay is investigated. • The sufficient and necessary condition for the coexistence of predator and prey populations is derived. • The approximate expression of the density function of the stochastic predator–prey model is obtained. • Several examples with simulations are presented to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper it is exhibited the bifurcation scenario concerning a typical singularity of planar piecewise smooth vector fields in two zones. This singularity is characterized by a quadratic contact of one vector field and a quartic contact of another vector field at the same point of the switching manifold. By means of a three parameter perturbation, we observe the presence of bifurcations like: Hopf Bifurcation, Sliding Hopf Bifurcation, the birth and annihilation of chaotic scenarios, among others. The two and three dimensional bifurcation diagrams are included. • quartic–quadratic fold singularity of planar piecewise smooth vector fields. • normal form of the quartic–quadratic fold singularity. • 3-parameter bifurcation of a planar piecewise smooth vector field. • Planar bifurcation diagrams. • tridimensional bifurcation diagram. [ABSTRACT FROM AUTHOR]

This paper investigates the bifurcations of homoclinic orbits to hyperbolic saddle points in a simplified railway wheelset model with cubic and quintuple nonlinear terms. Using Melnikov's method, the sufficient conditions for the existence of the supercritical and the subcritical pitchfork bifurcations of homoclinic orbits are proven. To determine the integrability of the variational equations around homoclinic orbits in the meaning of differential Galois theory, the corresponding Fuchsian second-order differential equation for the normal variational equation and the Riemann P function are obtained. It is shown that the coefficients of the linear terms and the cubic coupling terms play a very significant role on influencing the existence of homoclinic orbits. While, the cubic coupling terms have little effect on the size of the left-hand and right-hand potential wells of homoclinic orbits. These results are beneficial to explore the key mechanism of hunting stability of a simplified railway wheelset model. • The sufficient conditions for pitchfork bifurcations of homoclinic orbits are proven. • The integrability of variational equations around homoclinic orbits is determined. • Linear and cubic coupling term coefficients affect the existence of homoclinic orbits. [ABSTRACT FROM AUTHOR]

• A Light-fueled liquid crystal elastomer self-fluttering aircraft is proposed. • The system has two motion patterns: stationary and self-fluttering. • Triggering conditions, frequency, and amplitude of self-fluttering are adjustable. • The amplitude and frequency of self-fluttering can be designed by system parameters. In response to external stimuli, active materials are able to alter their shape or motion. Nonetheless, considerable challenges remain in the effective realization and control of active machines. Conventional control methods, involving sensor-based and external device control as well as electronic control, are intricate and tough to put into practice. Conversely, self-oscillators built upon active materials exhibit remarkable properties, including steady motion and straightforward regulation. Drawing inspiration from flying organisms, this paper puts forward an innovative light-fueled self-fluttering aircraft with a liquid crystal elastomer (LCE) engine, capable of sustained flight in the presence of steady illumination. The governing equations of the self-fluttering aircraft are derived in accordance with the well-known dynamic LCE model. The system is capable of experiencing a supercritical Hopf bifurcation, transitioning from a stationary pattern to a self-fluttering pattern, and the energy compensation mechanism of the self-fluttering pattern is revealed as well. Moreover, the quantitative investigation covers how distinct system parameters influence the Hopf bifurcation conditions and the self-fluttering amplitude and frequency, as well as the average flight velocity. With the benefits of customizable size, electronics-free and dependence on ambient energy, the proposed system may generate fresh insights into the practical use of active materials-based self-oscillators in micro actuators, soft robotics, bionic devices, and energy harvesting. [ABSTRACT FROM AUTHOR]

In this paper, we delve into double Hopf bifurcation induced by memory-driven directed movement in a spatial predator–prey model with Allee effect and maturation delay of predators. We first adopt a novel technique to handle the associated characteristic equation and thus obtain the crossing curves as well as the double Hopf points. We then calculate explicit formulae of normal form regarding non-resonant double Hopf bifurcation. We thus divide the dynamics of the developed model into several categories near the double Hopf bifurcation points. Our numerical and theoretical results both demonstrate that the model can exhibit various complex phenomena when the parameters are near the double Hopf bifurcation points. For example, the transition from one stable spatially inhomogeneous periodic orbit with mode-5 to another with mode-4 and the coexistence of them can be observed. • A spatial model with delays in both reaction and diffusion terms is considered. • An innovative technique is given to obtain crossing curves and bifurcation points. • The algorithm to compute normal form of double Hopf bifurcation is generalized. • Compared with only random diffusion, memory-based movement causes complex dynamics. [ABSTRACT FROM AUTHOR]

Travel between patches and behavior changes of susceptible individuals are two crucial factors affecting the disease dynamics for any emerging infectious disease. Therefore studies of their impacts on disease dynamics are of great importance, especially to public health policy-makers. In this paper, we propose an SIR model that incorporates these two factors into consideration. For our two-patch SIR model with infection force mediated by (past) disease surveillance, we identify the basic reproduction number and show how it is related to travel modified basic reproduction numbers. We also carry out stability and Hopf bifurcation analysis and obtain conditions under which the disease persists in one patch or both patches. Numerical simulations are carried out to illustrate how travel between two patches and disease surveillance mediated infection force affect the disease dynamics and thus provide us some helpful insights on proper control of emerging infectious diseases. • Build a two-patch SIR model with infection force mediated by (past) disease surveillance. • Identify the overall basic reproduction number and related it with several travel modified basic reproduction numbers (See Table 1). • Find that the travel of susceptible individuals alone can cause an endemic even if the disease dies out in each isolated patch. • Study the effects of the past disease surveillance on the spread of the disease. [ABSTRACT FROM AUTHOR]

In this paper, the qualitative properties of the fixed points in the non-hyperbolic cases, codimension-one bifurcations and weak resonances of a Cournot-Bertrand duopoly mixed competition model are explored. The two firms adopt different decision variables and different objective functions, which are more consistent with the actual economic market situation. The qualitative properties of all the fixed points in the non-hyperbolic cases are investigated using the reduction principle and the center manifold theorem. After that, all the potential codimension-one bifurcations, including transcritical bifurcation, supercritical or subcritical flip bifurcation and Neimark–Sacker bifurcation are analyzed using the bifurcation theory and the center manifold theorem. The direction, stability, and even the explicit approximate expression are derived for each type of bifurcation. By perturbing the closed invariant curve caused by the Neimark–Sacker bifurcation, the 2 : 5 weak resonance associated with Arnold's tongue is theoretically proved, and the absence of 1 : 6 and 5 : 6 weak resonances is further analyzed. A large number of numerical simulations show complete consistency with all theoretical analyses. Moreover, the continuation method is used to conduct numerical bifurcation analyses, further verifying the correctness of theoretical analyses, and testing more codimension 2 bifurcations, such as fold-flip bifurcation, generalized flip bifurcation and 1 : 2 strong resonance. In addition, the economic implications of these bifurcations are also explained accordingly. [ABSTRACT FROM AUTHOR]

In this paper, we present a complete parametric analysis on a nonlinear predator–prey system with the generalized Holling type III functional response and a strong Allee effect. We apply the hierarchical parametric analysis to derive explicit conditions for the existence and stability of equilibrium solutions in a 5-dimensional parameter space. Specifically, a detailed study on Hopf bifurcation is given to show bifurcation of multiple limit cycles, and complex dynamics of multistable states including bistable, tristable and tetrastable phenomena. We also conduct simulations and provide a biological interpretation of the multistable states under various conditions. • A predator–prey generalized Holling type III functional response and a strong Allee effect is investigated. • The hierarchical parametric analysis is applied to study the existence and stability of equilibrium solutions in a 5-dimensional parameter space. • A detailed study on Hopf bifurcation is given to show bifurcation of multiple limit cycles. • Complex dynamics of multistable states including bistable, tristable and tetrastable phenomena are explored. [ABSTRACT FROM AUTHOR]

In this paper, we use a cortex-basal ganglia resonance network to explore the possible Hopf bifurcation mechanism of beta oscillation. Different from traditional viewpoints, this model contains a direct inhibitory projection from the subthalamic nucleus (STN) to the cortex excitatory nuclei (EXN). First, we obtain the oscillation critical conditions of the delay by Hopf bifurcation analysis. Then, we explore the effects of some key parameters on Hopf bifurcation points. Delay is an indispensable factor in this model, oscillations will occur only when delay increases to a certain extent. Interestingly, the amplitude of oscillation increases with an increase in delay, but the trend of frequency is opposite; and heterogeneous delays have different effects on amplitude and frequency. The inhibitory coupling weight in the pathway " S T N → E X N " can reduce the firing activation level of cortical circuit, and the increase of coupling weights in the closed loop of the STN and globus pallidus external (GPe) can disinhibition this effect. On the stable boundary curve, the quantitative relationship of coupling weights between the direct and indirect projection from the basal ganglia to cortex is monotonously increasing. An increase in some critical coupling weights in the model, the bidirectional Hopf bifurcations (supercritical and subcritical) occur alternately, which can uniformly explain the origin mechanism of oscillations. In the inhibition-excitation closed loops, the excitatory and inhibitory connection weights have different effects on oscillations, the quantitative relationship of which is monotonically decreasing under Hopf bifurcation critical conditions. In the oscillation region, we observe that the amplitude is the smallest near bifurcation points and gradually increases with the evolution of oscillations. Finally, we find that the direct inhibitory projection from the STN to the EXN is a key pathway for external stimuli to control cortical oscillations. All theoretical analysis results compares well to numerical simulations, we hope that theoretical mechanisms obtained in this paper can inspire experimental research in Parkinson's disease. • The projection from the STN to the EXN can inhibit cortical circuit. • The quantitative relationship of various projections from BG to cortex is increasing. • The supercritical and subcritical Hopf bifurcations can occur alternately. • The quantitative relationship of different weights in closed loops is decreasing. • The oscillation amplitude is the smallest near bifurcation points. [ABSTRACT FROM AUTHOR]

In this paper an asymmetric planar continuous piecewise linear differential system with three zones x ˙ = y − F ( x ) , y ˙ = − g ( x ) is considered. The aim of this paper gives a completely study of limit cycles when this system satisfies such conditions and the uniqueness equilibrium does not lie in the central region. When ( x − x 0 ) g ( x ) > 0 for ∀ x ≠ x 0 and y = F ( x ) is a Z -shaped curve, it owns at most two limit cycles, which exist between a linear Hopf bifurcation surface and a double limit cycle bifurcation surface. Moreover, we prove the conjectures proposed by Ponce et al. [27]. When the uniqueness equilibrium lies in the central region, this system has exactly one limit cycles by others. Finally, some numerical examples are demonstrated. [ABSTRACT FROM AUTHOR]

Based on the study of scholars at home and abroad, this paper improves the three-dimensional IS-LM model in macroeconomics, analyzes the equilibrium point of the system and stability conditions, focuses on the parameters and complex dynamic characteristics when Hopf bifurcation occurs in the three-dimensional IS-LM macroeconomics system. In order to analyze the stability of limit cycles when Hopf bifurcation occurs, this paper further introduces the first Lyapunov coefficient to judge the limit cycles, i.e. from a practical view of the business cycle. Numerical simulation results show that within the range of most of the parameters, the limit cycle of 3D IS-LM macroeconomics is stable, that is, the business cycle is stable; with the increase of the parameters, limit cycles becomes unstable, and the value range of the parameters in this situation is small. The research results of this paper have good guide significance for the analysis of macroeconomics system. [ABSTRACT FROM AUTHOR]

This paper focuses on a general Gause-Kolmogorov-Type predator–prey model with direct predator-taxis and indirect prey-taxis. The global existence and boundedness of solutions to the system are proved. Our approach is applicable to more general situation. Some measures of success for the constant coexistence steady state are rigorously calculated, such as decreasing the indirect prey-taxis sensitivity or the release rate of stimulus, and increasing the predator-taxis sensitivity or the decay rate of stimulus. By choosing the indirect prey-taxis coefficient as the bifurcation parameter, the existence of Hopf and double-Hopf bifurcations is established. We find that both indirect prey-taxis and direct predator-taxis can promote the complexity of the spatiotemporal pattern, e.g. spatially nonhomogeneous periodic patterns with different spatial frequency, spatially nonhomogeneous quasi-periodic patterns. Finally, we apply our theoretical analyses to a Rosenzweig–MacArthur model with indirect prey-taxis and direct predator-taxis, spatially homogeneous periodic patterns. • We bond the effects of direct and indirect taxis in a predator-prey system. • We prove the global well-posedness of a predator-prey system with direct and indirect taxis. • We determine several beneficial strategies of success about the indirect and direct taxis. • We establish double-Hopf bifurcations and pattern formations for a general predator-prey system. • In an application, complex nonhomgeneous periodic and quasi-periodic patterns emerge. [ABSTRACT FROM AUTHOR]

In this paper, a previously developed 2-dimensional disease model is studied, which can be used for both epidemiologic modeling and in-host disease modeling. The main attention of this paper is focused on various dynamical behaviors of the system, including Hopf and generalized Hopf bifurcations which yield bistability and tristability, Bogdanov–Takens bifurcation, and homoclinic bifurcation. It is shown that the Bogdanov–Takens bifurcation and homoclinic bifurcation provide a new mechanism for generating disease recurrence, that is, cycles of remission and relapse such as the viral blips observed in HIV infection. [ABSTRACT FROM AUTHOR]

• We introduce a simple two-parameter quadratic three-dimensional system. • It only has six terms, two nonlinearities and one equilibrium. • A homoclinic flip bifurcation of case inward twist C in is present. • Chaotic behavior with Smale horseshoes and strange attractors is warranted. • It is the first example of a 3D vector field exhibiting the case C in. In this paper we consider a two-parameter quadratic three-dimensional system with only six terms and two nonlinearities. First we analyze the Hopf bifurcation of its only equilibrium detecting several degeneracies. With this information we numerically obtain various bifurcation diagrams of periodic orbits in which saddle-node and period-doubling bifurcations as well as homoclinic connections appear. A careful study of the homoclinic orbits, in a region of the two-parameter plane where the equilibrium is a real saddle, shows the presence of a homoclinic flip bifurcation of case C. Here this orbit changes from orientable to non-orientable, being the lowest codimension for a homoclinic bifurcation to a real saddle equilibrium that results in chaotic behavior. More specifically, we determine that it corresponds to the case inward twist C in , in such a way that, as far as we know, it is the first example of a 3D vector field exhibiting this case. [ABSTRACT FROM AUTHOR]

In this paper, based on the restoring force coupling method, a novel mathematical model is established to study the VC parameter vibration responses of the ball bearing-rotor system, and both the rotor radial and tilting motions have been considered. Firstly, the steady state analysis on the stiffness fluctuation is conducted to determine the equivalent linear resonant frequencies of the nonlinear ball bearing-rotor system. Secondly, both the two-way sweep-frequency and fixed frequency-analysis are adopted to study the different transient response mechanisms of the rotor VC parameter vibrations, Finally, the results show that not only the multiple resonant responses including the super-harmonic, sub-harmonic, Hopf bifurcation and chaos are generated for the rotor parametric vibration, but also the complex internal resonance phenomenon occurs when the rotor natural frequency ratio in different radial directions is close to 1:2. • A novel M-DOF model for the rotor VC vibration analysis is built. • The method for rotor equivalent linear resonance frequencies calculation is given. • The rotor transient response under different VC vibration mechanism is given. • The generation mechanism of the rotor multiple resonant responses is discussed. [ABSTRACT FROM AUTHOR]

Highlights • A two-dimensional prey-predator model in presence of habitat complexity and prey refuge with two discrete delays has been proposed and analyzed. • The proposed model system is persistent and globally asymptotically stable if the intrinsic growth rate of prey population is greater than a threshold value, which depends upon the habitat complexity and refuge. • The negative feedback delay and gestation delay both destabilize the behavior of the system. • The numerical experiments conducted shows that the Hopf-bifurcation is supercritical, the bifurcated periodic solution is stable and its period increases. Abstract In this paper, we consider a two-dimensional prey-predator system with two delays. One delay is for negative feedback of the prey population and another is for gestation delay of the predator population. The predator is partially dependent on the prey followed by Holling type-II functional response. Due to habitat complexity and prey refuge, the Holling type-II functional response is modified in this work. We discuss the boundedness, permanence, local and global asymptotic behavior of the non-delayed and delayed models. The existence of periodic solutions via Hopf-bifurcation with respect to both the delays is established. The stability and direction of Hopf-bifurcation is also analyzed by using Normal form theory and Centre manifold theory. Lastly, numerical simulations have been carried out to confirm the analytical findings. The main objective of this work is to balance the prey-predator relationship in the presence of habitat complexity, prey refuge and delays. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical model describing the interaction of malignant glioma cells, macrophages and glioma specific CD8+T cells is discussed. The biologically feasible equilibria and corresponding local stability are deduced. The bifurcations and related dynamical behaviors of this model are further studied thoroughly. The existence of transcritical bifurcation and saddle–node bifurcation is derived based on Sotomayor's theorem and Hopf bifurcation is well discussed. The codimension 2 bifurcation such as Bogdanov–Takens bifurcation is investigated using the normal form theory and center manifold theorem in more detail. Finally, numerical simulations are obtained to validate our analytical findings by varying the parameters. • The dynamics of a mathematical model describing the interaction of malignantglioma cells, macrophages and glioma specific CD8+T cells are discussed. • The co-existence of equilibria of the model are obtained by numerical simulation. • The existence of codimension 1 and codimension 2 bifurcations is investigated. • Extensive numerical simulations are obtained to validate our analytical findings. [ABSTRACT FROM AUTHOR]

Malware spreading, characterized by differential equations, in networked systems has become a common topic of study in the scientific literature. In this paper, we focus primarily on the propagation dynamics of malware in cyber–physical systems (CPSs). Although some work prior has been done to understand how the malware dynamics evolves in complex networks, these studies have restricted their attention to stability, bifurcation and oscillation based on ordinary differential equation models. Here, we address the question "how does one detect tipping phenomena which result in a total collapse of CPSs if the spread of malware is related to both time and space?" To this end, we use a reaction–diffusion equation to model spatio-temporal dynamic evolutions of malware in CPSs and accurately predict the emergence of tipping points due to the Turing instability and Hopf bifurcation for the first time. In order to further reveal the mechanism of tipping, the specific conditions of the Turing instability and Hopf bifurcation are given. It is confirmed that the introduction of space factor and transmission delay causes CPSs to change from stable to unstable, and then the tipping occurs. Moreover, the delay-dependent stability is discussed and the stability and direction of Hopf bifurcation are explored. These studies provide a strategic guidance for the prediction and control of malicious virus in CPSs. Finally, some numerical simulation examples are provided to substantiate the theoretical results. • A novel reaction-diffusion malware spreading model with multiple delays is proposed. • The basic reproductive ratio is calculated and the Turing instability induced by tipping is studied. • The tipping point of time delay is given and the direction of Hopf bifurcation is determined. • The spatial-temporal evolution of infected nodes in CPSs can be predicted quantitatively. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel multiscale transfer entropy (MTE) approach to quantify the information transfer of time series on multiple time scales. The MTE combines the advantages of both the multiscale analysis and the transfer entropy, which is able to identify directional, dynamical and scale-dependent information flows. To minimize finite size effects and to avoid spurious detection of causality, we resort to a refined time-delayed multiscale transfer entropy (TMTE) estimator given on overlapping coarse-graining. We also suggest several extensions of the TMTE, including an effective TMTE and a net TMTE. Synthetic simulations including the linear vector autoregressive (VAR) models, the long-range correlated ARFIMA processes, and the nonlinear Rössler systems are analyzed, and an application of the TMTE to daily closing prices and trading volumes of S&P 500 is studied. [ABSTRACT FROM AUTHOR]

In this paper, a generalized (2+1)-dimensional Camassa–Holm–Kadomtsev–Petviashvili (gCHKP) equation is investigated, which describes the role of dispersion in the formation of patterns in liquid drops. We succinctly construct its bilinear formalism. By further using homoclinic breather limit approach, some exact solutions including breather waves, rogue waves and solitary waves of the equation are well presented. Our results show that rogue waves can come from the extreme behavior of the breather solitary waves for the (2+1)-dimensional gCHKP equation. [ABSTRACT FROM AUTHOR]

This article considers a Cournot duopoly model in a continuous-time framework and analyze its dynamic behavior when the competitors are heterogeneous in determining their output decision. Specifically the model is expressed in the form of differential equations with discrete delays. The stability conditions of the unique Nash equilibrium of the system are determined and the emergence of Hopf bifurcations is shown. Applying some recent mathematical techniques (stability switching curves) and performing numerical simulations, the paper confirms how different time delays affect the stability of the economy. [ABSTRACT FROM AUTHOR]

This paper is comprehensively concerned with the dynamics of a class of high-dimension fractional ring-structured neural networks with multiple time delays. Based on the associated characteristic equation, the sum of time delays is regarded as the bifurcation parameter, and some explicit conditions for describing delay-dependent stability and emergence of Hopf bifurcation of such networks are derived. It reveals that the stability and bifurcation heavily relies on the sum of time delays for the proposed networks, and the stability performance of such networks can be markedly improved by selecting carefully the sum of time delays. Moreover, it is further displayed that both the order and the number of neurons can extremely influence the stability and bifurcation of such networks. The obtained criteria enormously generalize and improve the existing work. Finally, numerical examples are presented to verify the efficiency of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a cancer network including a protein module and a corresponding microRNA cluster that inhibits the synthesis of proteins. The existence of multiple steady states and their stability depending on the parameters are firstly determined. Bistability and dependency on the parameters, Hopf bifurcations and the corresponding properties like direction and stability of Hopf bifurcations are determined by computing the normal form on the center manifold. Then, the role of the delay in the process of synthesis of the protein is investigated. We show that the delay can stabilize the unstable equilibrium and destabilize the stable equilibrium. Some simulations are carried out to numerically illustrate the obtained theoretical results. Finally, the biological interpretation of the theoretical results is discussed. [ABSTRACT FROM AUTHOR]

In this paper, the influence of the fear effect and Leslie–Gower function on the dynamic behavior of the predator–prey model is considered. First, the well-posedness analysis of this model is demonstrated, and the existence and local stability of the equilibrium point are given. Then by using the bifurcation theory and selecting the appropriate bifurcation parameters, many types of bifurcation phenomena in the system are discovered, including transcritical bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation. Meanwhile, the numerical simulations on the basis of theoretical analysis are further carried out to intuitively show the influence of fear effect on population. As the degree of fear increases, the system will undergo multiple dynamic behaviors switching until the final extinction of prey population, while the predator population will survive due to the presence of substitute prey. In addition, the initial density of predator and prey can determine whether the solution of system tends to a coexisting steady state or a periodic oscillation. • A novel prey–predator model with the impact of fear is introduced. • The system can experience many types of bifurcation phenomena. • Numerical simulations and sensitivity analysis are performed. • Our results show that fear effect plays an important role on deriving the complex dynamics. [ABSTRACT FROM AUTHOR]

In this paper we study a three-dimensional tumor–immune system interaction model consisted of tumor cells, activated T cells, and immune checkpoint inhibitor anti-PD-1. Based on the uncontrollable character of tumor cells in the absence of immune response and treatment, the growth of tumor cells is assumed to be exponential. We discuss the distribution of equilibria qualitatively and study the stability of all possible equilibria with and without anti-PD-1 drug. When no drug is applied, the model has a tumor-free equilibrium and at most one tumorous equilibrium. Biologically, there exists a threshold d T 1 for the death rate d T of T cells: when d T ≥ d T 1 tumor cells will keep growing; when d T < d T 1 tumor cells may be eradicated for some positive initial values and keep growing for some other positive initial values. For the case with anti-PD-1 treatment, the model has at most five tumor-free equilibria and two interior equilibria. Our analysis indicates that there exists a threshold γ A 1 for the intravenous continuous injection γ A : when γ A ≤ γ A 1 the fate of tumor cells is the same as the case with no drug applied; when γ A > γ A 1 the model may exhibit bistable phenomena and periodic orbits. Furthermore, we establish the existence of local Hopf bifurcation around the interior equilibrium and determine the stability of the bifurcating periodic orbits. Our simulations show that the model exhibits a stable periodic orbit which implies the long term coexistence and balance of the tumor and immune system. • Study a tumor–immune system interaction model with an immune checkpoint inhibitor. • Discuss the number and stability of all possible equilibria. • Establish the existence of Hopf bifurcation and the stability of the periodic orbit. • Carry out numerical simulations to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

• A dynamical system global analysis underlines the importance of a meditated choice of the relevant policy parameters, saving ratio and share of abated emission. • Pollution induced poverty traps emerge for low level of saving ratio and share of abated emission. • Periodic behavior of the state variables represents early signal of the risk of being trapped in an unsatisfactory level of economic performance. In this paper we present a minimal integrated environment-economic growth model. The accumulations of capital and pollution are connected by reciprocal feedbacks. Pollution is an undesirable but inevitable by-product of production, whose efficiency is hindered by pollution. The nexus between pollution and production is embodied in a damage function. The evolution of capital is enriched by the specific technology adopted here, an S -shaped production function. The model dynamics is represented by a couple of nonlinear differential equations, whose long run behavior gives rise to multiple stationary points. A global analysis underlines the importance of a meditated choice of the relevant policy parameters. Pollution induced poverty traps emerge for low level of saving ratio and share of abated emission. Periodic behaviour of the economic and environmental variables represents an early signal of the imminent risk of being trapped in an unsatisfactory level of economic performance. Numerical examples corroborate the theoretical results of the paper. [ABSTRACT FROM AUTHOR]

In this paper we investigate stability conditions for small-amplitude periodic solutions emerging near symmetry-preserving Hopf bifurcations in a time-delayed fully-connected N-node PLL network. The study of this type of systems which includes the time delay between connections has attracted much attention among researchers mainly because the delayed coupling between nodes emerges almost naturally in mathematical modeling in many areas of science such as neurobiology, population dynamics, physiology and engineering. In a previous work it has been shown that symmetry breaking and symmetry preserving Hopf bifurcations can emerge in the parameter space. We analyze the stability along branches of periodic solutions near fully-synchronized Hopf bifurcations in the fixed-point space, based on the reduction of the infinite-dimensional space onto a two-dimensional center manifold in normal form. Numerical results are also presented in order to confirm our analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we perform spatiotemporal bifurcation analysis in a ratio-dependent predator–prey model and derive explicit conditions for the existence of non-constant steady states that emerge through steady state bifurcation from related constant steady states. These explicit conditions are numerically verified in details and further compared to those conditions ensuring Turing instability. We find that (1) Turing domain is identical to the parametric domain where there exists only steady state bifurcation, which implies that Turing patterns are stable non-constant steady states, but the opposite is not necessarily true; (2) In non-Turing domain, steady state bifurcation and Hopf bifurcation act in concert to determine the emergent spatial patterns, that is, non-constant steady state emerges through steady state bifurcation but it may be unstable if the destabilising effect of Hopf bifurcation counteracts the stabilising effect of diffusion, leading to non-stationary spatial patterns; (3) Coupling diffusion into an ODE model can significantly enrich population dynamics by inducing alternative non-constant steady states (four different states are observed, two stable and two unstable), in particular when diffusion interacts with different types of bifurcation; (4) Diffusion can promote species coexistence by saving species which otherwise goes to extinction in the absence of diffusion. [ABSTRACT FROM AUTHOR]

In this paper, we prove the equivalence of the multiple time scales (MTS) method and the center manifold reduction (CMR) method for deriving the normal form of Hopf bifurcation for delayed reaction–diffusion equations. First, we generalize the MTS method associated with functional differential equations to delayed reaction–diffusion equation for solving the normal form of Hopf bifurcation. This provides a user-friendly approach of analyzing the stability of time-periodic solutions, particularly, for the engineering researchers who are not familiar with CMR method. Second, we outline the CMR method associated with the computation of normal form of Hopf bifurcation for delayed reaction–diffusion equations. Third, we prove that the two methods can derive equivalent up to the third order normal form of Hopf bifurcation. Finally, different types of examples are presented, such as reaction–diffusion equations with Neumann boundary condition and Dirichlet boundary conditions, single delay and two delays, spatially homogeneous and spatially inhomogeneous periodic solutions. We obtain the equivalent Hopf bifurcation normal forms results derived by using the CMR and MTS methods for above examples, respectively. • We generalize MTS method to Hopf bifurcation of delayed reaction–diffusion equation. • We outline CMR method for Hopf bifurcation of delayed reaction–diffusion equations. • We prove the equivalence of the two methods up to the third order normal form. • Four typical examples are presented. [ABSTRACT FROM AUTHOR]

To analyze the effect of animal spirits on economic dynamics, we construct a simple Keynesian business-cycle model with animal spirits incorporated into it. Similar to the well-known Kaldor model, we assume that each firm possesses the investment function depending on demand, in which the animal spirits are connected with the degree of the propensity to invest. However, we would like to emphasize that, unlike the Kaldor model, even if the level of demand for each firm is the same, the degree of propensity to invest (i.e., the strength of animal spirits) depends on the change in demand. Thus, we assume that animal spirits in demand-upswing phases are stronger than those in demand-downswing phases. In each phase, the degree of dependence on demand is firm specific. Therefore, we adopt a statistical approach to connect such heterogeneous micro-investment behaviors with a macro-investment function. Through this approach, some kind of nonlinearity is naturally introduced into the macro-investment function. We demonstrate that the nonlinear macro-investment function yields persistent business cycles. We also demonstrate the occurrence of a generalized Hopf bifurcation. • We change of the title of the paper, that is, entitled a new title according to reviewer's comment. • We have made the text as easy to understand as possible including corrections of not only theoretical points but also grammatical and stylistic ones. • We also tried to reduce the number of pages. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the species interaction model with the ratio-dependent Holling III functional response and strong Allee effect. To explore nonhomogeneous solutions of the model, we consider the existence and non-existence of non-constant steady states and temporal bifurcation. Then we show the boundedness of the global positive solutions for the parabolic system and present the upper and lower bounds of positive solutions for the associated elliptic system. The non-existence and existence of the non-constant steady states of the elliptic system with the homogeneous Neumann boundary conditions are obtained by using the priori estimates, maximum principle and index theory. Furthermore, the existence and the direction of Hopf bifurcation are investigated via the stability analysis, center manifold theory and normal form reduction. Numerical simulations are carried out to verify our theoretical analysis and to illustrate that the ratio-dependent Holling III functional response and strong Allee effect have strong impact on dynamical behaviors of the species interaction systems. • Boundedness of parabolic and elliptic models is analyzed. • Nonconstant steady states are presented. • Hopf bifurcation with strong Allee effect is described. [ABSTRACT FROM AUTHOR]

In this paper, we consider the glycolysis system with symmetry and degeneration. Based on the detailed stability of constant steady state solution and symmetry of nonlinear glycolysis system, for the Hopf bifurcations, we obtain the existence, stability and bifurcation direction not only for the homogeneous and inhomogeneous periodic solutions, but also for the degenerate bifurcating periodic solutions. The approach we adopted is the Lyapunov–Schmidt reduction method instead of the center manifold theory. On the other hand, we derive the nonexistence of the steady state solutions for small input flux of substrate. Additionally, for the steady state bifurcations, main object is to discuss degenerate bifurcations by means of Lyapunov–Schmidt procedure and singularity theory because the classical Crandall–Rabinowitz theorem cannot be applied. Some numerical simulations are achieved to illustrate the analytical results, especially the degenerate bifurcation results. It is interesting to noticed that the degenerate bifurcations can lead to richer spatiotemporal patterns, including the meeting of two Hopf bifurcation branches and the coupling of two eigenfunction models. [ABSTRACT FROM AUTHOR]

In this paper, we systematically study Parkinson oscillation origin mechanism, oscillation amplitude and frequency characteristics in an improved cortex-basal ganglia (EXN-INN-STN-GPe) resonance model. Firstly, Hopf bifurcation critical conditions T 1 0 and T 2 0 for heterogeneous delays are derived, which are greatly affected by coupling weights. The delay is a key factor to affect the frequency of oscillation, many frequency bands (such as gamma oscillation, beta oscillation, alpha oscillation, etc.) appear with an increase in delay. According to T 1 0 and T 2 0 , the state of STN-GPe loop can be divided into four types: stable state, induced oscillation, spontaneous oscillation and compound oscillation, which represent different origin mechanism of oscillation. They have different amplitude characteristics, which increases with an increase in delay in most cases. The amplitude of the induced oscillation is smaller than the amplitude of the spontaneous oscillation in many cases. In the compound oscillation region, the EXN-STN antiphase oscillation mode can inhibit the oscillation amplitude in the STN-GPe. Induced oscillation may indicate that the oscillation origins in the cortex, which spreads to the STN-GPe loop with the evolution of disease. Self-feedback connections of EXN (W E E) and INN (W I I) play a very important role in affecting the amplitude of induced oscillation. The induced oscillation amplitude of the GPe is increased with an increase in W E E , but the trend is the opposite when it is affected by W I I. As an increase in coupling weight between the EXN and INN, the minimum of W E E and W I I required to induce oscillation are both increased. The theoretical analysis is in good agreement with the numerical simulation, the mechanism obtained may inspire experimental and new theoretical research in Parkinson's disease. • Hopf bifurcation critical conditions T 1 0 and T 2 0 for heterogeneous delays are obtained. • The delay is a key factor to affect the frequency of oscillation. • According to T 1 0 and T 2 0 , the state of STN-GPe loop can be divided into four types. • The amplitude of induced oscillation is smaller than that of spontaneous oscillation. • EXN-STN antiphase oscillation mode can inhibit oscillation amplitude in the STN-GPe. [ABSTRACT FROM AUTHOR]

A discrete-time predator–prey model with Allee effect is investigated in this paper. We consider the strong and the weak Allee effect (the population growth rate is negative and positive at low population density, respectively). From the stability analysis and the bifurcation diagrams, we get that the model with Allee effect (strong or weak) growth function and the model with logistic growth function have somewhat similar bifurcation structures. If the predator growth rate is smaller than its death rate, two species cannot coexist due to having no interior fixed points. When the predator growth rate is greater than its death rate and other parameters are fixed, the model can have two interior fixed points. One is always unstable, and the stability of the other is determined by the integral step size, which decides the species coexistence or not in some extent. If we increase the value of the integral step size, then the bifurcated period doubled orbits or invariant circle orbits may arise. So the numbers of the prey and the predator deviate from one stable state and then circulate along the period orbits or quasi-period orbits. When the integral step size is increased to a critical value, chaotic orbits may appear with many uncertain period-windows, which means that the numbers of prey and predator will be chaotic. In terms of bifurcation diagrams and phase portraits, we know that the complexity degree of the model with strong Allee effect decreases, which is related to the fact that the persistence of species can be determined by the initial species densities. [ABSTRACT FROM AUTHOR]

Understanding the effect of slowly varying control parameters in dynamical systems is important in many fields such as mechanics, biology, ecology and social sciences, where normally changes in parameters take place very slowly. When a control parameter becomes time varying, the system dynamics exhibits a delay in bifurcation, i.e., the system responds to the bifurcation scenario with a lag in real time. In this paper, we experimentally explore the delay associated with Hopf and pitchfork bifurcations in a parametrically driven nonlinear oscillator. For this study we choose a generic nonlinear oscillator, namely the parametrically driven Murali–Lakshmanan–Chua (PDMLC) oscillator. We identify and characterize the occurrence of delay in bifurcations in both the rising and falling edges of the external force and measure the delay associated with these bifurcations in both the edges. We show that the delay in Hopf and pitchfork bifurcations increase when the rate of change of control parameter decreases. We further show that the delay obeys a power law as a function of the external frequency. All the numerical simulation results are corroborated with the real-time electronic circuit experiment and we find a good qualitative agreement between the numerical and experimental results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Turing–Hopf bifurcation arising from the reaction–diffusion equations. It is a degenerate case and where the characteristic equation has a pair of simple purely imaginary roots and a simple zero root. First, the normal form theory for partial differential equations (PDEs) with delays developed by Faria is adopted to this degenerate case so that it can be easily applied to Turing–Hopf bifurcation. Then, we present a rigorous procedure for calculating the normal form associated with the Turing–Hopf bifurcation of PDEs. We show that the reduced dynamics associated with Turing–Hopf bifurcation is exactly the dynamics of codimension–two ordinary differential equations (ODE), which implies the ODE techniques can be employed to classify the reduced dynamics by the unfolding parameters. Finally, we apply our theoretical results to an autocatalysis model governed by reaction–diffusion equations; for such model, the dynamics in the neighbourhood of this bifurcation point can be divided into six categories, each of which is exactly demonstrated by the numerical simulations; and then according to this dynamical classification, a stable spatially inhomogeneous periodic solution has been found. [ABSTRACT FROM AUTHOR]

Diffusion is ubiquitous inside cells, and it is capable of inducing spontaneous pattern formation in reaction–diffusion systems on a spatially homogeneous domain. In this paper, we investigate the dynamics of a diffusive cancer network regulated by microRNA and obtain the condition that the network undergoes a Hopf bifurcation and a Turing pattern bifurcation. In addition, we also develop the amplitude equation of the network model by using Taylor series expansion, multi-scaling and further expansion in powers of a small parameter. As a result of these analyses, we obtain the explicit condition on how the dynamics of the diffusive cancer network evolve. These results reveal that this system has rich dynamics, such as spotted stripe and hexagon patterns. The bifurcation diagram helps us understand the biological mechanism in the cancer network. Finally, numerical simulations confirm our analytical results. [ABSTRACT FROM AUTHOR]

In this paper we consider a 3D three-parameter unfolding close to the normal form of the triple-zero bifurcation exhibited by the Lorenz system. First we study analytically the double-zero degeneracy (a double-zero eigenvalue with geometric multiplicity two) and two Hopf bifurcations. We focus on the more complex case in which the double-zero degeneracy organizes several codimension-one singularities, namely transcritical, pitchfork, Hopf and heteroclinic bifurcations. The analysis of the normal form of a Hopf-transcritical bifurcation allows to obtain the expressions for the corresponding bifurcation curves. A degenerate double-zero bifurcation is also considered. The theoretical information obtained is very helpful to start a numerical study of the 3D system. Thus, the presence of degenerate heteroclinic and homoclinic orbits, T-point heteroclinic loops and chaotic attractors is detected. We find numerical evidence that, at least, four curves of codimension-two global bifurcations are related to the triple-zero degeneracy in the system analyzed. • The 3D system considered is invariant under the change (x, y, z) → (−x, −y, z). • A double-zero degeneracy (a double-zero eigenvalue with geometric multiplicity two) is studied. • Transcritical, pitchfork, Hopf and heteroclinic bifurcations are involved. • Degenerate global connections, T-point heteroclinic loops and chaotic attractors are detected. • Four degenerate global connections related to the triple-zero bifurcation have been found. [ABSTRACT FROM AUTHOR]

This paper presents a bi-directional N-type locally-active memristor (LAM) model, which has two symmetrical locally-active regions with respect to the origin. For this memristor, the locally-active regions coincide with the edge of chaos regimes, where complex dynamic behaviors may occur. By deriving the small-signal admittance of the memristor, it is found that the LAM may be pure resistive, inductive or capacitive in terms of biasing voltages. When the LAM operates at the edge of chaos and connects with an external inductor in series, a second-order periodic oscillator is established. In this case, it is possible to move the poles of the system from the left-half plane to the right-half plane, thereby generating periodic oscillations. Complex dynamics of the second-order memristor-based oscillator is analyzed via Hopf bifurcation analysis and based on the theories of local activity and edge of chaos. Using the parasitic capacitor of the LAM in the second-order oscillator, a third-order chaotic oscillator is designed, which has a negative real pole and a pair of complex conjugate poles with positive real parts, satisfying the Shilnikov criterion. It is found that both periodic oscillation and chaos appear only at the edge of chaos of the LAM, thus verifying the complexity theorem, no-complexity theorem and edge of chaos theorem. Finally, circuit experiments are performed, which verify the dynamics and the effectiveness of the LAM model, the periodic oscillator and the chaotic oscillator. • A novel bi-directional N-type locally-active memristor model is built. • This memristor shows capacitive, inductive and resistive properties with different operating voltages via small-signal method. • The memristor-based second-order periodic and third-order chaotic oscillators are designed and realized. • The complexity theorem, no-complexity theorem and edge of chaos theorem are verified. [ABSTRACT FROM AUTHOR]

Abstract: Hopf–Hopf bifurcation is one of typical codimension-two bifurcations, which requires some rigid bifurcation conditions and occurs only in high-dimension systems. In this paper, a new critical criterion of this bifurcation is presented for a general discrete time system. Unlike the corresponding classical critical criterion (or the bifurcation definition), the new criterion is composed of a series of algebraic conditions explicitly expressed by the coefficients of the characteristic polynomial, which does not depend on eigenvalue computations of Jacobian matrix. This characteristic gives the advantage of the proposed criterion which is more convenient and efficient for detecting the existence of this type of codimension-two bifurcation or exploring the parameter mechanism of the bifurcation than the corresponding classical criterion. The equivalence between the proposed criterion and the corresponding classical criterion is rigorously proved. The bifurcation design problem of a three-degree-of-freedom vibro-impact system is used as example to show the effectiveness of the proposed criterion. [Copyright &y& Elsevier]

Abstract: On the basis of the geometric singular perturbation theory and the theory of delayed Hopf bifurcation in slow–fast systems with delay, the stability of nonlinear systems with slowly and periodically varying delay is investigated in this paper. Sufficient conditions ensuring asymptotic stability of those systems are obtained. Especially, though a time-varying delay usually increases complexity in the analysis of system dynamics and it usually deteriorates system stability as well, the study indicates that under certain conditions, the stability of the systems with a time-invariant delay only can be improved by incorporating a slowly and periodically varying part into the constant delay. Two illustrative examples are given to validate the analytical results. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate Hopf bifurcation and center stability of a predator–prey biological economic model. By employing the local parameterization method, Hopf bifurcation theory and the formal series method, we obtain some testable results on these issues. The economic profit is chosen as a positive bifurcation parameter here. It shows that a phenomenon of Hopf bifurcation occurs as the economic profit increases beyond a certain threshold. Besides, we also find that the center of the biological economic model is always unstable. Finally, some numerical simulations are given to illustrate the effectiveness of our results. [Copyright &y& Elsevier]

• The hunting cooperation has a destabilizing effect in Holling type II predator-prey model. • The hunting cooperation can induce the bistability phenomenon. • The hunting cooperation can undergo rich bifurcation phenomena. • The hunting cooperation can induce the rise of Turing instability. In this paper, the dynamics of a diffusive Holling type II predator-prey model with hunting cooperation are studied. For the local model, the stability of nonnegative equilibria, the detailed behavior of Hopf bifurcation, saddle-node bifurcation, and Bogdanov–Takens bifurcation are investigated. It is shown that under suitable conditions the model exhibits a supercritical and orbitally asymptotically stable limit cycle when the predator cooperation in hunting rate pass through a threshold value. For the reaction-diffusion model, we analyze the diffusion-driven Turing instability of both the positive equilibria and bifurcating periodic solutions, the existence, direction and stability of Hopf bifurcations. Our result shows that the hunting cooperation plays a crucial role in determining the stability and bifurcation behavior to the model, that is, it is beneficial to the coexistence of predator and prey, it can have a destabilizing effect, it can induce the bistability phenomenon, it can undergo not only the saddle-node bifurcation but also Bogdanov–Takens bifurcation, and it can induce the rise of Turing instability, which is a strong contrast to the case without hunting cooperation. Moreover, some numerical simulations are performed to visualize the complex dynamic behavior. [ABSTRACT FROM AUTHOR]

• Large time delay enhances the stability of Alternating Wave Solution (AW). • We combine the method of multiple scales and Floquet theory to study the stability of AW. • The normal form with third order is insufficient to determine the stability of AW. • The mechanism on how nonlinear gains affect the stability of AW is determined. In this paper, the profile and stability of alternating wave solution, which arises as a bifurcated periodic solution of equivariant Hopf bifurcation with amazing properties, are investigated for a Stuart-Landau system consisting of three oscillators. The method of multiple scales is used to compute the normal form equation up to fifth order. The Floquet theory is introduced because it is difficult to directly analyze the stability of the alternating wave solution. By applying a time-varying complex coordinate transformation which does not change the stability of the solution of normal form that represents the alternating wave, the multipliers that completely determine the stability of alternating wave solution are explicitly solved. As a result, the criteria on parameters such that stable alternating wave solutions can be observed are provided. Based on studies through examples, we show that the proposed scheme of analysis is effective and some results on how parameters influence the stability of the alternating wave solution can be summarized. Our analysis confirms Golubitsky's assertion that the alternating wave solution will not be stable immediately after the equivariant Hopf bifurcation. We also find that a large time delay and a complex nonlinear gain will enhance the stability of alternating wave solution. [ABSTRACT FROM AUTHOR]

• A new reaction-diffusion HFMD model with nonsmooth saturation treatment function is introduced. • The stability of equilibria, and the existence of Turing instability, Hopf bifurcation for our model are analyzed. • Simulations demonstrate the validity of theoretical results. • The model is used to fit the accumulated cases of HFMD in Guangxi Province, China from January, 2016 to May, 2019. In this paper, we propose a new reaction-diffusion model to investigate the spatial spread of hand, foot and mouth disease (HFMD). To reflect the actual factors which are limited medical resources for treatment and intensive treatment strategies which depend on a certain threshold, we also consider a continuous and nonsmooth treatment function in our model. First, existences of equilibria are determined by discussing the relation of parameters. Second, for the local stability of equilibria, the effect of the spatial diffusion including the variation of the size of the space domain and the diffusion coefficients of susceptible and infectious individuals on the stability is studied and Turing instability is demonstrated. Third, spatially homogeneous Hopf bifurcation and spatially homogeneous Hopf bifurcation are also studied by analyzing the corresponding characteristic equation. Finally, we use the model to fit the reported cases of HFMD in Guangxi Province, China from January, 2016 to May, 2019. Our results containing theoretical analysis and numerical simulations show that reducing the threshold I c with respect to intensive treatment and reducing the range of activities of infectious individuals can effectively control the spread of HFMD. [ABSTRACT FROM AUTHOR]

• A simple direct method to determine a base of the complementary spaces for the Lie transform is given. • The normal forms for vector field with double purely imaginary eigenvalues with geometric multiplicity one are considered. • Explicit formulas for the normal forms coefficients with three unfolding parameter is given. • Using this method, the normal form for the non-semisimple 1:1 resonance Hopf bifurcation can be given easily. In this paper, a new and efficient mechanism to compute the normal forms for 1:1 resonant Hopf bifurcation is developed. For a vector field given by ordinary differential equations, by assuming that eigenvalues at an equilibrium point are purely imaginary, double, and non-semisimple, the mechanism provides a direct method to calculate the coefficients for the normal forms. In particular, we present the following results: (1) a simple direct method to determine a basis of the complementary spaces for the Lie transform; (2) a simple direct method to determine the projection of any vector in H 4 3 to the complementary spaces; and (3) the normal forms for vector fields with double purely imaginary eigenvalues with geometric multiplicity one. In addition, explicit formulas for coefficients with three unfolding parameters are obtained and the normal forms for the non-semisimple 1:1 resonant Hopf bifurcation are obtained. [ABSTRACT FROM AUTHOR]

In the present article, we introduce an age-dependent predation system with prey harvesting, where we assume that the predators fertility function p (x) is a piecewise function concerning their maturation period τ. System is rewritten as a semilinear Cauchy problem with an operator which is not densely defined in its domain. With the help of the integrated semigroup theory, the Hopf bifurcation theorem, the center manifold argument and normal form theory for semilinear equations with non-dense domain, we demonstrate much richer novel dynamical phenomena than the existing ones. Qualitative analysis manifests that system undergoes a Hopf bifurcation around the positive equilibrium age distribution. Furthermore, the explicit formulae are provided to determine the direction the Hopf bifurcation and the stability of the bifurcating periodic solutions, which is the main result of this paper. Last but not least, numerical simulations are given to illustrate our analytical results. [ABSTRACT FROM AUTHOR]

• The aim is to establish the Hopf bifurcation theorem of rotating-periodic solutions in odd-dimensional systems or systems coupled by multiple odd-dimensional subsystems. • Hopf bifurcation theorem of rotating-periodic solutions can be used to analyze the mechanism of rotating waves in systems such as the unidirectionally coupled Lorenz systems. • The rotating waves can be periodic, quasi-periodic or other types. In this paper, we investigate the mechanism of rotating waves in a ring of unidirectionally coupled Lorenz systems. Rotating waves in our Lorenz systems are special cases of rotating-periodic solutions in nonlinear systems. Rotating-periodic solutions as a generalization of periodic solutions, have the form x (t + T) = Q x (t) with some orthogonal matrix Q. We establish the Hopf bifurcation theorem of rotating-periodic solutions in general odd-dimensional dynamical systems or systems coupled by multiple odd-dimensional subsystems. And by this bifurcation theorem, the existence of the rotating wave bifurcation is proved in the coupled Lorenz systems. In particular, the rotating waves can be periodic or quasi-periodic in our result. [ABSTRACT FROM AUTHOR]

• Effect of spatial average and delay on the predator-prey model is investigated. • Conditions for stability, steady state bifurcation and Hopf bifurcation are obtained. • Algorithm of normal form of steady state-Hopf bifurcation for system with spatial average and delay is derived. • Spatiotemporal dynamical classification near the steady state-Hopf bifurcation point is studied. In this paper, we discuss a diffusive predator-prey model with nonlocality and delay. Stability and bifurcation analysis suggest that the joint impacts of the nonlocal term and delay result in instability of the positive constant steady state. Moreover, steady state, Hopf and steady state-Hopf bifurcations and interactions of these bifurcations may occur under certain conditions. Normal forms of steady state, Hopf and steady state-Hopf bifurcations for a general reaction-diffusion model with nonlocal effects and delay are derived. In numerical simulations, spatially inhomogeneous steady states and periodic solutions and heteroclinic connections between these solutions are obtained. [ABSTRACT FROM AUTHOR]