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

Abstract: We investigate a population of individuals who play the Rock–Paper–Scissors (RPS) game. The players choose strategies not only by optimizing their payoffs, but also taking into account the popularity of the strategies. For the standard RPS game, we find an asymptotically stable polymorphism with coexistence of all strategies. For the general RPS game we find the limit cycles. Their stability depends exclusively on two model parameters: the sum of the entries of the RPS payoff matrix, and a sensitivity parameter which characterizes the personality of the players. Apart from the supercritical Hopf bifurcation, we found the subcritical bifurcation numerically for some intervals of the parameters of the model. [Copyright &y& Elsevier]

This paper will explore a fractional-order hepatitis B virus (HBV) infection model that takes into account cell-cell and virus-cell transmission and multiple treatment modalities. The desired control strategy is realized by means of a fractional order PD controller. Firstly, we calculated the basic regeneration number and equilibrium point of the HBV model. Afterwards, for the uncontrolled HBV virus system, the adequate conditions for both stability and Hopf bifurcation are systematically investigated via choosing the appropriate time delay as a parameter for bifurcation. Subsequently, under fractional order PD controller, the effect of a proposed controller on system stability and Hopf bifurcation is studied. The desired dynamic characteristics can be obtained afterwards. Finally, numerical simulations show that all three treatments significantly reduce R 0. The onset of oscillations can be delayed by decreasing the order of the fractional order. There are three control pathways for fractional order PD control, and the generation of bifurcation can also be delayed by changing the gain parameter. Using the above methods, the diffusion of HBV virus particles in the body can be effectively controlled. The conclusions drawn in this paper are extremely novel and have potential theoretical value for the future treatment of hepatitis B illness. [ABSTRACT FROM AUTHOR]

This paper makes a new breakthrough in deliberating the bifurcations of fractional-order bidirectional associative memory neural network (FOBAMNN). In the beginning, the corresponding bifurcation results are established according to self-regulating parameter, which is different from bifurcation outcomes available by using time delay as the bifurcation parameter, and greatly enriches the bifurcation results of continuous neural networks(NNs). The deived results manifest that a larger self-regulating parameter is more conducive to the stability of the system, which is consistent with the actual meaning of the self-regulating parameter representing the decay rate of activity. In addition to the innovation in the research object, this paper also has innovation in the procedure of calculating the bifurcation critical point. In the face of the quartic equation about the bifurcation parameters, this paper utilizes the methodology of implicit array to calculate the bifurcation critical point succinctly and effectively, which eschews the disadvantages of the conventional Ferrari approach, such as cumbersome formula and huge computational efforts. Our developed technique can be employed as a general method to solve the bifurcation point including the problem of dealing with the bifurcation critical point of delay. Ultimately, numerical experiments test the key theoretical fruits of this paper. [ABSTRACT FROM AUTHOR]

• Extended Goodwin model integrates endogenous technological change and labor supply. • Reinterpretation of induced innovation in terms of mechanization. • Application of 3D-Hopf bifurcation theorem reveals limit cycle dynamics. • Employment rate's effect on productivity as a bifurcation parameter defines cycle stability. • Numerical simulation for 10 OECD economies and sensitivity analysis. This paper extends the Goodwin model of distributive cycles by incorporating the simultaneous endogeneity of technical change and labor supply within a classical-Marxian framework. It reinterprets induced innovation, suggesting that firms optimize mechanization to maximize cost reduction, obtaining a micro-founded relationship between mechanization and the wage share. Additionally, it assumes a positive relationship between labor supply and the employment rate. The resulting three-dimensional dynamical system includes wage share, employment rate, and capital-output ratio as state variables. The Hopf bifurcation theorem reveals the emergence of limit cycles as the employment rate's effect on labor productivity (reserve-army-creation effect) approximates a critical value from below. Numerical simulations for 10 OECD countries illustrate the cyclical nature of the model and its consistency with empirical patterns. Furthermore, a sensitivity analysis explores the effect of parameters variations, emphasizing the social dimensions of productivity and labor supply as critical factors defining the stability of distributive cycles. [ABSTRACT FROM AUTHOR]

The red palm weevil is one of the most dangerous pests affecting date palm trees in the Gulf region and the Middle East. This paper investigates the control of the red palm weevil (RPW) in date palms using several techniques, namely stem injection, sex pheromone traps, and the Sterile Insect Technique. A mathematical model describing the spread of the red palm weevil is presented, and the boundedness of the solution for the RPW system is studied. The conditions for local stability, forward bifurcation, and Hopf bifurcation are also obtained. Additionally, this research highlights the significance of optimal control in managing RPW and simulates the model using the Forward-Backward Sweep numerical method. It is observed that the parameters of the sterile insect technique, the pheromone trap, and the mechanical injection play important roles in controlling the RPW. Mechanical Injection may be more effective in controlling the palm weevil compared to the other two methods. [ABSTRACT FROM AUTHOR]

• Frequency response of a strongly nonlinear time delay system is obtained. • Optimal time delays of control system are given. • First statement about the feedback control relationship between displacement and velocity. • Appropriate time delay parameters can make the system softening or hardening. In this paper, the strongly nonlinear vibration of a cantilever beam system with concentrated mass at an intermediate position controlled by displacement and velocity time delay is investigated. The nonlinear governing equation is studied using the homotopy analysis method. The effects of the time delay, displacement and velocity feedback gain coefficients, as well as frequency on the amplitude of the system were studied in detail. The results indicate that the velocity feedback control is not necessarily superior to displacement feedback control. Reasonable selection of the displacement and velocity time delay parameters can avoid the Hopf bifurcation, adjust the number, amplitude and stability of periodic solutions, and exhibit the softening behavior at low frequency and hardening spring characteristic at high frequency. The theoretical research in this paper will promote the application of homotopy analysis method in the field of time delay control, and serve as a theoretical reference for the design and optimization of cantilever beam control systems with concentrated mass. [ABSTRACT FROM AUTHOR]

This paper aims to study the multi-scale oscillation characteristics and stability of pumped storage unit(PSU) under primary frequency regulation(PFR) condition with low water head grid-connected. Firstly, a novel nonlinear mathematical model of pumped storage unit governing system(PSUGS) considering the nonlinear characteristics of diversion tunnel is established, and the nonlinear state equations under power control mode is deduced. On this basis, the bifurcation characteristics and dynamic response process of PSUGS system with coupled first-order, third-order and fifth-order synchronous generator are studied respectively, and the effects of different order synchronous generator on the nonlinear PSUGS system are compared. In addition, the transient characteristics of the nonlinear PSUGS system considering the fifth order synchronous generator under different water heads are studied. Finally, the sensitivity of the regulation system parameters of PSU under PFR with low water head is analyzed, and the effects of hydraulic, mechanical and electrical factors on the multi-scale oscillation characteristics and stability of the nonlinear PSUGS system are revealed. The results show that the stability of PSU considering the first-order synchronous generator is less than that considering the third-order and fifth-order synchronous generator. The stability of PSU considering the third-order synchronous generator is close to that considering the fifth-order synchronous generator. In the operation of PSU, the lower the water head, the smaller the stability of the system. Under PFR condition, the nonlinear PSUGS system meets the supercritical Hopf bifurcation. During PFR oscillation of the unit, the high-frequency wavelet is excited by the generator excitation system and is sensitive to the q-axis parameters of the generator. The low-frequency wavelet is mainly excited by the power grid and mapped with the water hammer effect in the dynamic process. The research results of this paper provide a theoretical basis for the stability analysis and optimal regulation of PSU under PFR condition with low water head. [ABSTRACT FROM AUTHOR]

There are many harmful effects of smoking. It not only engulfs the health and life of smokers, but also pollutes the air, endangers the life and health of others, and brings a heavy burden to public health. To this end, a delayed smoking model including potential smokers, occasional smokers, smokers, temporary quitters, permanent quitters and smokers with some illness, is investigated in the present paper. Firstly, local stability and existence of Hopf bifurcation of the model is conducted. Secondly, global exponential stability is explored. Lastly, we numerically simulate the correctness of the obtained theoretical results in the paper. [ABSTRACT FROM AUTHOR]

This paper extends the Marx-Keynes-Schumpeter model in Flaschel (2015) to study the social dimension of climate change. Agents are divided between those supporting and those opposing taxing Green House Gas (GHG) emissions. The composition of the population varies according to a continuous-time version of the discrete-choice approach. Conditional to the level of interaction between players, society chooses the respective carbon tax rate. Higher taxes reduce capital accumulation but support the development of energy-saving production techniques. Output growth and employment rates will be higher or lower depending on which effect prevails. Economic activity generates GHG emissions and determines the employment rate, which, in turn, endogenously feedback on environmental sentiments. Lower emissions reinforce sustainable attitudes, while falling employment increases households' concerns with more "urgent" needs, decreasing support for taxation. Hence, the model is compatible with a positive relationship between environmental attitudes and energy efficiency but not a clear association with output. A sufficiently strong response of attitudes to emissions combined with a "sentiments-autonomous" carbon tax may lead to the disappearance of the equilibrium in which most agents oppose regulation, controlling for multi-stability. Our 3-dimensional system admits endogenous persistent and bounded fluctuations, where the interaction between green attitudes and the macro-economy appears as a novel source of growth-cycle dynamics. [ABSTRACT FROM AUTHOR]

This paper illuminates the issue of bifurcations for a fractional-order bidirectional associative memory neural network(FOBAMNN) with four different delays. On account of the affirmatory presumption, the developed FOBAMNN is firstly transformed into the one with two nonidentical delays. Then the critical values of Hopf bifurcations with respect to disparate delays are calculated quantitatively by establishing one delay and selecting remaining delay as a bifurcation parameter in the transformed model. It detects that the stability of the developed FOBAMNN with multiple delays can be fairly preserved if selecting lesser control delays, and Hopf bifurcation emerges once the control delays outnumber their critical values. The derived bifurcation results are numerically testified via the bifurcation graphs. The feasibility of theoretical analysis is ultimately corroborated in the light of simulation experiments. The analytic results available in this paper are beneficial to give impetus to resolve the issues of bifurcations of high-order FONNs with multiple delays. [ABSTRACT FROM AUTHOR]

This paper expatiates the stability and bifurcation for a fractional-order neural network (FONN) with double leakage delays. Firstly, the characteristic equation of the developed FONN is circumspectly researched by employing inequable delays as bifurcation parameters. Simultaneously the bifurcation criteria are correspondingly extrapolated. Then, unequal delays-spurred-bifurcation diagrams are primarily delineated to confirm the precision and correctness for the values of bifurcation points. Furthermore, it lavishly illustrates from the evidence that the stability performance of the proposed FONN can be demolished with the presence of leakage delays in accordance with comparative studies. Eventually, two numerical examples are exploited to underpin the feasibility of the developed theory. The results derived in this paper have perfected the retrievable outcomes on bifurcations of FONNs embodying unique leakage delay, which can nicely serve a benchmark deliberation and provide a comparatively credible guidance for the influence of multiple leakage delays on bifurcations of FONNs. [ABSTRACT FROM AUTHOR]

To explore transmission dynamics of tuberculosis, a tuberculosis transmission model with vaccination and time delay is developed in current paper. Positivity and boundedness are analyzed. Local stability of tuberculosis-free equilibrium in respect of the time delay due to latent period of tuberculosis is analyzed and we have found threshold value of the time delay for the local stability of tuberculosis-free equilibrium. Then, local stability of tuberculosis-existence equilibrium following exhibition of Hopf bifurcation at the crucial value of the time delay due to latent period of tuberculosis is derived. It is shown that the developed model undergoes a Hopf bifurcation around the tuberculosis-existence when the time delay due to latent period of tuberculosis passes through the threshold value. Direction and stability of the Hopf bifurcation are investigated with the help of the normal form method and center manifold theory. Finally, numerical simulations are carried out in the justification of obtained analytical findings. The results obtained provide significant information for tuberculosis disease controlling. [ABSTRACT FROM AUTHOR]

This paper examines economic cycles that do not depend on exogenous economic actions. More precisely, the paper develops a positive model of government behavior in order to define the intertemporal fiscal policies that are optimal for a country and which, determine both the optimal budget level and the optimal level of environmental quality. For this purpose, we establish an optimal control model involving intertemporal subsidy strategies characteristically used by an authoritarian government similar to those found in central Europe. It will be shown that by applying the Hopf bifurcation theorem,a cyclical strategy – that is, waves of regulation, environmental subsidies alternating with deregulation and cuts in social programs – may represent an optimal policy. In this paper, we propose an extremely simple optimal control model which is applied to budget surpluses and environmental subsidies. We investigate the cyclical environmental policies as applied through the bifurcation theorem. A number of propositions are stated during the solution process. The first proposition sets the rules in the model parameters in order to establish the cyclical policies. A second proposition sets the relationship between the discount rate and the opportunity cost of capital, which is used to determine a taxing strategy which becomes the future optimal subsidy policy. [ABSTRACT FROM AUTHOR]

Abstract This paper shows the importance of considering the thermal state of a combustor to investigate or predict its thermoacoustic stability. This aspect is often neglected or regarded as less important than the effect of the operating parameters, such as thermal power or equivalence ratio, but under certain circumstances it can have a dramatic influence on the development of the instabilities. The paper presents experimental results collected from a combustor featuring a lean swirl-stabilized flame exhibiting thermoacoustic instability at some operating conditions. It is shown that this instability is caused by a change of the flame topology that is induced by the progressive increase of the wall temperature with the thermal power. This dependence of the instability on wall temperature leads to inertial effects and hysteresis when the operating condition is changed dynamically. A low-order model of the system reproducing this remarkable dynamics is proposed and validated against the experimental data. [ABSTRACT FROM AUTHOR]

In this paper, it is found that the systems alternate between oscillatory and steady states in studying the stability of a class of stochastic dynamical systems with fast time-varying periodic delays. We propose a new definition of bifurcation, State Flip (SF) bifurcation, by using the abrupt change in the dwell time ratio of two states. Taking the amplitude of the time-varying delay function δ as the bifurcation parameter, we investigate the dynamics of the evolution of a time-varying delay system driven by Gaussian white noise. Firstly, we convert systems with fast time-varying periodic delays into comparative systems with distributed delays, which in turn are converted into stochastic systems with constant delays for discussion. In order to reduce the stochastic delay equation to the average Itô equation, we adopted the stochastic averaging method and the generalized central manifold theory. It is used to capture the one-dimensional slow variables of a system near the critical state of the transition between two steady states during the evolution of the system. We then find that as the amplitude increases from weak to strong, the stochastic time-varying delay system changes from a two-period oscillation to alternating between oscillation and steady, and finally to a steady state. Lastly, we perform numerical simulations of the original stochastic time-varying delay system and the transformed two-dimensional stochastic constant time-delay system respectively to verify that the ideas and methods in this paper are reasonable and effective. • We have conducted an analysis of a class of stochastic system dynamics characterized by fast time-varying pointwise delay. • The method of transforming distributed delay systems into equivalent time-varying delay systems has been thoroughly investigated. • We have identified a novel bifurcation concept: state flip bifurcation. [ABSTRACT FROM AUTHOR]

Gene regulatory networks play an extremely important role in human life activities. This paper presents a novel fractional-order gene regulatory network model (NFGRNM). The model employs incommensurate fractional orders and introduces time delay arising from the binding of miRNAs to RNA. Time delay often triggers instability such as oscillation and bifurcation, so in this paper we introduce a generalized hybrid control to control the unstable dynamical behavior induced by time delay. First we derive the NFGRNM using fractional order theory, after which the equilibrium point of the network model is calculated. Next, a systematic dynamics analysis of the NFGRNM is performed to derive sufficient conditions to generate the Hopf bifurcation. Most importantly, we introduce the extended hybrid control for the periodic oscillations generated by the time delay in the NFGRNM. Finally, numerical simulation results show that the order of the fractional order can have a great impact on the dynamical behavior of the network model and that the hybrid control has a significant role in delaying (or advancing) the Hopf bifurcation. Since infectious diseases such as SARS-Cov-2, influenza A, and HIV are still prevalent today, the study of gene regulatory networks not only promotes the study of the dynamics of complex networks but also has profound implications for the study of virus transmission in the host and various life activities in humans. • The time lag generated by miRNA-RNA binding is used as a bifurcation parameter. • The Hopf bifurcation behavior of gene regulatory network models is investigated and generalized hybrid control is added. • The effect of fractional order and control parameters on the robustness of the system is investigated. • The effect of disproportionate fractional orders and control parameters on the dynamical behavior of system bifurcations. [ABSTRACT FROM AUTHOR]

This paper reports on the bifurcations of a single-species model with spatial memory environment. The occurrence conditions of the Hopf bifurcation and Turing bifurcation are given with/without the spatial memory effect. Especially, we show that the positive equilibrium enjoys the stability switches with spatial memory effect. [ABSTRACT FROM AUTHOR]

In this paper, a reaction–diffusion–advection model with nonlocal delay effect and nonlinear boundary condition is investigated. By employing the Lyapunov–Schmidt reduction method, we not only establish the existence, multiplicity and stability of spatially nonhomogeneous steady-state solutions, but also obtain some sufficient conditions ensuring the occurrence of a Hopf bifurcation at the steady-state solutions. It is observed that time delay determines the existence of Hopf bifurcation when the interior reaction term is stronger than the boundary reaction term. Finally, the general theoretical results are applied to a diffusive Logistic model with advection term under monostable nonlinear boundary condition and the effect of advection on Hopf bifurcation values is also considered. The results show that Hopf bifurcation is more likely to occur in the case of small advection. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamics of a directed acyclic neural network by edge adding control. We find that the local stability and Hopf bifurcation of the controlled network only depend on the size and intersection of directed cycles, instead of the number and position of the added edges. More specifically, if there is no cycle in the controlled network, the local dynamics of the network will remain unchanged and Hopf bifurcation will not occur even if the number of added edges is sufficient. However, if there exist cycles, then the network may undergo Hopf bifurcation. Our results show that the cycle structure is a necessary condition for the generation of Hopf bifurcation, and the bifurcation threshold is determined by the number, size, and intersection of cycles. Numerical experiments are provided to support the validity of the theory. [ABSTRACT FROM AUTHOR]

This paper examines a class of two-species reaction-diffusion-advection competition models with two time delays. A system of DDE equations was derived, both theoretically and numerically, using the Galerkin technique method. A condition is defined that helps to find the existence of Hopf bifurcation points. Full diagrams of the Hopf bifurcation points and areas of stability are investigated in detail. Furthermore, we discuss three different sources of delay on bifurcation maps, and what impacts of all these cases of delays on others free rates on the regions of the Hopf bifurcation in this model. We find two different stability regions when the delay time is positive (τ > 0), while the no-delay case (τ = 0) has only one stable region. Moreover, the effect of delays and diffusion rates on all free others parameters in this model have been considered, which can significantly impact upon the stability regions in both population concentrations. It is also found that, as diffusion increases, the time delay increases. However, as the delay maturation is increased, the Hopf points for both proliferation of the population and advection rates are decreased and it causes raises to the region of instability. In addition, bifurcation diagrams are drawn to display chosen instances of the periodic oscillation and two dimensional phase portraits for both concentrations have been plotted to corroborate all analytical outputs that investigated in the theoretical part. • To study the effect of diffusion with two different delay terms on the two-species advection competition system. • To drive theoretical outcomes of the DDE obtainable utilizing the Galerkin technique tool. • To find, theoretically, a condition that assists to obtain the existence of Hopf bifurcation points. • To construct the bifurcation diagram for the DDE and DPDE systems via numerical simulation to confirm theoretical results. [ABSTRACT FROM AUTHOR]

For many physical systems the transition from a stationary solution to sustained small amplitude oscillations corresponds to a Hopf bifurcation. For systems involving impacts, thresholds, switches, or other abrupt events, however, this transition can be achieved in fundamentally different ways. This paper reviews 20 such 'Hopf-like' bifurcations for two-dimensional ODE systems with state-dependent switching rules. The bifurcations include boundary equilibrium bifurcations, the collision or change of stability of equilibria or folds on switching manifolds, and limit cycle creation via hysteresis or time delay. In each case a stationary solution changes stability and possibly form, and emits one limit cycle. Each bifurcation is analysed quantitatively in a general setting: we identify quantities that govern the onset, criticality, and genericity of the bifurcation, and determine scaling laws for the period and amplitude of the resulting limit cycle. Complete derivations based on asymptotic expansions of Poincaré maps are provided. Many of these are new, done previously only for piecewise-linear systems. The bifurcations are collated and compared so that dynamical observations can be matched to geometric mechanisms responsible for the creation of a limit cycle. The results are illustrated with impact oscillators, relay control, automated balancing control, predator–prey systems, ocean circulation, and the McKean and Wilson–Cowan neuron models. [ABSTRACT FROM AUTHOR]

In this paper, an improved reduced order model for studying spatial xenon oscillations has been developed which can be utilized for stability analysis (linear as well as nonlinear). The model presented in this paper has been developed to overcome situations of unphysical results which were obtained by earlier models which used quasi static assumption when the system moves away from the equilibrium point. The earlier model consisted of multipoint neutron kinetics equations including xenon and iodine feedback. In order to eliminate unphysical behaviour in the earlier model, the quasi-static approach is discarded and the total power control is represented by an ordinary differential equation. It represents the action of a reactivity device governed by the deviation of the total power from the steady state power level. In the current approach, a PHWR is modelled by dividing it into two regions. The linear and nonlinear stability analysis was carried out using the new model and compared with the older results. Linear stability maps were constructed in different parameter planes and were found to be identical. However, there was a change in the nonlinear characteristics of the system and the results are presented in this paper. Moreover, the effect of control device on the system behaviour especially on total power variation is studied through numerical simulations. • Improved reduced order model for stability analysis of Xenon oscillations. • Quasi steady state approximation is not adequate for nonlinear analysis. • Linear stability boundary were found to be identical with model reduction. • Effect of model reduction on bifurcation characteristics is analyzed. • Interchange of type of Hopf bifurcation is observed. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with the existence of a limit cycle in a nonlinear ODE mathematical model which describes the interaction between three homogeneous populations. These take the form of pollinators, plants and herbivores. The interaction between the pollinators and plants is of mutualistic type given by a functional response of type II; meanwhile that for the plants–herbivores, from the demographic point of view, it can be seen as a predator–prey interaction. The resulting coupled nonlinear ODE system contains several parameters which have an important ecological interpretation. In the study we present here we give necessary conditions on the parameter values for the emergence of an attracting limit cycle and other which is unstable. These come from a supercritical and a subcritical Hopf bifurcation, respectively. Because of the amount of the involved parameters, almost any calculation becomes really massive. Thus, part of the analysis we present here intensively uses the Mathematica symbolic software. With the same software, by using a theorem authored by Kuznetsov, we calculated the first Lyapunov coefficient whose sign gives us the type of stability the limit cycle has. A number of numerical simulations also are included with the aim of showing the limit cycle in the three dimensional phase space. This is done in two main cases: when the ODE system has one positive equilibrium and when such system has two positive equilibria. The main part of the paper ends by exploring other set of parameters values which allows us to recover the previous dynamics but also gives us other interesting behaviors. [ABSTRACT FROM AUTHOR]

Abstract This paper is mainly concerned with an investigation of antigen-driven T-cell infection process through a mathematical model. Several mathematical models have been introduced in the literature in order to gain insights into the dynamics of the disease progression, however, the results considering the effect of multiple factors which include antigen-driven CD4 T-cell progress, latent infection stage, activation of CTLs response, role of antiretroviral therapies (ARTs) and possibilities of multiple time-delays during the infection process are not involved. Hence, the paper introduces a six-dimensional virus infection model by involving the above factors. Particularly, (i) the effect of activation of antigen-specific T-cells; (ii) the effect of the maturation of infected cells; (iii) effect of multiple time delays, that is, during the interaction between susceptible and infectious and during the activation of immune responses; which play a significant role in preventing and modulating the Human Immunodeficiency Virus (HIV). The main aim of the paper to analyze the local and global stability of the class of mathematical models regarding the effect of time delays which provides a better pathway to the infection progress. Finally, the overall contribution of the present work is listed as follows: (1) by constructing the suitable Lyapunov functional, the global stability of the intracellular delayed model is derived; (2) detailed Hopf-bifurcation analysis is discussed with respect to immune response delay. The numerical simulation is performed to validate the effectiveness and applicability of the theoretical predictions. [ABSTRACT FROM AUTHOR]

Abstract With rapid emergence of online social networks, the rumor transition has a growing significant impact on social stability and human lives. This paper investigates the local stability and optimal control of a rumor propagation model in online social network with changing number of total users, which is proposed in Ref. Dong et al. (2018). In addition, we set up a model with a delay based on the reality, and analyze the sufficient conditions for Hopf bifurcation. Finally, the rumor spreading model is simulated numerically. The results of simulation are shown the process of rumor propagation and prove the correctness of the theoretical analysis. Highlights • This paper presents an important improvement in theory and numerical analysis for Ref. Dong et al. (2018). • Bifurcation and stability switches are proved in a strict mathematical way. • Optimal control strategy of rumor propagation is investigated and compared in this paper. [ABSTRACT FROM AUTHOR]

Highlights • A general tridiagonal two-layer neural network model with delay is proposed. • A new method of Hopf bifurcation analysis is introduced by matrix decomposition. • The conditions obtained are simpler than traditional Hurwitz discriminant method. Abstract Firstly, a general tridiagonal two-layer neural network model with 2 n -neuron is proposed, where every layer has time delay. A new method of Hopf bifurcation analysis is introduced by matrix decomposition in this paper. Through factoring the tridiagonal matrix, the characteristic equation of the neural network model is simplified. Secondly, by studying the eigenvalue equations of the related linear system for the special six-neuron (three neurons per layer) two-layer neural network model, the sufficient conditions for experiencing the Hopf bifurcation are obtained. The conditions obtained by the new method proposed in this paper are simpler and more practical than those obtained by the traditional Hurwitz discriminant method. Next, based on the normal form method and the center manifold theorem, the explicit formulae about the stability of the bifurcating periodic solution and the direction of the Hopf bifurcation are established. Finally, the main results obtained in this paper are illustrated by three numerical simulation examples. [ABSTRACT FROM AUTHOR]

• Goodwin's model with delayed investment function. • Cyclical growth model with time delays and delay dependent coefficients. • Existence of Hopf bifurcations analyzed by choosing time delays as bifurcation parameters. • Chaotic dynamics. In this paper, we reconsider the Goodwin 1967 growth-cycle model, where the antagonistic relationship between wages and profits is assimilated to the prey-predator conflict modeled by Volterra in 1931. Here we propose an extension of Goodwin's basic model by adding two important elements of the business cycle theory: (i) a finite time delay between investment orders and deliveries of finished capital goods, as theorized by Kalecki (1935); (ii) a delayed reaction of real wages to the unemployment levels, as suggested by Chiarella (1990). Both these delays preserve the two-dimensionality of the original model, but it becomes a delayed differential equation system, with two discrete time delays and one-delay dependent parameters. The qualitative study of the system shows that without lags the economic meaningful equilibrium is structurally stable. Nevertheless, as soon the time delays become positive, that equilibrium loses its stability and, according to the combinations of parameters and length of the lags, either periodic or non-periodic (chaotic) fluctuations arise. Numerical simulations supporting the economic analysis show that, in the very long run, a "strange attractor" depicts the dynamic behavior of the system. [ABSTRACT FROM AUTHOR]

In this paper, it is proved that the KdV–Burgers equation with a monostable source term of Fisher–KPP type has small-amplitude periodic traveling wave solutions with finite fundamental period. These solutions emerge from a subcritical local Hopf bifurcation around a critical value of the wave speed. Moreover, it is shown that these periodic waves are spectrally unstable as solutions to the PDE, that is, the Floquet (continuous) spectrum of the linearization around each periodic wave intersects the unstable half plane of complex values with positive real part. To that end, classical perturbation theory for linear operators is applied in order to prove that the spectrum of the linearized operator around the wave can be approximated by that of a constant coefficient operator around the zero solution, which intersects the unstable complex half plane. • The Burgers–Korteweg–de Vries equation with a monostable source is considered. • It is proved that small-amplitude periodic waves do exist. • The periodic waves emerge from a subcritical Hopf bifurcation around a critical value of the wave speed and have finite fundamental period. • It is shown that these periodic waves are spectrally unstable as solutions to the PDE. • The Floquet spectrum of the linearized operator around a periodic wave intersects the unstable complex half plane. [ABSTRACT FROM AUTHOR]

This paper investigates the Andronov–Hopf and Bogdanov–Takens bifurcations in a three-dimensional Hindmarsh–Rose system with switching dynamics. A switching policy is applied to the slow recovery variable, dividing the system into two smooth subsystems by setting the switching threshold to zero. The presence of visible or invisible fold and cusp singularities in the sliding vector field is examined. By examining conditions satisfied by the trace and determinant, this study establishes the existence of Andronov–Hopf and Bogdanov–Takens bifurcations. The necessary and sufficient conditions for the occurrence of Andronov–Hopf and Bogdanov–Takens bifurcations are derived. Through numerical simulations, we have obtained the irregular limit cycle phase portrait with sawtooth-shaped and the corresponding time series of spike peaks gradually attenuating within the firing period, these new firing phenomena attributed to the switching policy. We have gathered statistics on the spike counts of membrane potential during firing events within a specific range of parameter d. Thus, we can achieve a wide variety of bursting patterns through this switching system. Furthermore, single-parameter and two-parameter bifurcation diagrams are generated by varying the key parameters b and d , we found that the chaotic and periodic regions intertwine with each other, with chaotic regions existing within the periodic regions, and periodic states found within the chaotic regions. The number of spikes during a firing period in neurons can provide important clues about the neuronal activity state, information processing, and potential diseases. In summary, this study demonstrates that the Filippov Hindmarsh–Rose system, characterized by its slow variable switching threshold, significantly influences the firing behavior of the system. • The study derived the necessary and sufficient conditions for Hopf and Bogdanov-Takens bifurcations. • The irregular limit cycle phase portrait with sawtooth-shaped attributed to the switching policy. • Choosing the slow recovery variable as the switching threshold enriched various bursting states. [ABSTRACT FROM AUTHOR]

Somitogenesis, the sequential formation of a periodic pattern along the anteroposterior axis of vertebrate embryos, is one of the most obvious examples of the segmental patterning processes that take place during embryogenesis and also one of the major unresolved events in developmental biology. In this paper, we investigate the effect of diffusion on pattern formation use a modified two dimensional model which can be used to explain somitogenesis during embryonic development. This model is suitable for exploring a design space of somitogenesis and can explain many aspects of somitogenesis that previous models cannot. In the present paper, by analyzing the local linear stability of the equation, we acquired the conditions of Hopf bifurcation and Turing bifurcation. In addition, the amplitude equation near the Turing bifurcation point is obtained by using the methods of multi-scale expansion and symmetry analysis. By analyzing the stability of the amplitude equation, we know that there are various complex phenomena, including Spot pattern, mixture of spot–stripe patterns and labyrinthine. Finally, numerical simulation are given to verify the correctness of our theoretical results. Somitogenesis occupies an important position in the process of biological development, and as a pattern process can be used to investigate many aspects of embryogenesis. Therefore, our study helps greatly to cell differentiation, gene expression and embryonic development. What is more, it is of great significance for the diagnosis and treatment of human diseases to study the related knowledge of model biology. [ABSTRACT FROM AUTHOR]