Search

A comprehensive theoretical and numerical analysis of the dynamical features of a fractional-order delay financial risk system(FDRS) is presented in this paper. Applying the linearization method and Laplace transform, the critical value of delay when Hopf bifurcation first appears near the equilibrium is firstly derived in an explicit formula. Comparison simulations clarify the reasonableness of fractional-order derivative and delay in describing the financial risk management processes. Then we employ persistent homology and six topological indicators to reveal the geometric and topological structures of FDRS in delay interval. Persistence barcodes, diagrams, and landscapes are utilized for visualizing the simplicial complex's information. The approximate values of delay when FDRS undergoes different periodic oscillations and even chaos are determined. The existence of periodic windows within the chaotic interval is correctly decided. The results of this paper contribute to capturing intricate information of underlying financial activities and detecting the critical transition of FDRS, which has promising and reliable implications for a deeper comprehension of complex behaviors in financial markets. • Determine delay τ 0 when Hopf bifurcation appears. • The effects of fractional orders and parameters on τ 0 are elucidated. • Topological features are visualized by simplicial complex in phase space. • Six indicators based on persistent homology identify varied oscillations. • A fractional-order delay system is reasonable to describe financial activities. [ABSTRACT FROM AUTHOR]

This paper aims to introduce a technique that utilizes both dynamical mechanisms and machine learning to reduce dimensionality in high-dimensional complex systems. Specifically, the method employs Hopf bifurcation theory to establish a model paradigm and use machine learning to train location parameters. The effectiveness of the proposed method is evaluated by testing the Van Der Pol equation and it is found that it possesses good predictive ability. In addition, simulation experiments are conducted using a hunting motion model, which is a well-known practice in high-speed rail, demonstrating positive results. To ensure the robustness of the proposed method, we tested it on noisy data. We introduced simulated Gaussian noise into the original dataset at different signal-to-noise ratios (SNRs) of 10 db, 20 db, 30 db, and 40 db. All data and codes used in this paper have been uploaded to GitHub. [ABSTRACT FROM AUTHOR]

The paper proposes a technique to accurately and efficiently locate periodic steady-state solutions of electric power systems. This technique is based on an enhanced version of the time-domain shooting method (TDSM) and the probe-insertion technique (PIT). The latter has been successfully applied to low-power electronic circuits but it is innovative for the study of electromechanical steady-state periodic behavior of power systems. With this aim, the paper discusses the inherent criticalities of the conventional formulation of power system models (PSMs). Then, a novel formulation is proposed to accommodate the hypotheses and mathematical requirements of the TDSM and PIT. The effectiveness and numerical efficiency of the proposed model and technique are discussed through two case studies based on the IEEE 14-bus and WSCC 9-bus systems. [ABSTRACT FROM AUTHOR]

In this paper, a class of virus-to-cell HIV model with intracellular delays, saturated incidence and Logistic growth is proposed to characterize the interaction between two types HIV strains, i.e., wild-type and drug-resistant strains. First, a series of threshold criteria on the locally and globally asymptotic stability of (infection-free, dominant, coexistence) equilibria are discussed based on the basic reproduction number R 0. Furthermore, a detailed Hopf bifurcation analysis is performed on the coexistence equilibrium using two delays as bifurcation parameters. We find that the Hopf bifurcations induced by double-strains are evidently different and more complicated than that of single strain, the former switches from stability (periodic branches) to un-stability (chaos) more frequently and earlier than the latter since double-strains would yield more pairs of imaginary roots in the characteristic equations. Meanwhile, the total viral load of double-strains would be higher than that of single-strain as well. The emergence of drug resistance imposes either negative or positive influences on the survival of wild-type strain, which would further facilitate the transmission of HIV. [ABSTRACT FROM AUTHOR]

In this paper, a model with spatial diffusion and mixed delays is presented to describe the spread of echinococcosis between dogs and livestock. Firstly, the local stability is investigated using the Routh–Hurwitz criterion. Furthermore, when considering time delays as bifurcation parameters, the conditions for the occurrence of Hopf bifurcation are discussed based on the linear approximation of the nonlinear system. Lastly, the fixed-time stability is studied under the implementation of appropriate control measures. Numerical simulations are provided to give a better understanding of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a diffusive consumer–resource model in which the consumer is involved with spatiotemporal memory and the resource has maturation delay. Firstly, for the case without maturation delay, the influence of the spatiotemporal distributed memory on the stability of the positive steady state is investigated. It has been shown that memory delay can induce Turing bifurcation for the negative memory-based diffusion coefficient and induce Hopf bifurcation for the positive memory-based diffusion coefficient. Then the joint effect of the memory delay and the maturation delay on the stability of the positive steady state is investigated and it has been shown that the joint effect of the memory delay and the maturation delay can induce more complicated spatiotemporal dynamics via Turing–Hopf bifurcation and double Hopf bifurcation. Finally, we apply our theoretical analysis results to a consumer–resource model with Holling-II type functional response to illustrate the existence of the different spatiotemporal patterns. • Propose a diffusive resource-consumer model with distributed memory and maturation delay. • Investigate the joint effect of the memory delay and the maturation delay on the spatiotemporal dynamics. • Find the spatially inhomogeneous spatial patterns and spatially inhomogeneous periodic patterns. [ABSTRACT FROM AUTHOR]

This paper mainly studies the stability and Hopf bifurcation criteria of hub-based genetic regulatory networks with multiple delays and bidirectional couplings. The hub-structured network is an important motif in complex networks, which provides a new view angle on structure to describe the regulation mechanism between genes (including both mRNAs and proteins). It is well known that hubs play a leading role in characterizing the network dynamical behaviors. However, the dynamics of hubs or star-coupled systems is not well understood. In this paper, we first examine the existence of the positive equilibria in this type of genetic networks. By analyzing the associated characteristic equation, we present sufficient conditions of biochemical parameters for delay-independent local stability in hub-coupled genetic regulatory networks. Then we investigate their Hopf bifurcation when such networks lose their stability. Specific conditions for delay-dependent stability and Hopf bifurcations in genetic networks with hub structure are derived. It is found that the dynamics of hub-structured genetic regulatory networks has no direct relationship with single time delay or individual connection, but instead depends on the sum of all delays among all genes and the product of the connection strengths between all genes. Finally, some simulation examples are provided to substantiate our analysis. [ABSTRACT FROM PUBLISHER]

In this paper, a cortex-basal ganglia model about Parkinson's disease with two time delays is studied, and the critical conditions for Hopf bifurcation are derived. The results show that time delays can change the state of basal ganglia. The basal ganglia is stable when the delays are small. However, when the time delay is greater than the corresponding bifurcation critical point, different types of oscillations occur in the basal ganglia. The larger the time delays, the more active the neuronal clusters in the basal ganglia. Furthermore, the bidirectional Hopf bifurcation is found by studying the connection weights between different neural nuclei. Finally, the influence of connection weight and time delay which are related to the internal segment of the globus pallidus on its oscillation is discussed. Research shows that reducing the connection weight and the corresponding time delay in excitatory neuronal clusters, or increasing the connection weight and decreasing the corresponding time delay in inhibitory neuronal clusters, can improve the oscillation of Parkinson's disease. [ABSTRACT FROM AUTHOR]

In this article, we analyze the conditions for the structural stability of a fractional order IS-LM-AS dynamic model with adaptive expectations. It is a generalization of our previous research lately published in the literature. We also present the conditions that the structural parameters of the model must meet for the economic system to present a periodic movement when the critical value of the fractional order of the system, q * , guarantees the presence of a Hopf bifurcation of degenerate type. The theoretical analysis is complemented with numerical simulations of the phase portraits in R 3 and of the temporal trajectories of the solutions of the model in MATLAB software. Finally, it is important to highlight that unlike the results of our previous research, the qualitative results found in this paper show that all the structural parameters of the model are essential in determining its global asymptotic stability and Hopf bifurcation. • A fractional economic model generalizes an integer-order model recently reported. • The memory effect in numerical simulations of economic systems is analyzed. • The fractional order when a Hopf bifurcation occurs is determined. • All the parameters of the model are essential in determining its global stability. [ABSTRACT FROM AUTHOR]

In this paper, we present a delayed deterministic and stochastic S I R I C V models to investigate the effects of the white noise intensities and the waning immunity of vaccinated individuals in the evolution of the disease. For the deterministic S I R I C V model, the basic reproduction number R 0 and the equilibrium points are calculated. The local stability of equilibrium points is analyzed. Particularly, when R 0 < 1 the disease-free equilibrium is locally stable for any positive value of τ. Furthermore, when R 0 > 1 , the local stability and sufficient conditions to ensure the occurrence of Hopf bifurcation for the endemic equilibrium point are established by considering the time delay τ as a bifurcation parameter. For the stochastic S I R I C V model, the conditions of the extinction and persistence of the disease are given by using the stochastic basic reproduction numbers R 0 s and R 0 s ∗. Numerical simulations are presented to enhance our analytical results and contrast the deterministic and stochastic models. [ABSTRACT FROM AUTHOR]

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

S-type locally-active memristor (LAM) has a great potential for brain- inspired neuromorphic computing, where the S-type LAM-based oscillator is a fundamental building block. Concerning the S-type LAM, this paper constructs a material-independent model in simple mathematical expression, which can be relatively easily analyzed. By biasing the memristor into the locally- active region, and connecting it with a capacitor, a second-order oscillator can be built. The small-signal equivalent circuit of the memristor and its frequency response are applied to determine the period oscillation frequency range and compensation capacitance. Hopf bifurcation theory is used to analyze oscillation mechanism of the second-order circuit and appropriate capacitance. By adding an extra inductor into the second-order oscillator, a novel third-order chaotic circuit is developed, where a saddle-focus is derived to create chaos. Its dynamic characteristics are investigated via Lyapunov exponents, bifurcation diagrams, dynamic route map, and so on. The local activities of the single memristor, second-order oscillator, and third-order chaotic circuit are verified through the mathematical analysis. Finally, physical circuit realizations of the S- type LAM-based oscillators, including the memristor emulator, are presented. Both simulation and experimental results demonstrate the practicability of the proposed mathematical model and the validity of the theoretical analysis. [ABSTRACT FROM AUTHOR]

It is well known that spatial memory exists in animal movement modelling. Since spatial memory is related to the past information, then delay arises. In view of the protection of ecological environment, more and more scholars are concerned about the effects of toxic substances on it. A prey–predator system with memory-induced cross-diffusion and toxin in predator is considered in this paper. First, we discuss the fundamental dynamics in detail. Then considering the memory-induced cross-diffusion coefficient and the averaged memory period delay in predator as the controlling parameters, we get that n -mode Hopf bifurcations exist at the positive steady state. Stability switches are generated, and there are spatially nonhomogeneous periodic solutions. Namely if the cross-diffusion coefficient is small, the populations always coexist. If the cross-diffusion coefficient is moderate, two kinds of critical values of the averaged memory period of Hopf bifurcations occur and stability switches may be induced by the delay. If the cross-diffusion coefficient becomes larger, one kind of critical value of the averaged memory period of Hopf bifurcation occur. From simulations, the memory-based cross-diffusion and toxin have vital influences on stability. The moderate toxin can be good for population coexistence. When the averaged memory period is small, the toxin does not change the stability. Once the averaged memory period is bigger, the toxin can change the stability. Moreover, it shows that the averaged memory delay could switch the stability of the system. [ABSTRACT FROM AUTHOR]

The evolution of financial system depends not only on the current state, but also on the previous state. Due to "long-term memory" and "non-locality" of the fractional derivative, fractional-order model can effectively characterize the dynamic features of financial process. An incommensurate fractional-order delay financial system (FDFS) is considered in this paper. Based on linearization and Laplace transformation, the characteristic equation of linearized system of FDFS is obtained. The critical value of the time delay for the occurrence of Hopf bifurcation is determined through the discussions of the eigenvalues of the characteristic equation and the transversality condition. A periodic pulse delay feedback controller is added to the FDFS to control the Hopf bifurcation and to regulate the stability domain of the system. Two illustrative examples are provided to validate our theoretical results. Moreover, numerical simulations demonstrate that the increase of the fractional-order can induce chaos in FDFS, which is detected by 0 − 1 test for chaos. This paper contributes to a better understanding of the dynamic behavior of financial market, forecasting financial risk and implementing effective financial regulation. [ABSTRACT FROM AUTHOR]

Although marine ecosystem is a highly complex phenomenon with many non-linearly interacting species, dissolved oxygen and plankton among these have perhaps the most fundamental relationship not only for the protection of marine environment but also for continuation of life on Earth. This paper deals with a generic diffusive model of dissolved oxygen, phytoplankton and zooplankton species, for which constant time delays are incorporated in growth response of phytoplankton and in the gestation time of zooplankton. We mainly focus on the stability analysis of the coexisting states and the existence of Hopf bifurcation through the characteristic equation, where time delay and oxygen production rate are considered as control parameters for all cases. Studying the effect of both time delays on a stable system, we show destabilisation of the system and irregularity in the spatio-temporal dynamical regimes, leading to chaotic oscillations. Although both delay terms have a destabilising effect, our findings indicate that time delay in zooplankton gestation may induce sharp strongly irregular pattern, whereas time delay in phytoplankton growth gives rise to more regular but higher frequency oscillations for oxygen–plankton interactions. The findings of this paper may provide new insights into main environmental issues including global warming and climate change. [ABSTRACT FROM AUTHOR]

• A new car-following model is proposed to describe the dynamic behavior of autonomous vehicle. • The bifurcation characteristic of traffic flow composed of autonomous vehicle is analyzed. • A controller that considers multi-step memory and multi-step prediction effect is designed. • A hybrid control strategy is proposed, which combines multi-step prediction and memory mechanism with PID. This paper is committed to capturing the dynamic behaviors of homogenous flow of autonomous vehicles (AVs), and exploring the control strategies to improve traffic conditions, which can alleviate traffic congestion and improve traffic efficiency. Firstly, a car-following model of AVs considering real-time driving state is established. Secondly, based on bifurcation theory and stability theory, bifurcation analysis is carried out and the relationship between bifurcation and stability is revealed. In order to suppress the bifurcation and improve the stability, a controller considering multi-step prediction and memory mechanism (MPM) is designed, and the root trajectories for eigenvalues and stable time length of the model controlled by MPM controller are calculated. In response to the limitations of the MPM controller, a hybrid controller including the MPM controller and PID controller is further proposed, and it is found that the model controlled by hybrid controller has greater range of stable bifurcation parameter and stable time length, which means better ability of bifurcation suppression. Finally, the capabilities of the controller proposed in this paper are effectively demonstrated by numerical experiments in MATLAB and simulation experiments in the ROS-Gazebo environment. [ABSTRACT FROM AUTHOR]

Recently, by replacing the parallel LC “resonator” in Chua's circuit with a lossless transmission line that is terminated by a short circuit, a nonlinear transmission line oscillator is proposed and is named as a “time-delayed Chua's circuit.” In a transmission line oscillator, a linear wave travels along a piece of the transmission line and interacts with terminating electrical components, i.e., a boundary-value problem for a hyperbolic partial differential equation with boundary conditions. Furthermore, a fixed time delay arises due to the propagation time through the transmission line. Delay-induced dynamics are particularly important in technologies where high-speed computers must take into account transmission line effects including propagation delay to prevent undesirable self-oscillations. In this paper, the equation describing Chua's circuit with a lossless transmission line is reduced to a differential-difference equation with one delay (i.e., neutral-type differential equation). By using suitable Lyapunov functionals, we reduce the problem of stability and uniform ultimate boundedness to a scalar ordinary differential inequality. Soon afterwards, we derive conditions for global exponential stability of an equilibrium point. The oscillation of small amplitude in the system are studied by perturbation theory, and a general approach is presented for equations of this type for finding the expansion of the oscillation to any order in terms of the coefficient of the fundamental frequency. The frequency-amplitude relations are derived to second order. Finally, we also give some numerical examples to verify the correctness of our analytic results by computer simulations. [ABSTRACT FROM AUTHOR]

The mammalian auditory system is able to process sounds over an extraordinarily large dynamic range, which makes it possible to extract information from very small changes both in sound amplitude and frequency. Evidently, response of the cochlea is essentially nonlinear, where it operates within Hopf bifurcation boundaries to maximize tuning and amplification. This paper presents a set of piecewise linear (PWL) and multiplierless piecewise linear (MLPWL1 and MLPWL2) active cochlear models, which mimic a range of behaviors, similar to the biological cochlea. These proposed models show similar dynamical characteristics of the Hopf equation for the active nonlinear artificial cochlea. Accordingly, a compact model structure is proposed upon which a 2-D cochlea is developed. The proposed models are investigated, in terms of their digital realization and hardware cost, targeting large scale implementation. Hardware synthesis and physical implementation on a FPGA show that the proposed models can reproduce precise active cochlea behaviors with higher performance and considerably lower computational costs in comparison with the original model. Results indicate that the MLPWL1 model has a lower computational overhead, precision, and hardware cost, while the PWL model has a higher precision and dynamically tracks the original model. On the other hand, the MLPWL2 model outperforms the others in terms of accuracy, dynamical tracking of the original model and implementation cost. The gain variations of the original, PWL, MLPWL1, and MLPWL2 models are 230, 100, 105, and 230 dB, respectively. The mean normalized root mean square errors (NRMSEs) of the PWL, MLPWL1, and MLPWL2 models are 0.11%, 11.97%, and 0.34%, respectively, as compared to the original cochlear model. [ABSTRACT FROM AUTHOR]

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]

Three-phase power-factor-correction (PFC) power supplies are commonly used to convert ac power from a three-phase grid to a regulated dc voltage with unity input power factor. The output voltage regulation is normally achieved by an outer voltage feedback loop and a sinusoidal pulse-width-modulated (SPWM) inner current loop. However, the non-ideal power grid may drive the converter to enter a low-frequency instability region. In this paper, a low-frequency instability phenomenon in three-phase PFC power supplies is reported. The converter can also be regarded as exhibiting a Hopf-type bifurcation in which the dc output voltage oscillates at 150 Hz, and a large amount of harmonics appear on the line current. We develop an averaged model of the grid-converter system to predict the low-frequency instability. The effects of circuit and grid parameters can be studied with this model. Additionally, as a result of the emergence of low-frequency oscillation, the stability boundary leading to a catastrophic bifurcation changes dramatically. Simulation results also provide design-oriented boundaries of operation. This phenomenon is confirmed by experimental measurement. [ABSTRACT FROM PUBLISHER]

In this paper, a delayed fractional-order predator–prey system with trans-species infectious diseases is proposed and the corresponding control strategy is implemented via fractional-order proportional-derivative (PD) control. Firstly, for the uncontrolled fractional-order predator–prey system, explicit conditions of stability and Hopf bifurcation are established by selecting time delay as the bifurcation parameter. The predator–prey system will lose its stability and a family of oscillations will emerge when the time delay passes through the critical value. Secondly, under the fractional-order PD control, the influences of the controller on the system stability and bifurcation are investigated. It is demonstrated that the Hopf bifurcation can be postponed or advanced, and the desired dynamic can be achieved by choosing appropriate control gain parameters. In addition, the impacts of fractional order and control parameters on dynamics are explored. Finally, some numerical simulations are depicted to validate the theoretical results. [ABSTRACT FROM AUTHOR]

The influence of asymptomatic patients on disease transmission has attracted more and more attention, but the mechanism of some factors affecting disease transmission needs to be studied urgently. Considering the self-healing rate of asymptomatic patients, the cure rate of symptomatic patients, the transformation rate from asymptomatic to symptomatic and the infection delay, a type of infectious disease dynamics model S I s I a S with asymptomatic infection and infection delay is established in this paper. It is found that both the infection delay and the difference size between the cure rate and the self-healing rate not only affect the minimum value of the total number of patients in the persistent state of the disease, but also lead to disease extinction to be controlled by the proportion of symptomatic patients in patients. Moreover, the infection delay can lead to local Hopf bifurcation of periodic solutions. By using the normal form and center manifold theory the direction of Hopf bifurcations and the stability of bifurcated periodic solutions are discussed. At last, sensitivity analysis shows that the infection delay can change the correlation of the proportion of symptomatic patients in patients and the transformation rate to the total number of patients. [ABSTRACT FROM AUTHOR]

Researches have shown that in addition to direct virus-to-cell infection, viral particles can also be transferred from a productively-infected cell to an uninfected cell through the formation of virological synapses. In order to reduce the viral load in infected individuals, different classes of antiretroviral drugs have been developed, including reverse transcriptase inhibitor (RTI), integrase inhibitor (II), protease inhibitor (PI) and so on. In this paper, we incorporate the mitotic proliferation of target cells which is described by the logistic term, both virus-to-cell infection and cell-to-cell transmission, the intracellular delay and RTI-based therapy into an in-host HIV infection model. Through mathematical analysis, we find that the model undergoes a backward bifurcation when the turn-over rate coefficient of productively-infected cells is smaller than its mitotic proliferation rate coefficient. When the turn-over rate coefficient of productively-infected cells is greater than its mitotic proliferation rate coefficient, the existence of Hopf bifurcation at the chronic-infection equilibrium with and without the intracellular delay is established, respectively. Numerical simulations suggest that the dynamics of the model be sensitive to parameter values and initial conditions, which may be of great significance to control HIV infection. We also show numerical evidence to support the fact that the smaller the therapy efficacy, the higher the viral load in infected individuals. [ABSTRACT FROM AUTHOR]

In this paper, a fractional-order recurrent neural network is proposed and several topics related to the dynamics of such a network are investigated, such as the stability, Hopf bifurcations, and undamped oscillations. The stability domain of the trivial steady state is completely characterized with respect to network parameters and orders of the commensurate-order neural network. Based on the stability analysis, the critical values of the fractional order are identified, where Hopf bifurcations occur and a family of oscillations bifurcate from the trivial steady state. Then, the parametric range of undamped oscillations is also estimated and the frequency and amplitude of oscillations are determined analytically and numerically for such commensurate-order networks. Meanwhile, it is shown that the incommensurate-order neural network can also exhibit a Hopf bifurcation as the network parameter passes through a critical value which can be determined exactly. The frequency and amplitude of bifurcated oscillations are determined. [ABSTRACT FROM PUBLISHER]

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

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

This paper considers an extended predator–prey model, in which the fear of predators reduces prey reproduction and the search speed of predators depends on prey density. First, our mathematical analysis shows that high levels of fear can stabilize the coexistence steady state, while low levels would result in periodic oscillation. Comparing to the model ignoring fear where supercritical bifurcation occurs, Hopf bifurcation in our model can be both supercritical and subcritical, which leads to bi-stability and two limit cycles. Second, our analysis demonstrates that a relatively small search speed of predators can promote stability of the coexistence steady state, while a large speed would lead to periodic oscillation. Comparing to the model with invariant search speed where Hopf bifurcation takes place, Bautin bifurcation occurs in our model, which results in tri-stability and three limit cycles. While the paradox of enrichment always takes place in the Holling-type II predation model, it does not occur here when the search speed is small. Even when the speed is large, the prey species can adapt by enhancing their fear level and stabilize the system effectively. Third, our analysis shows that enhancing prey's sensitivity to predation risk or slowing the predator search speed, can stabilize the coexistence steady state, while a low sensitivity and a high speed will lead to periodic oscillation. Numerical simulations confirm and extend our results. • A predation model with both fear cost and variable search speed is studied. • Bautin bifurcation is proven in the model with tri-stability and three limit cycles. • Both supercritical and subcritical Hopf bifurcations are shown. • Conditions under which the paradox of enrichment does not occur are given. • Enhancing prey's sensitivity to risk and/or slowing the search speed can stabilize the system. [ABSTRACT FROM AUTHOR]

In this paper, we explore an ecoepidemic model of a prey–predator type in which the prey population is infected by an SI type disease with two infectious routes (horizontal and vertical). Specifically, we consider that both susceptible and infected preys have emergent carrying capacities, which implies that both preys are involved in intraspecific and interclass competitions. In addition, we assume that the susceptible prey gathers together to defend the predator. It is found that the proposed system exhibits abundant dynamics and admits nine equilibria: three boundary equilibria, three plane equilibria, and three interior equilibria. At least one of the three populations can be eliminated from the ecosystem based on the stability of the previous two types of equilibria. According to the stability of the interior equilibria, the given ecosystem can allow the coexistence of susceptible prey–infected prey–predator. We then investigate the stability and direction of Hopf bifurcation. A sensitivity analysis is also conducted to show the effects of the six parameters which are relevant to the emergent carrying capacity and infectious routes on the studied system. Numerical simulations are conducted to verify our theoretical conclusions. [ABSTRACT FROM AUTHOR]

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

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

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]

The paper is formulated with the Holling–Tanner prey–predator model with Beddington–DeAngelis functional response including prey harvesting. Gestational time delay of predator and the dynamic stability of time delay preventing system are incorporated into the system of our paper. The equilibria of the proposed system are determined and the existence of interior equilibrium point for the proposed system is described. Local stability of the system with the magnitude of time delay at the interior equilibrium point is discussed. Thereafter, the direction and the stability of Hopf bifurcation are established with the help of normal theory and center manifold theorem. Furthermore, profit function is calculated with the help of bionomic equilibrium and it is optimized using optimal control. Finally, some numerical simulations are introduced to verify the validity of analytic results of our proposed model. [ABSTRACT FROM AUTHOR]

In this paper, looking at some new contributions to the Kalecki business cycle theory, we re-examine his 1935 model concerning the gestation period between orders and deliveries of capital goods. The model gives rise to a delay differential equation (DDE) with delay dependent coefficients depending on the time delay. The model has only one equilibrium point. Proved that a unique stability switch exists, we study the emergence of the Hopf bifurcation and the direction, stability and period of the bifurcating periodic solutions. We derive an explicit formula for determining the properties of the Hopf bifurcation by using the first Lyapunov coefficient. To confirm our analytic results, we consider two types of non-linear functions, all consistent with Kalecki's hypotheses: two S-shaped functions and one fractional function. Some comments dealing with the economic implications of our analysis are also included. • Kalecki's model of the investment cycle. • Model with time delays and delay dependent coefficients. • Existence of Hopf bifurcations analyzed by choosing time delays as bifurcation parameters. [ABSTRACT FROM AUTHOR]

Locally-active memristor (LAM) is one of the promising candidates of artificial neurons, indicating it has potential applications in neuromorphic computing. Quantitative theoretical analysis on LAMs can provide benefits for designing related oscillator circuits and systems. This study begins with the aim of assessing the importance of DC V-I characteristic in the performance of LAMs by using small-signal analysis method. The DC V-I curve of the LAM is specified by two parameters involving resistance (conductance) and differential resistance (differential conductance). In addition to these two static parameters, we extract a crucial dynamic parameter to describe the behavior of the LAM. Theoretical analysis demonstrates that the performance of generic current-controlled and voltage-controlled LAMs is closely associated with three crucial parameters, i.e., the above two static and one dynamic parameters. Hence, only based on these three parameters, can one derive the small-signal equivalent circuit of LAMs and determine the oscillation frequency range and condition for simple LAM-based oscillators. By applying the presented universal dynamics analysis results, we further propose a modified mathematical model with higher accuracy to mimic the quasi-static and oscillating behaviors of a real Nb2O5 device, and provide some fundamental guidance for the design of LAM-based high-frequency oscillators. [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]

Online social networks (OSNs), as multilingual communication platforms, provide great convenience for rumor spreading. However, a great deal of negative impacts on individuals, society and economy will be caused during the spread of rumor. In order to explore the rules of rumor propagation on online social networks and formulate the corresponding control strategies, in this paper, we propose new 2I2SR rumor propagation models with and without time-delay based on multilingual environment. Firstly, we calculate the basic reproduction number which completely determines the local and global stability of the rumor free equilibrium. Secondly, our theoretical analysis reveals the existence and stability of the rumor equilibria, and some critical conditions are obtained for the existence of Hopf bifurcation by selecting critical parameters and delay as bifurcation parameters. Furthermore, by using Pontryagin's Maximum principle, a real-time optimization method that minimizes the cost of restraining rumor is proposed to eliminate the rumor within an expected time period. Finally, some numerical simulations and a practical application are given to prove the correctness of the theoretical results. [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]

Nonlinear dynamics and control of the galloping of elastic structures is an important research topic that has been addressed in the present paper. Some recent research has used high-frequency excitation in controlling galloping oscillations under steady wind. However, galloping under unsteady wind is potentially more dangerous because of the presence of an additional parametric instability on top of self-excited oscillation. No scientific study is thus far available in the literature concerning the control of galloping under unsteady wind. The present paper investigates the efficacy of high-frequency excitation, both parametric and direct, for controlling unsteady galloping. Analytical and numerical methods are employed to understand the dynamics of the system with and without high-frequency excitation. A detailed nonlinear analysis of the problem reveals rich dynamic characteristics of the system under different types of possible aerodynamic damping profiles. Overall, it is shown that both direct and parametric high-frequency excitation can effectively control the galloping oscillation. Direct excitation can completely eliminate the periodic as well as quasiperiodic galloping arising due to unsteady wind. However, parametric excitation can only limit the oscillation amplitude. • Effect of high-frequency excitations (HFE) on periodic and quasiperiodic galloping under unsteady wind flow is studied. • Method of direct partition of motion (MDPM) and method of multiple scales (MMS) are used. • Direct HFE can mitigate the periodic and quasiperiodic oscillation. • Parametric HFE can reduce the amplitude of the galloping response by shifting the resonance region. [ABSTRACT FROM AUTHOR]

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

In this paper, the subject of bifurcation for a fractional order predator–prey system involving two nonidentical delays is nicely discussed. The critical values of delays for Hopf bifurcation are exactly calculated for the proposed model by using two nonidentical delays as bifurcation parameter, respectively. In addition, the effects of fractional order and additional delay on the bifurcation point are delicately explored. It detects that the stability performance is extremely pulverized with the enhancement of fractional order and another delay. This hints that the generation of Hopf bifurcation can be advanced as fractional order and another delay increase. The final numerical simulations gauge the correctness of the developed theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper proposes a multiple-delayed predator–prey system with habitat complexity and harvesting effort, and investigates the dynamical behavior including stability properties and Hopf bifurcation. Firstly, stability of equilibrium points and the existence of Hopf bifurcation are investigated and some critical conditions which guarantee the corresponding results are obtained based on mathematical view. Secondly, the explicit formulae for determining the direction, stability and period of the bifurcating periodic solutions are derived by using the center manifold theory and the normal form theory. Finally, in order to verify the theoretical results, some numerical simulations are done to illustrate the results. It is observed that the level of abundance of prey and predator populations depends on the gestation delay if the gestation delay exceeds some critical values. [ABSTRACT FROM AUTHOR]

This paper deals with the stability and bifurcation of equilibria in a new chaotic fractional-order system in the sense of the Caputo fractional derivative with the chaos entanglement function. We derive conditions under which the system undergoes a Hopf bifurcation and obtain critical parameter value in the Hopf bifurcation. Moreover, the linear feedback control technique is used to control and stabilize the system to equilibrium point in order to eliminate the chaotic vibration. We then design control laws to synchronize two identical chaotic fractional-order systems. Furthermore, by means of numerical simulation, we support the validity of analytical results and reveal more dynamical behaviors consisting chaos, local bifurcation, limit cycles, quasiperiodic and asymptotic stability behaviors. We further emphasize that the order of fractional derivative plays significant roles as the chaos controlling parameter and the Hopf bifurcation parameter. [ABSTRACT FROM AUTHOR]

Nonlocal reaction–diffusion model is an important area to study the dynamics of the individuals which compete for resources. In this paper, we consider a predator–prey model with herd behavior and nonlocal prey competition. We investigate the effects of nonlocal competition on dynamics of the system in the bounded region when the kernel function takes 1 | Ω | and derive the conditions that the nonlocal system undergoes Hopf bifurcation and Turing bifurcation. Then we discuss the influence of nonlocal competition on the stability of the positive constant equilibrium in unbounded region when the kernel function takes a step kernel function. Our result shows that nonlocal competition can destabilize the stability of the predator–prey system. • The dynamics of the nonlocal predator–prey system is studied. • The conditions for Hopf and Turing bifurcations are derived. • Effect of the nonlocal competition term on the stability is analytically investigated. • Effect of the kernel function on the stability is numerically investigated. [ABSTRACT FROM AUTHOR]

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

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

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

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

Abstract Life in the Sahelian zone is very difficult due to lack of resources on a permanent basis. This scarcity of resources leads the pastor, with their flock to organize themselves to better resist the difficult conditions. Pastoralism is of paramount importance for this region. There is an urgent need to understand this way of life in order to help pastors to better organize themselves. In this paper, we propose mathematical models that consist of three food trophic chains. We consider and formalize, in a given location, interactions between browsers, grazers and forage resources. We first analyze the resulting system without delays and in a second time take into account two positive and discrete time-delays. In our model, delays denote an average time required in order that food consumed by browsers and grazers are beneficial for them. Qualitative analysis of the delays-free model shows several equilibria as well as various situations of multistability. Considering delay as parameter, we investigate the effect of delay on the stability of equilibria. It is found that the delays can lead the system dynamic behavior to exhibit stability switches, and Hopf bifurcation occurs when the delay crosses some critical value. By applying the normal form theory and the center manifold theorem, the explicit formula which determines the stability and direction of the bifurcating periodic solutions is determined. Finally based on a nonstandard numerical scheme, we provide some numerical simulations in order to illustrate our qualitative results and support discussions. [ABSTRACT FROM AUTHOR]

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