Search

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

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]

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

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

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]

In this paper, a four-species food web model is formulated to investigate the influence of fear effect and nonlinear top predator harvesting on the dynamical behaviors. The global stability of the system at the interior equilibrium is explored by Li–Muldowney geometric approach. By applying the Sotomayor's theorem, it is shown that the system undergoes transcritical bifurcation and pitchfork bifurcation. The conditions for the occurrence of Hopf bifurcation are established, and the stability of the bifurcating limit cycle is discussed by normal form theory. Finally, the numerical simulations are carried out. It is observed that the system presents chaotic dynamics, periodic window and chaotic attractors. Two distinct routes to chaos are discovered, one is period doubling cascade and the other is the generation and destruction of quasi-periodic states. Moreover, it is found that the fear effect can suppress fluctuations in population density to stabilize the system. [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]

Neuromorphic computing can solve computationally hard problems with energy efficiencies unattainable for von Neumann architectures. A locally-active memristor, which possesses the capability to amplify infinitesimal fluctuations in energy and can be used to generate neuromorphic behaviors, is a natural candidate for constructing an electronic equivalent of biological neurons. This paper identifies some unknown neuromorphic dynamics of the Chua corsage memristor (CCM), and shows that the CCM, when biased at the edge of chaos domain, can exhibit rich dynamics of biological neurons. Using Chua’s theories of local activity and edge of chaos, we demonstrate that under the destabilizing of the input voltage and the circuit parameters (inductance or capacitance), two CCM-based circuits can produce thirteen types of neuromorphic behaviors either on, or near the edge of chaos domain via supercritical or subcritical Hopf bifurcation. In addition, we give the conditions to test the edge of chaos of the CCM and the CCM-based circuit only by using the poles and the zero of their admittance functions. [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]

In this paper, we establish a delayed virotherapy model including infected tumor cells, uninfected tumor cells and free virus. In this model, both infected and uninfected tumor cells have special growth patterns, and there are at most two positive equilibria. We mainly analyze the stability and Hopf bifurcation of the model under different time delays. For the model without delay, we study the Hopf and Bogdanov–Takens bifurcations. For the delayed model, by center manifold theorem and normal form theory of functional differential equation, we study the direction of Hopf bifurcation and stability of the bifurcated periodic solution. Moreover, we prove the existence of Zero-Hopf bifurcation. Finally, some numerical simulations show the results of our theoretical calculations, and the dynamic behaviors near Zero-Hopf and Bogdanov–Takens point of the system are also observed in the simulations, such as bistability, periodic coexistence and chaotic behavior. • A delayed virotherapy model including infected and uninfected tumor cells is established. • We mainly analyze the stability and Hopf bifurcation of the model under different time delays. The system also undergoes Bogdanov–Takens and Zero-Hopf bifurcation. • In this paper, some interesting dynamic phenomena are simulated, such as bistability, periodic coexistence and chaotic behavior. [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]

This paper proposes the stability and Hopf bifurcation for a quaternion-valued neural network(QVNN) with leakage delay and communication delay. Due to the higher order of the characteristic equation, the order of characteristic equation is reduced via the matrix block theory. Simultaneously, taking the leakage delay and communication delay as bifurcation parameter respectively, a number of conditions ensuring the stability of Hopf bifurcation periodic solutions are established. It lavishly illustrates that the stability of QVNN can be demolished more when considering the leakage delay as bifurcation parameter. Finally one simulations example is given to validate the theoretical results. [ABSTRACT FROM AUTHOR]

Selection of an appropriate therapy threshold to restrain the virus load is still challenging on structured treatment interruptions (STIs) for HIV. In this paper, we ponder that how the sliding control and multistability to regulate the treatment effect through comprehensive dynamics of a virus-immune model. Firstly, based on piecewise therapy, we propose a delayed reaction–diffusion virus-immune model under the homogeneous Neumann boundary condition. Secondly, the existence and stabilities of five kinds of equilibria as well as the direction and stability of spatial Hopf bifurcation at regular equilibrium are investigated. Thirdly, the sliding domain and the boundary node bifurcations are addressed by theoretical analysis. Finally, we appraise the effects of therapy threshold, sliding domain and multistability on HIV therapy by simulations, and further seek out the appropriate therapy threshold for infected patients with given physiological parameters and present the corresponding principles. Our explorations will provide evidence for HIV and other disease therapies. [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]

Given the critical role of time delays in epidemic modeling, this paper delves into the dynamics and finite-time optimal stabilization of a novel epidemic system characterized by such delays. Our findings reveal that time delays significantly influence both the system's dynamics and the formulation of an optimal control strategy. Specifically, the system's endemic equilibrium point remains locally asymptotically stable under mild conditions for small time delays. However, exceeding critical delay thresholds induces Hopf bifurcation. To closely regulate the system's state towards the equilibrium point and minimize control costs, we propose an optimal control problem. We derive the explicit form of the optimal control strategy employing Pontryagin's Maximum Principle. Finally, numerical simulations further confirm the theoretical results obtained in this paper. • A time-delayed SEAIR epidemic model with a general incidence function is studied. • We focus on discussing the effects of time delays on dynamics and optimal control. • The theoretical results are well verified by the numerical simulation results. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of a robust UAV tracking, surveillance and landing of a mobile ground target. The translational and angular dynamics of the vehicle are affected by bounded uncertainties; a Quasi-Integral Sliding Mode control is designed to obtain robustness from nearly the initial time. The flying mission considers three different dynamics of movement: the take-off to the desired altitude, the relative circular surveillance motion around the mobile ground target and eventually precise landing over the ground vehicle. This paper introduces a novel dynamic motion planning generator to perform such tracking maneuvers. It is based on the solution of a second order nonlinear differential equation, whose solution is force to move in a set of new parameterized 'Bifurcation Sliding Mode Surfaces' that exploit the Hopf Bifurcation properties to change the dynamic around the equilibrium point. A temporal switching technique is introduced for changing between three different bifurcation sliding surfaces at different time intervals. To illustrate that the quadcopter effectively performs the desired maneuvers, a computer animation is provided at the end of the paper. [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]

In this paper, an SIR epidemic model considering hospital resources and vaccination is established and the rich dynamics and complex bifurcations are investigated. Firstly, the existence of disease-free equilibrium and endemic equilibria is explored. It is founded that when the vaccination rate is not high, the number of endemic equilibrium points changed easily with the number of hospital resources and vaccination, resulting in transcritical bifurcation and saddle–node bifurcations. Secondly, different types singularities such as degenerate saddle–node of codimension 1 at the disease-free equilibrium, and cusp or focus type Bogdanov–Takens singularities of codimension 3 at endemic equilibria are presented. Thirdly, bifurcation analysis at these equilibria is investigated, and it is found that the system undergoes a sequence of bifurcations, including Hopf, degenerate Hopf bifurcation, homoclinic bifurcation, the cusp type Bogdanov–Takens bifurcation of codimension 2, and the focus type Bogdanov–Takens bifurcation of codimension 3 which are the organizing centers for a series of bifurcations with lower codimension. And the system shows very rich dynamics such as the existence of multiple coexistent periodic orbits, homoclinic loops. Finally, numerical simulations are presented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

Rumors do the social's perception seal, leaving people untouched with the true thing. Since the harmfulness of rumors is well known, in order to cut off the net of rumors and make the society run in order, it is necessary to have a deeper understanding of the spread of rumors. In this paper, the non-smooth reaction–diffusion system with considering the encouraging effect of secondary propagation of Internet platform on rumor propagation is used to study the dynamic system of rumor propagation. Firstly, the existence of the non-negative solution is proved by using the theory of upper and lower solutions for mixed monotone system. Secondly, the value of the basic reproduction number is calculated and the existence of positive equilibrium are discussed. Thirdly, the backward bifurcation and the saddle-node bifurcation are investigated. Fourthly, the stability of rumor propagation equilibrium and Hopf bifurcation are analyzed theoretically. Finally, some numerical simulations which proves the validity of the above theory are provided. [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]

A correction is noted to the previously published paper (Zou and Lv, 2017). Regularity, positive recurrence and stationary distribution are considered in this paper. A complete threshold analysis of strong stochastic persistence and extinction is investigated. Stochastic Hopf bifurcation is considered from the viewpoint of numerical simulations. [ABSTRACT FROM AUTHOR]

Memristors can be employed to mimic biological neural synapses or to describe electromagnetic induction effects. To exhibit the threshold effect of electromagnetic induction, this paper presents a threshold flux-controlled memristor and examines its frequency-dependent pinched hysteresis loops. Using an electromagnetic induction current generated by the threshold memristor to replace the external current in 2-D Hindmarsh–Rose (HR) neuron model, a 3-D memristive HR (mHR) neuron model with global hidden oscillations is established and the corresponding numerical simulations are performed. It is found that due to no equilibrium point, the obtained mHR neuron model always operates in hidden bursting firing patterns, including coexisting hidden bursting firing patterns with bistability also. In addition, the model exhibits complex dynamics of the actual neuron electrical activities, which acts like the 3-D HR neuron model, indicating its feasibility. In particular, by constructing the fold and Hopf bifurcation sets of the fast-scale subsystem, the bifurcation mechanisms of hidden bursting firings are expounded. Finally, circuit experiments on hardware breadboards are deployed and the captured results well match with the numerical results, validating the physical mechanism of biological neuron and the reliability of electronic neuron. [ABSTRACT FROM AUTHOR]

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

This paper concentrates on dynamics and stability analysis of two predators–one prey mathematical model with competition between predators and fading memory in one predator. The investigation of the constructed model shows that there exist five equilibria, e.g. trivial extinction state of all populations, extinction of both predators state, extinction of first or second predator state and coexisting state. Investigating the eigenvalues of characteristic polynomial, conditions for the local stability around each equilibrium are also determined depending on the parameter space. Analytical formulations are complemented with numerical simulations, where time simulations and single parameter numerical continuation of each variable are performed with respect to model parameters and multiple sub-and super-critical Hopf bifurcations, period doubling bifurcation and transcritical bifurcation are detected for different values of memory related parameter. Our results show that fading memory and competition between predators have substantial impact on the existence and dynamics of all three populations and may shed lights on further understanding of interacting species in ecology. • A model comprising one prey and two competitive predators with fading memory is constructed. • Fading memory and competition between predators have a large impact on the dynamics. • Fading memory in one predator may affect the existence and nature of various bifurcations. [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]

In the present paper, we consider a reaction–diffusion system, based on the Leslie–Gower predator–prey model with Allee effect in both the predator and prey populations. The model is a generalization of the case where the individual population is subject to the Allee effect and it is flexible such that one can easily obtain the scenario with the Allee effect only on the prey population or, predator population. The non-spatial model was analyzed by using the positivity of solutions, uniformly boundedness and local stability of the interior equilibrium. The sufficient conditions for the instability with zero-flux boundary conditions are obtained for the spatial model. We have derived the analytical conditions for Hopf and Turing bifurcation on the spatial domain. We also observe that the system exhibits complex dynamics like Bogdanov–Takens (BT) bifurcation and generalized Hopf (GH) bifurcation. Our numerical simulations reveal that depending on the strength of Allee effects, the model dynamics exhibit stripes, spots, cold–hot spots–stripes coexistence and cold–hot spots patterns. We obtain very interesting characteristics of the reaction–diffusion model which reflects the fact that the system can develop patterns both inside and outside of the Turing parameter domain. Our study may enrich the field of the Allee effect and will help us to acquire a better understanding of the predator–prey interaction in a real environment. [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]

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

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

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

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

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

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

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

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

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

The current study presents complex dynamics of an SIR epidemic model that incorporates a saturated type incidence rate as well as treatment. We provide here rigorous results for asymptotic stability of equilibrium states of the proposed system. Several bifurcations including Hopf, Generalized Hopf, saddle–node, transcritical and Bogdanov–Takens are also discussed. The stability of bifurcated periodic solutions is verified with the help of first Lyapunov number. Extensive numerical simulations are performed to validate these results. In a numerical example it is observed that if the saturation factor increases slowly, then the unique endemic equilibrium state is asymptotically stable for a certain range. The further increase in the value of saturation parameter, the endemic equilibrium state loses its stability and periodic solutions appear through Hopf bifurcation. It is also observed that the increase in saturation parameter beyond Hopf bifurcation threshold, results in regaining the stability of the endemic equilibrium state, which forms an interesting dynamical phenomenon in the bifurcation diagram named as an endemic bubble. It is pointed out that in the case of two endemic equilibrium states, one of these two is always saddle, whereas, the other one becomes unstable through Hopf bifurcation. In this scenario, the periodic solution is initially stable and it becomes unstable through generalized Hopf bifurcation. In numerical example for Bogdanov–Takens bifurcation two pairs of feasible bifurcation thresholds exist for the same set of parameters value. The bifurcation diagrams and equilibrium surfaces are also plotted to observe the combined effects of medication and saturation parameters. • In this paper an SIR epidemic model with a saturated type incidence rate as well as treatment is studied. • Stability and bifurcation of all feasible equilibrium states are discussed. • An endemic bubble is observed through Hopf bifurcation in case of unique endemic equilibrium. • Generalized Hopf bifurcation in case of two endemic equilibrium is pointed out. • In case of Bogdanov–Takens bifurcation there are two pairs of feasible bifurcation thresholds. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the local stability and bifurcation of a delay-coupled genetic regulatory networks consisting of two modes with the hub structure. By analyzing the equilibrium equation, the number of the positive equilibria is discussed in both the cases that there are inhibition coupling and activation coupling in the networks. It is revealed that multiple equilibria could exist in the developed genetic networks and the number of the equilibria could be distinct under the two cases of delayed-coupling. For the equilibrium, the conditions of the coupling-delay-independent stability and the saddle-node bifurcation are derived with respect to the biochemical parameters. The coupling-delay-dependent stability and the Hopf bifurcation criteria on the biological parameters and the coupling delay are also given. Moreover, the complexity of the algorithm used in this paper is analyzed. The numerical simulations are made to certify the obtained results. The multistability of the developed genetic regulatory networks is displayed. The different effects of the coupling delay on the stability of the genetic networks under different biochemical parameters are shown. [ABSTRACT FROM AUTHOR]

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

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

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