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When playing games in groups, it is an advantage for individuals to have accurate statistical information on the strategies of their opponents. Such information may be obtained by remembering previous interactions. We consider a rock-paper-scissors game in which agents are able to recall their last m interactions, used to estimate the behavior of their opponents. At critical memory length, a Hop! bifurcation leads to the formation of stable limit cycles. In a mixed population, agents with longer memories have an advantage, provided the system has a stable fixed point, and there is some asymmetry in the payoffs of the pure strategies. However, at a critical concentration of long memory agents, the appearance of limit cycles destroys their advantage. By introducing population dynamics that favors successful agents, we show that the system evolves toward the bifurcation point. [ABSTRACT FROM AUTHOR]

The role of mutation, which is an error process in gene evolution, in systems of cyclically competing species has been studied from various perspectives, and it is regarded as one of the key factors for promoting coexistence of all species. In addition to naturally occurring mutations, many experiments in genetic engineering have involved targeted mutation techniques such as recombination between DNA and somatic cell sequences and have studied genetic modifications through loss or augmentation of cell functions. In this paper, we investigate nonlinear dynamics with targeted mutation in cyclically competing species. In different ways to classic approaches of mutation in cyclic games, we assume that mutation may occur in targeted individuals who have been removed from intraspecific competition. By investigating each scenario depending on the number of objects for targeted mutation analytically and numerically, we found that targeted mutation can lead to persistent coexistence of all species. In addition, under the specific condition of targeted mutation, we found that targeted mutation can lead to emergences of bistable states for species survival. Through the linear stability analysis of rate equations, we found that those phenomena are accompanied by Hopf bifurcation which is supercritical. Our findings may provide more global perspectives on understanding underlying mechanisms to control biodiversity in ecological/biological sciences, and evidences with mathematical foundations to resolve social dilemmas such as a turnover of group members by resigning with intragroup conflicts in social sciences. [ABSTRACT FROM AUTHOR]

A self-propelled motor driven by the enzymatic reaction of catalase adsorbed onto a filter paper floating on an aqueous solution of H2O2 was used to study nonlinear behavior in the motor's motion. An increase in the concentration of H2O2 resulted in a change from no motion to irregular oscillatory motion, periodic oscillatory motion, and continuous motion. The mechanisms underlying oscillation and mode bifurcation are discussed based on experimental results on O2 bubble formation and growth on the underside of the motor. [ABSTRACT FROM AUTHOR]

One century after the seminal work by Leonor Michaelis and Maud Menten devoted to the theoretical study of the enzymatic reactions, in this paper, we give an overview of the most recent trends concerning the mathematical modeling of several enzymatic mechanisms, focusing on its asymptotic analysis, which needs the use of advanced mathematical tools, such as center manifold theory, normal forms, and bifurcation theory. Moreover, we present some perspectives, linking the models here presented with similar models, arising from different research fields. [ABSTRACT FROM AUTHOR]

In this paper periodic trajectories of dynamical systems presenting discontinuities are studied. The considered model consists of two distinct linear differential systems, each one containing a single equilibrium point of centre type. Each system is defined on disjoint regions of the plane, the separation line is a union of two half-straight lines contained on the coordinate axes. The obtained differential system is non-smooth, so we apply Filippov's theory to study the transitions from one dynamical system to another. The combination of the two linear plus the Filippov system acting on the separation line generates a nonlinear regime observed by the presence of limit cycles, sliding and tangential periodic trajectories as well as the coexistence of such objects. In theorem we establish the location, stability and hyperbolicity of limit cycles for certain classes of the considered model. In theorem we perform the global analysis of a representative model through bifurcation theory to analyse the birth of limit cycles, sliding periodic trajectories and tangential ones. We also provide some results addressing the coexistence of periodic trajectories and two potential physical interpretations of the model considered in the paper, one addressing nonlinear oscillations and the other considering slow-fast systems of neuron models. The main techniques employed to obtain the results are first integrals, Poincaré half return maps, and elements of bifurcation theory. [ABSTRACT FROM AUTHOR]

Abstract In this study, we investigate a discrete-time version of logit dynamics, as applied to the rock–paper–scissors (RPS) game. First, we show that around the Nash equilibrium point, an attracting closed invariant curve appears due to the Neimark–Sacker bifurcation. Next, near the resonance point, we find a period-three attracting cycle, which can be thought of as a counterpart to the cyclically stable set in the RPS game with best response dynamics. Moreover, we show that the cycle can coexist with an attracting closed invariant curve, a period-three saddle cycle, and the attracting or repelling Nash equilibrium point. Finally, we use the codimension-two bifurcation theory to specify the set of heteroclinic bifurcations that destroy the coexistence of the attractors. Highlights • The rock–paper–scissors game in discrete-time logit dynamics is analyzed. • An attracting closed invariant curve around the Nash equilibrium appears. • The coexistence of period three cycles and attracting Nash equilibrium is proved. • The coexistence of period three cycles and closed invariant curve is proved. • Heteroclinic cycle of RPS game in discrete-time logit dynamics is conjectured. [ABSTRACT FROM AUTHOR]

The vehicles are prone to accidents during cornering on a wet or low friction coefficient roads if the longitudinal velocity (Vx) and steering angle (δ) are increased beyond a certain limit. Therefore, it is of major concern to analyze the behaviour and define the stability boundary of the vehicle for such scenarios. In this paper, stability analysis of a 2 degrees of freedom nonlinear bicycle model replicating a car model including lateral (sideslip angle β) and yaw (yaw rate r) dynamics only operating on a wet surface road has been performed. The stability is analysed by utilizing the phase plane method and bifurcation analysis. The obtained converging and diverging nature of the trajectories (β, r) depicts the stable and unstable equilibrium points in the phase plane. The movement of these points results in the transition of the stability known as bifurcation due to the change in the control parameters (Vx, δ). The Matcont/Matlab is utilized to obtain the bifurcation diagrams and the nature of bifurcations. The obtained results show that a saddle node (SNB) and a subcritical Hopf bifurcation (HB) exists for steering angle (±0.08 rad) and higher than (±0.08 rad) with Vx = (10 - 40) m/s respectively. The SNB and HB denotes the spinning of the vehicle and sliding of the vehicle respectively, thus generating a unstable behaviour. A stability boundary is obtained representing the stable and unstable range of parameters. [ABSTRACT FROM AUTHOR]

This paper considers a banking loan model using a difference equation with a nonlinear deposit interest rate. The construction of the model is based on a simple bank balance sheet composition and a gradient adjustment process. The model produces two unstable loan equilibriums and one stable equilibrium when the parameter corresponding to the deposit interest rate is situated between its transcritical and flip bifurcations. Some numerical simulations are presented to align with the analytical findings, such as the bifurcation diagram, Lyapunov exponent, cobweb diagram, and contour plot sensitivity. The significance of our result is that the banking regulator may consider the lower and upper bounds for setting the nonlinear interest rate regulation and provide a control regulation for other banking factors to maintain loan stability. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of process faults in the Lorenz system that can affect any of the system parameters and cause the system to exhibit various behaviors. In this paper, the homoclinic orbits in the Lorenz system are described and then the occurrence of process faults in the system is investigated that can cause a homoclinic explosion, bifurcation, change of fixed point, or even instability in the system. In such systems, where a small change in one of the parameters causes large changes in the behavior of the system, to prevent disaster in industrial systems and also to stop the propagation of faults in the system, the faults must be identified as soon as possible. In this paper, the states in the system are estimated by using a reduced-order observer, and the faults are detected. The purpose of this article is to recognize the change in behavior of this system in the face of this type of fault and to express the importance of timely detection and identification of faults in the system so as not to lead to failure and disaster in industrial systems. Finally, the effect of process fault, disturbance, and sensor fault are investigated simultaneously and the states and faults in the system are estimated. [ABSTRACT FROM AUTHOR]

In this paper, we were concerned with the global behavior of positive solutions for third-order semipositone problems with an integral boundary condition y ‴ + β y ″ + α y ′ + λ f (t , y) = 0 , t ∈ (0 , 1) , y (0) = y ′ (0) = 0 , y (1) = χ ∫ 0 1 y (s) d s , where α ∈ (0 , ∞) and β ∈ (− ∞ , ∞) are two constants, λ , χ are two positive parameters, and f ∈ C ([ 0 , 1 ] × [ 0 , ∞) , R) with f (t , 0) < 0 . Our analysis mainly relied on the bifurcation theory. In this paper, we were concerned with the global behavior of positive solutions for third-order semipositone problems with an integral boundary condition where and are two constants, are two positive parameters, and with . Our analysis mainly relied on the bifurcation theory. [ABSTRACT FROM AUTHOR]

This paper consists of four parts. It begins by using the authors' generalized Schauder formula, [41] , and the algebraic multiplicity, χ , of Esquinas and López-Gómez [15,14,31] to package and sharpening all existing results in local and global bifurcation theory for Fredholm operators through the recent author's axiomatization of the Fitzpatrick–Pejsachowicz–Rabier degree, [42]. This facilitates reformulating and refining all existing results in a compact and unifying way. Then, the local structure of the solution set of analytic nonlinearities F (λ , u) = 0 at a simple degenerate eigenvalue is ascertained by means of some concepts and devices of Algebraic Geometry and Galois Theory, which establishes a bisociation between Bifurcation Theory and Algebraic Geometry. Finally, the unilateral theorems of [31,33] , as well as the refinement of Shi and Wang [53] , are substantially generalized. This paper also analyzes two important examples to illustrate and discuss the relevance of the abstract theory. The second one studies the regular positive solutions of a multidimensional quasilinear boundary value problem of mixed type related to the mean curvature operator. [ABSTRACT FROM AUTHOR]

In this paper, we consider a diffusive–advective predator–prey system in a spatially heterogeneous environment subject to a hostile boundary condition, where the interaction term is governed by a Holling type II functional response. We investigate the existence and global attractivity of both trivial and semi-trivial steady-state solutions and the existence and local stability of coexistence steady-state solutions, depending on the size of a key principal eigenvalue. In addition, we show that the effect of advection on the principal eigenvalue is monotonic for small advection rates, depending on the concavity of the resource distribution. For arbitrary advection rates, we consider two explicit resource distributions for which we can say precisely the behaviour of the principal eigenvalue as it depends on advection, highlighting that advection can either improve or impair a population's ability to persist, depending on the characteristics of the resource distribution. We present some numerical simulations to demonstrate the outcomes as they depend on the advection rates for the full predator–prey system. These insights highlight the intimate relationship between environmental heterogeneity, directed movement, and the hostile boundary. The methods employed include upper and lower solution techniques, bifurcation theory, spectral analysis, and the comparison principle. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a discrete-time prey-predator model. The model is formulated by using the piecewise constant argument method for differential equations and taking into account Holling type III. The existence and local behavior of equilibria are studied. We established that the system experienced both Neimark-Sacker and perioddoubling bifurcations analytically by using bifurcation theory and the center manifold theorem. In order to control chaos and bifurcations, the state feedback method is implemented. Numerical simulations are also provided for the theoretical discussion. [ABSTRACT FROM AUTHOR]

In this paper, a three dimensional discrete eco-epidemiological model with Holling type-III functional response is proposed. Boundedness of the solutions of the system is analyzed. Existence condition and stability of all fixed points are discussed for the proposed model. Furthermore, we obtained the transcritical bifurcation surfaces of the system by bifurcation theory. Based on the explicit criteria for the Neimark Sacker bifurcation and flip bifurcation, we obtained that the system undergoes these two types of bifurcations at the positive fixed point. Then we apply a hybrid control strategy that based on both parameter perturbation and a state feedback strategy to control the Neimark-Sacker bifurcation. Finally, some numerical simulations are carried out to support the analytical results. [ABSTRACT FROM AUTHOR]

Bursting is a crucial form of firing in neurons, laden with substantial information. Studying it can aid in understanding the neural coding to identify human behavioral characteristics conducted by these neurons. However, the high-dimensionality of many neuron models imposes a difficult challenge in studying the generative mechanisms of bursting. On account of the high complexity and nonlinearity characteristic of these models, it becomes nearly impossible to theoretically study and analyze them. Thus, this paper proposed to address these issues by focusing on the midbrain dopamine neurons, serving as the central neuron model for the investigation of the bursting mechanisms and bifurcation behaviors exhibited by the neuron. In this study, we considered the dimensionality reduction of a high-dimensional neuronal model and analyzed the dynamical properties of the reduced system. To begin, for the original thirteen-dimensional model, using the correlation between variables, we reduced its dimensionality and obtained a simplified three-dimensional system. Then, we discussed the changing characteristics of the number of spikes within a burst by simultaneously varying two parameters. Finally, we studied the co-dimension-2 bifurcation in the reduced system and presented the bifurcation behavior near the Bogdanov-Takens bifurcation. [ABSTRACT FROM AUTHOR]

This paper investigates the intricate connection between visual perception and the mathematical modeling of neural activity in the primary visual cortex (V1). The focus is on modeling the visual MacKay effect [D. M. MacKay, Nature, 180 (1957), pp. 849--850]. While bifurcation theory has been a prominent mathematical approach for addressing issues in neuroscience, especially in describing spontaneous pattern formations in V1 due to parameter changes, it faces challenges in scenarios with localized sensory inputs. This is evident, for instance, in MacKay's psychophysical experiments, where the redundancy of visual stimuli information results in irregular shapes, making bifurcation theory and multiscale analysis less effective. To address this, we follow a mathematical viewpoint based on the input-output controllability of an Amari-type neural fields model. In this framework, we consider sensory input as a control function, a cortical representation via the retino-cortical map of the visual stimulus that captures its distinct features. This includes highly localized information in the center of MacKay's funnel pattern "MacKay rays." From a control theory point of view, the Amari-type equation's exact controllability property is discussed for linear and nonlinear response functions. For the visual MacKay effect modeling, we adjust the parameter representing intra-neuron connectivity to ensure that cortical activity exponentially stabilizes to the stationary state in the absence of sensory input. Then, we perform quantitative and qualitative studies to demonstrate that they capture all the essential features of the induced after-image reported by MacKay. [ABSTRACT FROM AUTHOR]

In this paper, we have delved into the intricate dynamics of a discrete-time Hepatitis B virus (HBV) model, shedding light on its local dynamics, topological classifications at equilibrium states, and pivotal epidemiological parameters such as the basic reproduction number. Our analysis extended to exploring convergence rates, control strategies, and bifurcation phenomena crucial for understanding the behavior of the HBV system. Employing linear stability theory, we meticulously examined the local dynamics of the HBV model, discerning various equilibrium states and their topological classifications. Subsequently, we identified bifurcation sets at these equilibrium points, providing insights into the system’s stability and potential transitions. Further, through the lens of bifurcation theory, we conducted a comprehensive bifurcation analysis, unraveling the intricate interplay of parameters that govern the HBV model’s behavior. Our investigation extended beyond traditional stability analysis to explore chaos and convergence rates, enriching our understanding of the dynamics of the understudied HBV model. Finally, we validated our theoretical findings through numerical simulations, confirming the robustness and applicability of our analysis in real-world scenarios. By integrating biological and epidemiological insights into our mathematical framework, we offered a holistic understanding of the dynamics of HBV transmission dynamics, with implications for public health interventions and disease control strategies. [ABSTRACT FROM AUTHOR]

The modified Gardner equation is widely used to describe the supernonlinear proliferation of ion-acoustic waves and quantum electron-positron ion magneto plasmas. This paper focuses on the investigation of the modified Gardner equation with Kuramoto-Sivashinsky perturbation. The existence results of nonlinear and supernonlinear ion-acoustic solitary and periodic waves are established by employing the geometric singular perturbation theory, invariant manifold theory, and bifurcation theory. The supernonlinear solitary wave is a new class of solitary waves, which was proposed by Dubinov and Kolotkov [12]. In this work, the existence of the novel type of ion-acoustic solitary and periodic waves in the perturbed modified Gardner equation has been proven for the first time. Through rigorous mathematical analysis and numerical simulations, we further substantiate the validity of our proposed methods and model. Overall, this study enhances the understanding of the nonlinear and supernonlinear dynamics of ion-acoustic waves in the modified Gardner equation, while also providing a solid foundation for future investigations and applications in plasma physics and related disciplines. [ABSTRACT FROM AUTHOR]

Deterministic (ordinary differential equation) models for the transmission dynamics of vector-borne diseases that incorporate disease-induced death in the host(s) population(s) are generally known to exhibit the phenomenon of backward bifurcation (where a stable disease-free equilibrium of the model coexists with a stable endemic equilibrium when the associated reproduction number of the model is less than unity). Further, it is well known that, in these models, the phenomenon of backward bifurcation does not occur when the disease-induced death rate is negligible (e.g., if the disease-induced death rate is set to zero). In a recent paper on the transmission dynamics of visceral leishmaniasis (a disease vectored by sandflies), titled “A Mathematical Study to Control Visceral Leishmaniasis: An Application to South Sudan,” published in Bulletin of Mathematical Biology, Vol. 79, Pages 1110-1134, 2017, Ghosh et al. (2017) stated that their deterministic model undergoes a backward bifurcation even when the disease-induced mortality in the host population is set to zero. This result is contrary to the well-established theory on the dynamics of vector-borne diseases. In this short note, we illustrate some of the key errors in the Ghosh et al. (2017) study. [ABSTRACT FROM AUTHOR]

The prospect of finding generic early warning signals of an approaching tipping point in a complex system has generated much recent interest. Existing methods are predicated on a separation of timescales between the system studied and its forcing. However, many systems, including several candidate tipping elements in the climate system, are forced periodically at a timescale comparable to their internal dynamics. Here we find alternative early warning signals of tipping points due to local bifurcations in systems subjected to periodic forcing whose time scale is similar to the period of the forcing. These systems are not in, or close to, a fixed point. Instead their steady state is described by a periodic attractor. We show that the phase lag and amplification of the system response provide early warning signals, based on a linear dynamics approximation. Furthermore, the power spectrum of the system's time series reveals the generation of harmonics of the forcing period, the size of which are proportional to how nonlinear the system's response is becoming with nonlinear effects becoming more prominent closer to a bifurcation. We apply these indicators to a simple conceptual system and satellite observations of Arctic sea ice area, the latter conjectured to have a bifurcation type tipping point. We find no detectable signal of the Arctic sea ice approaching a local bifurcation. [ABSTRACT FROM AUTHOR]

In this paper, we use a semi-discretization method to consider the predator–prey model of Leslie type with ratio-dependent simplified Holling type IV functional response. First, we discuss the existence and stability of the positive fixed point in total parameter space. Subsequently, through using the central manifold theorem and bifurcation theory, we obtain sufficient conditions for the flip bifurcation and Neimark–Sacker bifurcation of this system to occur. Finally, the numerical simulations illustrate the existence of Neimark–Sacker bifurcation and obtain some new dynamical phenomena of the system—the existence of a limit cycle. Corresponding biological meanings are also formulated. [ABSTRACT FROM AUTHOR]

At every point p on a smooth n -manifold M there exist n + 1 skew-symmetric tensor spaces spanning differential r -forms ω with r = 0 , 1 , ⋯ , n . Because d ∘ d is always zero where d is the exterior differential, it follows that every exact r -form (i.e., ω = d λ where λ is an r − 1 -form) is closed (i.e., d ω = 0 ) but not every closed r -form is exact. This implies the existence of a third type of differential r -form that is closed but not exact. Such forms are called harmonic forms. Every smooth n -manifold has an underlying topological structure. Many different possible topological structures exist. What distinguishes one topological structure from another is the number of holes of various dimensions it possesses. De Rham's theory of differential forms relates the presence of r -dimensional holes in the underlying topology of a smooth n -manifold M to the presence of harmonic r -form fields on the smooth manifold. A large amount of theory is required to understand de Rham's theorem. In this paper we summarize the differential geometry that links holes in the underlying topology of a smooth manifold with harmonic fields on the manifold. We explore the application of de Rham's theory to (i) visual, (ii) mechanical, (iii) electrical and (iv) fluid flow systems. In particular, we consider harmonic flow fields in the intracellular aqueous solution of biological cells and we propose, on mathematical grounds, a possible role of harmonic flow fields in the folding of protein polypeptide chains. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the dynamic properties of a class of three-dimensional SIR models. First, we use the complete discriminant theory of polynomials to obtain the parameter conditions for the topological types of each fixed point. Second, by employing the centre manifold theorem and bifurcation theory, we prove that the system may undergo transcritical, flip and Neimark–Sacker bifurcations at fixed points. Finally, we simulated all the codimension-1 bifurcations. We found that the system also has codimension-2 bifurcations, i.e. 1:2, 1:3 and 1:4 strong resonances. [ABSTRACT FROM AUTHOR]

With the deep extension of coal mining in China, fault water inrush has become one of the major disasters threatening the safety production of coal mine. Based on the control equations of steady state and non-Darcy seepage in fractured rock mass, the multi-parameter nonlinear dynamic seepage equations of fractured rock mass are established in this paper. Based on the nonlinear dynamics theory, the function of the state variable in the system is derived, and the influence of the gradual change of non-Darcy flow factors on the structural stability of seepage system is studied. The research achievements show that there are three branches in the equilibrium state of the seepage system. Specifically, the stability of the equilibrium state changes abruptly near the limit parameter. The seepage dynamic system of fractured rock mass has the delayed bifurcation, and the coal mine disaster such as fault water inrush occurs easily at the bifurcation point. The research results are of great significance to enrich the theory of fault water inrush in coal mine, and to reveal the disastrous mechanism of fault water inrush and guide its prevention and control technology in coal mine, which can provide the theoretical reference for predicting the water seepage stability in fractured rock mass. [ABSTRACT FROM AUTHOR]

The paper's primary objective is to examine the dynamic behavior of an integer and noninteger predator–prey system with a Holling type IV functional response in the Caputo sense. Our focus is on understanding how harvesting influences the stability, equilibria, bifurcations, and limit cycles within this system. We employ qualitative and quantitative analysis methods rooted in bifurcation theory, dynamical theory, and numerical simulation. We also delve into studying the boundedness of solutions and investigating the stability and existence of equilibrium points within the system. Leveraging Sotomayor's theorem, we establish the presence of both the saddle-node and transcritical bifurcations. The analysis of the Hopf bifurcation is carried out using the normal form theorem. The model under consideration is extended to the fractional reaction–diffusion model which captures non-local and long-range effects more accurately than integer-order derivatives. This makes fractional reaction–diffusion systems suitable for modeling phenomena with anomalous diffusion or memory effects, improving the fidelity of simulations in turn. An adaptable numerical technique for solving this class of differential equations is also suggested. Through simulation results, we observe that one of the Lyapunov exponents has a negative value, indicating the potential for the emergence of a stable-limit cycle via bifurcation as well as chaotic and complex spatiotemporal distributions. We supplement our analytical investigations with numerical simulations to provide a comprehensive understanding of the system's behavior. It was discovered that both the prey and predator populations will continue to coexist and be permanent, regardless of the choice of fractional parameter. [ABSTRACT FROM AUTHOR]

This paper examines the dynamic behavior of a particular category of discrete predator-prey system that feature both fear effect and refuge, using both analytical and numerical methods. The critical coefficients and properties of bifurcating periodic solutions for Flip and Hopf bifurcations are computed using the center manifold theorem and bifurcation theory. Additionally, numerical simulations are employed to illustrate the bifurcation phenomenon and chaos characteristics. The results demonstrate that period-doubling and Hopf bifurcations are two typical routes to generate chaos, as evidenced by the calculation of the maximum Lyapunov exponents near the critical bifurcation points. Finally, a feedback control method is suggested, utilizing feedback of system states and perturbation of feedback parameters, to efficiently manage the bifurcations and chaotic attractors of the discrete predator-prey model. [ABSTRACT FROM AUTHOR]

In this paper, we examined the codimension-two bifurcation analysis of a two-dimensional discrete epidemic model. More precisely, we examined the codimension-two bifurcation analysis at an endemic equilibrium state associated with 1: 2, 1: 3 and 1: 4 strong resonances by bifurcation theory and series of affine transformations. Finally, theoretical results were carried out numerically. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the dynamic behaviors of fractional- order logistic model with Allee effects in Caputo–Fabrizio sense. First of all, we apply the two-step Adams–Bashforth scheme to discretize the fractional-order logistic differential equation and obtain the two-dimensional discrete system. The parametric conditions for local asymptotic stability of equilibrium points are obtained by Schur–Chon criterion. Moreover, we discuss the existence and direction for Neimark–Sacker bifurcations with the help of center manifold theorem and bifurcation theory. Numerical simulations are provided to illustrate theoretical discussion. It is observed that Allee effect plays an important role in stability analysis. Strong Allee effect in population enhances the stability of the coexisting steady state. In additional, the effect of fractional-order derivative on dynamic behavior of the system is also investigated. [ABSTRACT FROM AUTHOR]

In this paper, a new generalized non-monotonic and saturated incidence rate was introduced into a susceptible-infected-susceptible (SIS) epidemic model to account for inhibitory effect and crowding effect. The dynamic properties of the model were studied by qualitative theory and bifurcation theory. It is shown that when the influence of psychological factors is large, the model has only disease-free equilibrium point, and this disease-free equilibrium point is globally asymptotically stable; when the influence of psychological factors is small, for some parameter conditions, the model has a unique endemic equilibrium point, which is a cusp point of co-dimension two, and for other parameter conditions the model has two endemic equilibrium points, one of which could be weak focus or center. In addition, the results of the model undergoing saddle-node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation as the parameters vary were also proved. These results shed light on the impact of psychological behavior of susceptible people on the disease transmission. [ABSTRACT FROM AUTHOR]

This research paper investigates the Kadomtsev-Petviashvii-Benjamin-Bona-Mahony equation. The new Kudryashov and generalized Arnous methods are employed to obtain the generalized solitary wave solution. The phase plane theory examines the bifurcation analysis and illustrates phase portraits. Finally, the external perturbation terms are considered to reveal its chaotic behavior. These findings contribute to a deeper understanding of the dynamics of the Kadomtsev-Petviashvii-Benjamin-Bona-Mahony wave equation and its applications in real-world phenomena. [ABSTRACT FROM AUTHOR]

For food chain system with three populations, direct predation is the basic interaction between species. Different species often have different predation functional responses, so a food chain system with Holling-II response for middle predator and Beddinton–DeAngelis response for top predator is proposed. Apart from direct predation, predator population can significantly impact the survival of prey population by inducing the prey's fear, but the impact often possesses a time delay. This paper is concentrated to explore how the fear and time delay affect the system stability and the species persistence. By use of Lyapunov functional method and bifurcation theory, the positiveness and boundedness of solutions, local and global behavior of species, the system stability around the equilibrium states and various kinds of bifurcation are investigated. Numerically, some simulations are carried out to validate the main findings and the critical values of the bifurcation parameters of fear and conversion rate are obtained. It is observed that fear and delay can not only stabilize, but also destabilize the system, which depends on the magnitude of the fear and delay. The system varies from unstable to stable due to the continuous increase of the prey's fear by middle predator. Small fear induced by top predator or small delay of the prey's fear can stabilize the system, while they are sufficiently large, the system stability is to be destroyed. Simultaneously, the conversion rate can also change the stability and even make the species to be extinct. Some rich dynamics like multiple stabilities and various types of bistability behaviors are also exhibited, which results in the convergence of the species from one stable equilibrium to another. [ABSTRACT FROM AUTHOR]

During the operation of a DC microgrid, the nonlinearity and low damping characteristics of the DC bus make it prone to oscillatory instability. In this paper, we first establish a discrete nonlinear system dynamic model of a DC microgrid, study the effects of the converter sag coefficient, input voltage, and load resistance on the microgrid stability, and reveal the oscillation mechanism of a DC microgrid caused by a single source. Then, a DC microgrid stability analysis method based on the combination of bifurcation and strobe is used to analyze how the aforementioned parameters influence the oscillation characteristics of the system. Finally, the stability region of the system is obtained by the Jacobimatrix eigenvaluemethod. Grid simulation verifies the feasibility and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Memristor is a kind of nonlinear resistance with memory function, which is a typical nonlinear circuit component. In this paper, the problem of a simple chaotic circuit based on the memristor was studied. We constructed a new simple circuit of multiple memorizing components, summarized the mathematical model and found seven types of chaotic attractors without reference to classical circuits that substituted memristors for elements in the original circuit. Traditional methods of dynamic analysis are used to analyze the equilibrium point and stability of the system. Furthermore, we analyzed the dynamical behavior with the varying coefficient of the system specifically. In order to reduce the deviation from the actual physical circuit system as much as possible so as to facilitate the follow-up research of scholars, we solved the problem of Hopf bifurcation in this system under the condition of time delay through the use of the canonical form and Hopf bifurcation theory. Theoretical analysis and simulation results show that there existed the state transition in seven chaotic attractors of system and the bifurcation was stable when we changed the parameter τ slightly. As a small unit circuit, this paper lays a foundation for the research and control of large-scale system with memorizing components. [ABSTRACT FROM AUTHOR]

In Western societies, school pedagogies tend to be biased in favour of talk and emphasise the links between talking, thinking and learning. Thus talk is often privileged over silence as the basis for learning activities in classrooms, sustained by theories of learning which afford priority to talk. Such cultural bias towards talk means that by contrast, silence can be perceived negatively and construed as a form of 'non-participation'. Through a systematic literature review of journal articles relating to silence as a pedagogical approach published between 2000 and 2021, this article reappraises the role and value of silence in school education. Some of the apparent paradoxes of silence as a pedagogical approach, different types and uses of silence in the classroom, cultural dimensions of silence and the relationships between silence, power and critical pedagogy are examined. The pedagogical importance of silence as a participatory approach to learning emerges as a significant point for educators and the paper offers some suggestions for potential applications in classroom practice. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the following Schrödinger-Poisson system with nonhomogeneous boundary conditions \begin{equation*} \begin {cases} -\frac {1}{2}\triangle {u}+\phi u =\omega u, & \text {in }\Omega, \\ -\triangle {\phi }=4\pi u^2, & \text {in } \Omega,\\ \phi =h,u=0, & \text {on } \partial \Omega, \end{cases} \end{equation*} where \Omega is a smooth and bounded domain in \mathbb {R}^3, h is a given nonnegative regular function on \partial \Omega and \omega \in \mathbb {R}. By using variational method and bifurcation theory, we obtain the existence of positive solutions to the above system for \omega larger than some positive constant. [ABSTRACT FROM AUTHOR]

The bifurcation theory for homoclinic networks with singular and nonsingular equilibriums is a key to understand the global dynamics of nonlinear dynamical systems, which will help one determine the dynamical behaviors of physical and engineering nonlinear systems. In this paper, the appearing and switching bifurcations for homoclinic networks through equilibriums in planar polynomial dynamical systems are studied. The appearing and switching bifurcations are discussed for the homoclinic networks of nonsingular and singular sources, sinks, saddles with singular saddle-sources, saddle-sinks, and double-saddles in self-univariate polynomial systems. The first integral manifolds for nonsingular and singular equilibrium networks are determined. The illustrations of singular equilibriums to networks of nonsingular sources, sinks and saddles are given. The appearing and switching bifurcations are studied for homoclinic networks of singular and nonsingular saddles and centers with singular parabola-saddles and double-inflection saddles in crossing-univariate polynomial systems, and the first integral manifolds of such homoclinic networks are determined through polynomial functions. The illustrations of singular equilibriums to networks of nonsingular saddles and centers are given. This paper may help one understand higher-order bifurcation theory in nonlinear dynamical systems, which is completely different from the classic bifurcation theories. [ABSTRACT FROM AUTHOR]

This paper explores solitary wave solutions arising in the deformations of a hyperelastic compressible plate. Explicit traveling wave solution expressions with various parameters for the hyperelastic compressible plate are obtained and visualized. To analyze the perturbed equation, we employ geometric singular perturbation theory, Melnikov methods, and invariant manifold theory. The solitary wave solutions of the hyperelastic compressible plate do not persist under small perturbations for wave speed c > − β k 2 . Further exploration of nonlinear models that accurately depict the persistence of solitary wave solution on the significant physical processes under the K-S perturbation is recommended. [ABSTRACT FROM AUTHOR]

Aiming at the wedge friction damper for freight train bogie, considering Dankowicz dynamic friction, the mechanical model of a three-degree-of-freedom inclined impact vibration system with gap and dynamic friction, is simplified. The mechanical model of the system is established, the motion equation of the system is obtained, and the motion state and conditions of the system are analyzed. The Poincare map is constructed by selecting a fixed collision section, and the response of the system is solved by the fourth-order Runge-Kutta numerical method with variable step size. The transition process of the system motion and the phenomenon of sticking and chatter are analyzed by numerical simulation when the external excitation frequency changes. The results show that: 1) Under certain parameters, with the change of excitation frequency, the system undergoes periodic doubling bifurcation, inverse periodic doubling bifurcation, Grazing bifurcation and Hopf bifurcation, and there is a "periodic bubble" phenomenon in the system motion. When the system excitation frequency is between 3.35–3.55, 4.425–6.12, and 7.34–7.758, the system motion has chatter and sticking phenomena; when the system excitation frequency is between 1.75–3.24 and 3.92–4.425, the sticking phenomenon disappears, and only the chatter phenomenon exists. 2) When other parameters remain unchanged, and the mass ratio decreases from 1.15 to 0.85, nonlinear dynamic phenomena such as the transition between periodic bubbles and chaotic bubbles will be found. In this paper, the bifurcation and chaos characteristics of the impact vibration system of the wedge friction damper are studied, and the rich friction-induced vibration forms such as chatter and sticking are revealed, which provides a reference for improving the stability of vehicle operation and the selection of parameters in vehicle vibration reduction design in engineering practice. [ABSTRACT FROM AUTHOR]

In this paper, we propose a Holling-IV predator-prey system considering the perturbation of a slow-varying environmental capacity parameter. This study aims to address how the slowly varying environmental capacity parameter affects the behavior of the system. Based on bifurcation theory and the slow-fast analysis method, the critical condition for the Hopf bifurcation of the autonomous system is given. The oscillatory behavior of the system under different perturbation amplitudes is investigated, corresponding mechanism explanations are given, and it is found that the motion pattern of the non-autonomous system is closely related to the Hopf bifurcation and attractor types of the autonomous system. Meanwhile, there is a bifurcation hysteresis behavior of the system in bursting oscillations, and the bifurcation hysteresis mechanism of the system is analyzed by applying asymptotic theory, and its hysteresis time length is calculated. The final study found that the larger the perturbation amplitude, the longer the hysteresis time. These results can provide theoretical analyses for the prediction, regulation, and control of predator-prey populations. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate two important problems related to nilpotent center conditions and bifurcation of limit cycles in switching polynomial systems. Due to the difficulty in calculating the Lyapunov constants of switching polynomial systems at non-elementary singular points, it is extremely difficult to use the existing Poincaré-Lyapunov method to study these two problems. In this paper, we develop a higher-order Poincaré-Lyapunov method to consider the nilpotent center problem in switching polynomial systems, with particular attention focused on cubic switching Liénard systems. With proper perturbations, explicit center conditions are derived for switching Liénard systems at a nilpotent center. Moreover, with Bogdanov-Takens bifurcation theory, the existence of five limit cycles around the nilpotent center is proved for a class of switching Liénard systems, which is a new lower bound of cyclicity for such polynomial systems around a nilpotent center. [ABSTRACT FROM AUTHOR]

A high spike-timing precision characterized by a small variation in interspike intervals of neurons is important for information processing in various brain functions. An experimental study on fast-spiking interneurons has shown that inhibitory autapses functioning as negative self-feedback can enhance spike-timing precision. In the present paper, bifurcation and negative self-feedback mechanisms for the enhanced spike-timing precision to stochastic modulations are obtained in two theoretical models, presenting theoretical explanations to the experimental finding. For stochastic spikes near both the saddle-node bifurcation on an invariant cycle (SNIC) and the subcritical Hopf (SubH) bifurcation with classes 1 and 2 excitabilities, respectively, enhanced spike-timing precision appears in large ranges of the conductance and the decaying rate of inhibitory autapses, closely matching the experimental observation. The inhibitory autaptic current reduces the membrane potential after a spike to a level lower than that in the absence of inhibitory autapses and the threshold to evoke the next spike, making it more difficult for stochastic modulations to affect spike timings, and thereby enhancing spike-timing precision. In addition, firing frequency near the SubH bifurcation is more robust than that near the SNIC bifurcation, resulting in a higher spike-timing precision for the SubH bifurcation. The bifurcation and negative self-feedback mechanisms for the enhanced spike-timing precision present potential measures to modulate the neuronal dynamics or the autaptic parameters to adjust the spike-timing precision. [ABSTRACT FROM AUTHOR]

In this paper, the stability and bifurcation of a two–dimensional p53 gene regulatory network without and with time delay are taken into account by rigorous theoretical analyses and numerical simulations. In the absence of time delay, the existence and local stability of the positive equilibrium are considered through the Descartes' rule of signs, the determinant and trace of the Jacobian matrix, respectively. Then, the conditions for the occurrence of codimension–1 saddle–node and Hopf bifurcation are obtained with the help of Sotomayor's theorem and the Hopf bifurcation theorem, respectively, and the stability of the limit cycle induced by hopf bifurcation is analyzed through the calculation of the first Lyapunov number. Furthermore, codimension-2 Bogdanov–Takens bifurcation is investigated by calculating a universal unfolding near the cusp. In the presence of time delay, we prove that time delay can destabilize a stable equilibrium. All theoretical analyses are supported by numerical simulations. These results will expand our understanding of the complex dynamics of p53 and provide several potential biological applications. [ABSTRACT FROM AUTHOR]

In this paper, the bifurcation theory of planar dynamical systems is employed to investigate the mixed Korteweg-de Vries (KdV) equation. Under different parameter conditions, the bifurcation curves and phase portraits of corresponding Hamiltonian system are given. Furthermore, many different types of exact traveling waves are obtained, which include hyperbolic function solution, triangular function solution, rational solution and doubly periodic solutions in terms of the Jacobian elliptic functions. Furthermore, as all parameters in the representations of exact solutions are free variables, the solutions obtained show more complex dynamical behaviors, and could be applicable to explain diversity in qualitative features of wave phenomena. [ABSTRACT FROM AUTHOR]

In this paper, the Lyapunov-Schmidt reduction is used to investigate the bifurcation of periodic travelling wave solutions of a perturbed (1+1)-dimensional dispersive long wave equation. We demonstrate that the bifurcation equation corresponding to the original problem is supplied by a nonlinear system of two cubic algebraic equations. As the bifurcation parameters change, this system has only one, three, or five regular real solutions. The linear approximation of the solutions to the main problem has been discovered. [ABSTRACT FROM AUTHOR]

Purpose: Van der Pol (VDP) oscillator is a typical self-excited system with observable limit cycle oscillations (LCOs). To suppress LCOs, a VDP oscillator coupled nonlinear energy sink (NES) is proposed in this paper. To achieve the optimal vibration reduction effect, the sensitive parameters of the VDP-NES system are investigated and its vibration reduction mechanism is revealed. Methods: First, the dynamic model of the coupled NES system is established by utilizing Newton's law. Then we reveal the vibration reduction mechanism based on stability and bifurcation theory, which may be a new angle of view to investigate the VDP oscillator with NES. Finally, the influence of different parameters on the vibration reduction effect is discussed based on numerical simulation. Results: The LCOs of VDP oscillator are caused by Hopf bifurcation, and its amplitude is closely related to the eigenvalue of the equilibrium point. When the NES is coupled, the unstable equilibrium point becomes stable, then the LCOs are completely suppressed. Although the stability of the equilibrium point of the system with NES keeps invariant, the real part of the eigenvalue decreases. As a result, the LCOs are partially suppressed, and the amplitude decreases by about 70%. In addition, the self-excited intensity and damping ratio are sensitive to the vibration reduction effect. Conclusion: Coupling NES can have a better vibration reduction effect on the LCOs, and vibration reduction is closely related to Hopf bifurcation and the equilibrium point of the system. The research shows that putting off the Hopf bifurcation to widen the parameter range of the stable equilibrium point can obtain the optimal vibration control effect. At the same time, decreasing the real part of the eigenvalue also can reduce the amplitude of LCOs. It may provide some ideas for vibration control strategies in practical engineering. [ABSTRACT FROM AUTHOR]