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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]

In this paper, a diffusive zooplankton–phytoplankton model with time delay and hunting cooperation is established. First, the existence of all positive equilibria and their local stability are proved when the system does not include time delay and diffusion. Then, the existence of Hopf bifurcation at the positive equilibrium is proved by taking time delay as the bifurcation parameter, and the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are investigated by using the center manifold theorem and the normal form theory in partial differential equations. Next, according to the theory of Turing bifurcation, the conditions for the occurrence of Turing bifurcation are obtained by taking the intraspecific competition rate of the prey as the bifurcation parameter. Furthermore, the corresponding amplitude equations are discussed by using the standard multi-scale analysis method. Finally, some numerical simulations are given to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we proposed a Holling-III predator–prey model considering the perturbation of slow-varying, carrying capacity parameters. The study aims to address how the slow changes in carrying capacity influence the dynamics of the model. Based on the bifurcation theory and the slow–fast analysis method, the existence and the equilibrium of the autonomous system are explored, and then, the critical condition of Hopf bifurcation and transcritical bifurcation is established for the autonomous system. The slow–fast coupled nonautonomous system has quasiperiodic oscillations, single Hopf bursting oscillations, and transcritical–Hopf bursting oscillations within a certain range of perturbation amplitude variation if the carrying capacity perturbation amplitude crosses some critical values, such that the predator–prey management is challenging for the extinction of predator populations under the critical value. The motion pattern of the nonautonomous system is closely related to the transcritical bifurcation, Hopf bifurcation and attractor type of the autonomous system. Finally, the effects of changes in parameters related to predator aggressiveness on system behavior are investigated. These results show how crucial the predator–prey control is for varying carrying capacities. [ABSTRACT FROM AUTHOR]

In this paper, we consider a spatially size–stage-structured population dynamics model with resting phase. The primary objective of this model is to study size structure, stage structure, resting phase and spatial location simultaneously in a single population system. At first, we reformulate the problem as an abstract nondensely defined Cauchy problem. Then, taking advantage of the integrated semigroup and bifurcation theories, we investigate the stability and Hopf bifurcation at the positive equilibrium of the model. Finally, numerical simulations are presented as evidence to support our analytical results. A discussion of related problems is also presented briefly. [ABSTRACT FROM AUTHOR]

One of the promising applications of locally-active memristors (LAMs) is to construct oscillators for oscillatory neural networks. By using two current-controlled (CC) LAMs, a fully CC LAM-based oscillator is designed in this paper. The oscillator principle originates from the small-signal inductive and capacitive impedance characteristics of two different CC LAMs, and thus extra reactance element is not required in the circuit. Based on bifurcation theory and small-signal analysis method, conditions of the equilibrium point instability are quantitatively derived. Theoretical analysis indicates that the circuit oscillation is dependent on three critical parameters. Then, according to the conditions of the equilibrium point instability, parameters design methods of the two LAMs are proposed, including the static and dynamic parameters. A simple NbOx CC LAM model is taken as an example to conduct detailed simulation analysis. The simulation results verify the feasibility of the proposed circuit and analysis methods. Finally, the effects of the LAM model parameters on the oscillator performance are investigated, which is helpful for optimal design of the oscillator. [ABSTRACT FROM AUTHOR]

This paper investigates a spatiotemporal SI epidemiological model under homogeneous Neumann boundary conditions. First, the long-time behavior of the solutions is described, a priori estimates of nonconstant positive solutions are given, and the nonexistence of nonconstant positive steady states is proved by the energy method. Second, the Turing instability of the positive constant steady-state is discussed, and the existence of nonconstant positive steady states is shown by using the degree theory. Moreover, applying the bifurcation theory, we establish the local and global structures of the steady-state bifurcation from simple eigenvalues, and describe some conditions for determining the direction of bifurcation, where the techniques of space decomposition and implicit function theorem are adopted to deal with the local structure of the steady-state bifurcation from double eigenvalues. Finally, some analysis results are supplemented by numerical simulations. [ABSTRACT FROM AUTHOR]

This paper studies the Lengyel–Epstein chemical reaction system with nonlinear functional response and no-flux boundary conditions. We first investigate the existence of steady-state bifurcation solutions of the system. Then, the stability of bifurcation solutions is analyzed. Meanwhile, some spatial patterns induced by steady-state bifurcation are simulated numerically and depicted graphically. It is well known that the classical bifurcation theory in nonlinear dynamical systems is based on simple eigenvalues. However, this is not always the case. Sometimes, the kernel of an objective operator is of two or more dimensions. For such cases, there is no existing theory to deal with them. In this paper, by using the space decomposition technique and implicit function theorem, we analyze the bifurcation phenomenon of the system in the case of a two-dimensional kernel of some certain objective operator. The results show that the chemical reaction between activator iodide and inhibitor chlorite can proceed stably under certain conditions. [ABSTRACT FROM AUTHOR]

In this paper, we consider a single-species reaction–diffusion–advection population model with nonlinear boundary condition in heterogenous space. We not only investigate the existence, nonexistence and stability of positive steady-state solutions through a linear elliptic eigenvalue problem by means of variational approach, but also verify the existence of steady-state bifurcations at zero solution through Crandall and Robinowitz bifurcation theory and discuss the stability of bifurcations, which can lead to Allee effect when the bifurcation is subcritical. [ABSTRACT FROM AUTHOR]

This paper presents a discrete predator–prey interaction model involving hibernating vertebrates, with detailed analysis and simulation. Hibernation contributes to the survival and reproduction of organisms and species in the ecosystem as a whole. In addition, it also constitutes a wise sharing of time, space, and resources with others. We have created a new predator–prey model by integrating the two species, Holling-III and Holling-I, which have a bifurcation within a specified parameter range. We discovered that this system possesses the stability of fixed points as well as several bifurcation behaviors. To accomplish this, the center manifold theorem and bifurcation theory are applied to create existence conditions for period-doubling bifurcations and Neimark–Sacker bifurcations, which are depicted in the graph as distinct structures. Examples of numerical simulations include bifurcation diagrams, maximum Lyapunov exponents, and phase portraits, which demonstrate not just the validity of theoretical analysis but also complex dynamical behaviors and biological processes. Finally, the Ott–Grebogi–Yorke (OGY) method and phases of chaos control bifurcation were used to control the chaos of predator–prey model in hibernating vertebrates. [ABSTRACT FROM AUTHOR]

Bifurcation of the local Gierer–Meinhardt model is analyzed in this paper. It is found that the degenerate Bogdanov–Takens bifurcation of codimension 3 exists in the model, except for the saddle-node bifurcation and the Hopf bifurcation. That was not reported in the literature about this model. The existence of equilibria, their stability, the bifurcation and the induced complicated and interesting dynamics are explored in detail, by using stability analysis, normal form method and bifurcation theory. Numerical results are also presented to validate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, the bounded invariant surfaces of a generalized Langford system are discussed. Firstly, by the first integrals of systems restricted in the Poincaré sections of a periodic orbit, the accurate expressions of a heteroclinic orbit, a family of invariant tori and a heteroclinic invariant ellipsoid are given near a periodic orbit. Then, applying the successor functions to compute the periods of periodic orbits for the systems in the Poincaré sections, we present the parameter conditions for the existence of periodic orbits with any periods on these invariant tori. Finally, using the averaging theory and the theory of the Poincaré bifurcation and by determining the monotonicity of the ratio of two Abelian integrals, we give the conditions respectively such that the system has a unique invariant torus and a unique heteroclinic invariant ellipsoid near a zero-Hopf equilibrium. [ABSTRACT FROM AUTHOR]

The oscillatory dynamics of a delayed housefly model is analyzed in this paper. The local and global stabilities at the non-negative equilibria are obtained via analyzing the distribution of eigenvalues and Lyapunov–LaSalle invariance principle, and the model undergoes the supercritical Hopf bifurcation and the transient oscillation. Based on Wu's global Hopf bifurcation theory, the existence of the global bifurcation is established under certain conditions. [ABSTRACT FROM AUTHOR]

Electric energy routers (EERs) can effectively deal with the energy management issues caused by the access of multi-sources and multi-loads. Different from the energy transmission form in the existing series architecture EER (SA-EER) for low-voltage distribution networks, a new series-parallel architecture EER (SPA-EER) has the advantages of high-power transmission capability and reactive power flexible operation. However, if controller parameters change beyond their critical stability boundaries, SPA-EER will produce slow- and fast-scale bifurcation behaviors, which are manifested as low- and high-frequency oscillations of port voltages or port currents in varying degrees. To avoid the system instability caused by controller parameter changes, how to discriminate the stability regions of controller parameters for SPA-EER is a key research content in this paper. To this end, the stroboscopic mapping discrete model of SPA-EER system is first established, and system bifurcation behaviors are analyzed based on bifurcation theory. Furthermore, the critical stability boundaries of controller parameters are discriminated by using Jacobian matrix, root locus and bifurcation diagram. After that, a time delay feedback control (TDFC) is employed to suppress system bifurcation behaviors. Finally, the correctness of bifurcation analysis and the effectiveness of TDFC are verified by a hardware-in-the-loop (HIL) experimental platform. [ABSTRACT FROM AUTHOR]

This paper is concerned with the spatiotemporal heterogeneity in a modified Leslie–Gower predator–prey system with Beddington–DeAngelis functional response and prey-taxis. Using Crandall–Rabinowitz bifurcation theory, we investigate the steady-state bifurcation of the nonlinear system by choosing the prey-tactic sensitivity coefficient as a bifurcating parameter. It is rigorously proved that a branch of nonconstant solution exists near the positive equilibrium when the prey-tactic sensitivity is repulsive. Moreover, we study the existence, direction and stability of periodic orbits around the interior constant equilibrium by selecting the intrinsic growth rate of the prey as a bifurcating parameter. A priori estimates play a critical role in the verification procedure. Some numerical simulations are carried out to support our main theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a Beddington–DeAngelis prey–predator model with fear effect, refuge and harvesting is investigated. First, the positivity of solutions and boundedness of system are given. Then, the existence and local stability of equilibria of such system are obtained. Next, not only different codimension-one bifurcations, such as saddle-node bifurcation, transcritical bifurcation and Hopf bifurcation take place, but also Bogdanov–Takens bifurcation of codimension-two occurs as predicted by the center manifold theorem and bifurcation theory. Finally, some numerical simulations are carried out to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

We consider the well-known Lozi map, which is a 2D piecewise linear map depending on two parameters. This map can be considered as a piecewise linear analog of the Hénon map, allowing to simplify the rigorous proof of the existence of a chaotic attractor. The related parameter values belong to a part of the parameter plane where the map has two saddle fixed points. In the present paper, we investigate a different part of the parameter plane, namely, the vicinity of the curve related to a center bifurcation of the fixed point. A distinguishing property of the Lozi map is that it is conservative at the parameter value corresponding to this bifurcation. As a result, the bifurcation structure close to the center bifurcation curve is quite complicated. In particular, an attracting fixed point (focus) can coexist with various attracting cycles, as well as with chaotic attractors, and the number of coexisting attractors increases as the parameter point approaches the center bifurcation curve. The main result of the present paper is related to the rigorous description of this bifurcation structure. Specifically, we obtain, in explicit form, the boundaries of the main periodicity regions associated with the pairs of complementary cycles with rotation number 1 / n. Similar approach can be applied to other periodicity regions. Our study contributes also to the border collision bifurcation theory since the Lozi map is a particular case of the 2D border collision normal form. [ABSTRACT FROM AUTHOR]

In this paper, we apply bifurcation theory to consider four predator–prey systems which include the Allee effect, and show that the species having a strong Allee effct may affect their predation and hence extinction risk. It is shown that the models with the Allee effect exhibit more complex dynamical behaviors compared with that without the Allee effect. In particular, two models with no Allee effect do not have Hopf bifurcation, but can have Hopf bifurcation with the Allee effect; and one model, which does not have Bogdanov–Takens bifurcation if no Allee effect is involved, can have Bogdanov–Takens bifurcation of codimension two. Especially, for one model with Holling type II functional response of the predator to the prey, the Allee effect not only completely changes the stability of the equilibrium at the origin, but also changes the supercritical Hopf bifurcation arising from an interior equilibrium to subcritical Hopf bifurcation with very limited parameter values to yield unstable limit cycles, and further increases the system's stability. [ABSTRACT FROM AUTHOR]

Impulsive control strategies have been widely used in cancer treatment and linear impulsive control has always been considered in previous studies. We propose a novel tumour-immune model with nonlinear killing rate as state-dependent feedback control, which can better reflect the saturation effects of the tumour and immune cell mortalities due to chemotherapy, and its dynamic behaviors are investigated. The paper aims to discuss the transcritical and subcritical bifurcations of the model. To begin with, the threshold conditions for tumour eradication and tumour persistence in the model without pulse interventions are provided. We define the Poincaré map of the proposed model and then address the existence and orbital asymptotically stability of the model's tumour-free periodic solution. Furthermore, by using the bifurcation theory of the discrete one-parameter family of maps, which is determined by the Poincaré mapping, we investigate the model's transcritical and subcritical pitchfork bifurcations with respect to the key parameter. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the bifurcations and exact traveling wave solutions of the Lakshmanan–Porsezian–Daniel model with parabolic law nonlinearity. Employing the bifurcation theory of planar dynamical system, we obtain the phase portraits of the corresponding traveling wave system. The existence of the two singular straight lines leads to the dynamical behavior of solutions with two scales. Corresponding to some special level curves, we give the explicit exact solutions under different parameter conditions, containing peakon solutions, solitary wave solutions, kink and anti-kink wave solutions, periodic wave solutions and periodic peakon solutions. Moreover, we collect all these traveling solutions into a theorem. Finally, we discuss a small perturbation of the LPD equation and persistence of the traveling waves. [ABSTRACT FROM AUTHOR]

Consider a piecewise linear dynamical system in ℝ 3 , divided by a switching plane, with the Teixeira Singularity (TS). Under the antiparallelism hypothesis, it is known that the system undergoes the so-called pseudo-Hopf bifurcation (pH). In this paper, we consider a particular position of the antiparallel vectors to yield a two-parametric unfolding of the TS. The bifurcation analysis gives us, besides the curve of pH bifurcation points, a curve of saddle-node bifurcation points for crossing limit cycles. We call this phenomenon the pseudo-Bautin bifurcation, since it exhibits the exact same behavior as the Bautin bifurcation for smooth dynamical systems. [ABSTRACT FROM AUTHOR]

This paper analyzes the properties of the nonequilibrium singular point in one-dimensional elementary catastrophe. For this analysis, the Kosambi–Cartan–Chern (KCC) theory is applied to characterize the dynamical system based on differential geometrical quantities. When both the nonlinear connection and deviation curvature are zero, that is, when the geometric stability of the KCC theory is neutral, two bifurcation curves are obtained: one is the known curve with an equilibrium singular point, and the other is a new curve with a nonequilibrium singular point. The two singular points are distinguished based on the vanishing condition of the Berwald connection. Applied to the ecosystem described by the Hill function, the absolute value of the cuspidal curvature of the nonequilibrium singular point is larger than that of the equilibrium singular point. The ecological interpretation of this result is that the range of bistability of the ecosystem in the nonequilibrium state is greater than that in the equilibrium state. The type of singular points in equilibrium and nonequilibrium bifurcation curves are not necessarily the same. For instance, there is a combination in which even if the former has one cusp, the latter may show various types, depending on the parametric space. These results demonstrate that there are cases where simply shifting the system from the equilibrium to nonequilibrium state expands the range of bistability and changes the type of singularity. Although singularity analysis is often performed near the equilibrium point, nonequilibrium analysis, i.e. analysis based on the KCC theory, provides a useful perspective for analyzing singularity theory according to the bifurcation phenomenon. [ABSTRACT FROM AUTHOR]

This paper elucidates the bifurcation mechanism of an equidistant economy in spatial economics. To this end, we derive the rules of secondary and further bifurcations as a major theoretical contribution of this paper. Then we combine them with pre-existing results of direct bifurcation of the symmetric group S N [Elmhirst, 2004]. Particular attention is devoted to the existence of invariant solutions which retain their spatial distributions when the value of the bifurcation parameter changes. Invariant patterns of an equidistant economy under the replicator dynamics are obtained. The mechanism of bifurcations from these patterns is elucidated. The stability of bifurcating branches is analyzed to demonstrate that most of them are unstable immediately after bifurcation. Numerical analysis of spatial economic models confirms that almost all bifurcating branches are unstable. Direct bifurcating curves connect the curves of invariant solutions, thereby creating a mesh-like network, which appears as threads of warp and weft. The theoretical bifurcation mechanism and numerical examples of networks advanced herein might be of great assistance in the study of spatial economics. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to study the oscillatory behaviors of gene expression networks in quorum-sensing system with time delay. The stability of the unique positive equilibrium and the existence of Hopf bifurcation are investigated by choosing the time delay as the bifurcation parameter and by applying the bifurcation theory. The explicit criteria determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are developed based on the normal form theory and the center manifold theorem. Numerical simulations demonstrate good agreements with the theoretical results. Results of this paper indicate that the time delay plays a crucial role in the regulation of the dynamic behaviors of quorum-sensing system. [ABSTRACT FROM AUTHOR]

In this paper, we present a new generic local bifurcation in planar Filippov systems. The new bifurcation presented here is a visible–invisible two-fold bifurcation which does not appear in the list published by Kuznetsov et al. in 2003. [ABSTRACT FROM AUTHOR]

In this paper, we mainly investigate an infected-age structured HIV model with logistic growth for target cells, which simultaneously takes the latent factor and different transmission modes into consideration. On the basis of integrated semigroup theory and Hopf bifurcation theory for semilinear equations in a nondense domain, we reach a conclusion that the system undergoes a Hopf bifurcation around the positive equilibrium, which implies that there exists a nontrivial periodic oscillation phenomenon when the Hopf bifurcation parameter τ passes through some critical values. Namely, the HIV infection will periodically erupt when the mature period τ crosses the critical value τ k (k = 0 , 1 , 2 , ...), that is, HIV virus will always survive in the patient's body. In order to verify the fact that the biological maturation period τ does play an essential role in our system, we also offer the relevant numerical simulation results by selecting several decisive parameters. [ABSTRACT FROM AUTHOR]

Nonlinear bifurcations and instabilities of autonomous nonconservative systems, mainly involving the fluid loading of a solid or structure, are reviewed and described in this accessible, pictorial overview. In contrast to the earlier papers in this series (focusing on the instability of elastic deformable systems, and low-order periodically-forced mechanical systems), we focus on a handful of case studies in which the loss of stability is primarily driven by nonconservative forces, i.e. path-dependent forces not associated with a potential. Many systems involving fluid-structure interaction can lose stability under changing conditions in which there is a net flow of energy from the fluid to the structure, sometimes resulting in growing oscillatory behavior. Again, the generic manifestations of instability typically occur within the framework of bifurcation theory. Progression is from simple local bifurcations to more complex global events, and all are related to instructive and intriguing applications. Hopf bifurcations are presented in the context of the galloping and flutter of cables and pipes. Next, Neimark bifurcations appear in aircraft applications involving the free-play fluttering of aerofoils and the wing rock of the Harrier jump-jet. Turning to ships in wind and waves, a homoclinic saddle connection governs the surging and surf-riding of a vessel in stern seas, while an omega flow explosion can compromise the course-keeping of a passenger ferry in a side wind. Recent work on the dynamic step-buckling of a spherical shell illustrates the role of a center manifold, and the paper ends with a careful study of dissipation-generated instabilities, drawing on the historical struggles to understand the evolution of spinning liquid planets. [ABSTRACT FROM AUTHOR]

The spatiotemporal dynamics of a space-time discrete toxic phytoplankton-zooplankton model is studied in this paper. The stable conditions for steady states are obtained through the linear stability analysis. According to the center manifold theorem and bifurcation theory, the critical parameter values for flip bifurcation, Neimark–Sacker bifurcation and Turing bifurcation are determined, respectively. Besides, the numerical simulations are provided to illustrate theoretical results. In order to distinguish chaos from regular behaviors, the maximum Lyapunov exponents are shown. The simulations show new and complex dynamics behaviors, such as period-doubling cascade, invariant circles, periodic windows, chaotic region and pattern formations. Numerical simulations of Turing patterns induced by flip-Turing instability, Neimark–Sacker Turing instability and chaos reveal a variety of spatiotemporal patterns, including plaque, curl, spiral, circle, and many other regular and irregular patterns. In comparison with former results in literature, the space-time discrete version considered in this paper captures more complicated and richer nonlinear dynamics behaviors and contributes a new comprehension on the complex pattern formation of spatially extended discrete phytoplankton-zooplankton system. [ABSTRACT FROM AUTHOR]

Creep groan is the undesirable vibration observed in the brake pad and disc as brakes are applied during low-speed driving. The presence of friction leads to nonlinear behavior even in simple models of this phenomenon. This paper uses tools from bifurcation theory to investigate creep groan behavior in a nonlinear 3-degrees-of-freedom mathematical model. Three areas of operational interest are identified, replicating results from previous studies: region 1 contains repelling equilibria and attracting periodic orbits (creep groan); region 2 contains both attracting equilibria and periodic orbits (creep groan and no creep groan, depending on initial conditions); region 3 contains attracting equilibria (no creep groan). The influence of several friction model parameters on these regions is presented, which identify that the transition between static and dynamic friction regimes has a large influence on the existence of creep groan. Additional investigations discover the presence of several bifurcations previously unknown to exist in this model, including Hopf, torus and period-doubling bifurcations. This insight provides valuable novel information about the nature of creep groan and indicates that complex behavior can be discovered and explored in relatively simple models. [ABSTRACT FROM AUTHOR]

This paper deals with a kind of reaction–diffusion–advection model which depicts a predator–prey ecosystem in rivers or streams. We obtain a complete classification on the dynamical behavior of the system in the parameter space of the predator's mortality rate μ and the prey's intrinsic growth rate r. More precisely, both species fail to survive when r is small. With the increase of r , there exist two critical values r ∗ and σ ∗ such that both species can coexist in the long run when r > r ∗ and 0 < μ < σ ∗ , otherwise the prey survives alone. Finally, with the aid of numerical simulations, we investigate the effects of diffusion and advection on the global dynamics of the system by computing these two critical values. Numerical simulations illustrate that the diffusion of both predators and prey would benefit the invasion of the predators, and the advection would be unfavorable to the survival of the predators. Moreover, numerical simulations also suggest that the unique positive steady state is globally asymptotically stable among all non-negative and nontrivial initial data. [ABSTRACT FROM AUTHOR]

A hexagonal lattice is a promising and plausible spatial platform for economic agglomeration in spatial economic models. This paper aims at the elucidation of agglomeration mechanisms for the replicator dynamics on this lattice. Attention is paid to the existence of invariant solutions that retain their spatial patterns when the bifurcation parameter changes. Such existence is a special feature of the replicator dynamics, which is widely used in economics. A theoretical procedure to find invariant patterns is proposed and possible invariant patterns are advanced and classified. Among a plethora of theoretically possible invariant patterns, those which actually become stable for a spatial economic model are investigated numerically. The major finding of this paper, is the demonstration of equilibrium curves of invariant patterns that are connected by those of noninvariant ones to form a complicated mesh-like structure, just like the threads of warp and weft. It would be an important scientific mission to elucidate the mechanism of this complicated structure, and contribute to the study in economic geography. [ABSTRACT FROM AUTHOR]

This paper presents a method to investigate exact traveling wave solutions and bifurcations of the nonlinear time-fractional partial differential equations with the conformable fractional derivative proposed by [Khalil et al., 2014]. The method is based on employing the bifurcation theory of planar dynamical systems proposed by [Li, 2013]. For the fractional PDEs, up till now, there is no related paper to obtain the exact solutions by applying bifurcation theory. We show how to use this method with applications to two fractional PDEs: the fractional Klein–Gordon equation and the fractional generalized Hirota–Satsuma coupled KdV system, respectively. We find the new exact solutions including periodic wave solution, kink wave solution, anti-kink wave solution and solitary wave solution (bright and dark), which are different from previous works in the literature. This approach can also be extended to other nonlinear time-fractional differential equations with the conformable fractional derivative. [ABSTRACT FROM AUTHOR]

In this paper, we further study a financial market model established in our earlier paper. The model dynamics is driven by a two-dimensional piecewise linear discontinuous map, which is investigated analytically and numerically for one-sided fixed points being flip saddle and two-sided fixed points being attractors. The existence of chaotic orbit is explained by using the theory of homoclinic intersection between stable and unstable manifolds of the flip saddle invariant set. The structure of chaotic attractor is disclosed. It consists of finite segments rooted on both sides of the x -axis which are unstable manifolds of flip saddle invariant set. The basins and their structural changes of bounded attractors and coexisting attractors are presented by contact bifurcation theory and numerical simulations. The border collision bifurcation (BCB for short) curves are calculated and coexisting multiattractors are disclosed by overlapping periodicity regions. The results can deepen our understanding of financial markets and dynamical systems. [ABSTRACT FROM AUTHOR]

Many studies have been done on different aspects of biped robots such as motion, path planning, control and stability. Dynamical properties of biped robot on a sloping surface such as equilibria and their stabilities, bifurcations and basin of attraction are investigated in this paper. Basin of attraction is an important property since it can determine the unseen conditions which affect the attractor of the system with multistabilities. By the help of basin of attractions, the paper claims that the strange attractors of compass-gait robot are hidden. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of stability and bifurcations of homoclinic loops for planar vector fields. For a given homoclinic loop, a sequence of loop numbers can be defined such that the stability and bifurcations of the loop are determined by the first nonzero term of the sequence. Formulas for the first several loop numbers were established in the past. In this paper, we will introduce general formulas for the loop numbers for both the single and double homoclinic loops. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the lateral dynamics of a railway wheelset suspended under a moving car with linear springs and dry friction dampers. Both theoretical and numerical methods are used to complement each other. The car runs on an ideal, straight and perfect track with a constant speed. A nonlinear relation between the creepages and the creep forces is used in this paper. The nonsmoothness of this model is due to the dry friction dampers. The speed is selected as the bifurcation parameter. The one-dimensional bifurcation diagram, which gives a general view of the dynamics of the system, is presented. Both symmetric and asymmetric periodic motions, quasi-periodic motions and chaotic motions are found. In addition to bifurcations that can exist in both smooth and nonsmooth systems, a kind of sliding bifurcations that are unique to nonsmooth systems is found. Bifurcation diagrams, phase portraits, Poincaré sections and Lyapunov exponents are presented to ensure that no contradictory results are given. The influence of the conicity of the wheel tread on the Hopf bifurcation type is examined. [ABSTRACT FROM AUTHOR]

In this paper, a predator–prey model with prey-stage structure and prey-taxis is proposed and studied. Firstly, the local stability of non-negative constant equilibria is analyzed. It is shown that non-negative equilibria have the same stability between ODE system and self-diffusion system, and self-diffusion does not have a destabilization effect. We find that there exists a threshold value ξ 0 such that the positive equilibrium point of the model becomes unstable when the prey-taxis rate ξ < ξ 0 , hence the taxis-driven Turing instability occurs. Furthermore, by applying Crandall–Rabinowitz bifurcation theory, the existence, the stability and instability, and the turning direction of bifurcating steady state are investigated in detail. Finally, numerical simulations are provided to support the mathematical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a class of three-dimensional eco-epidemiological models where prey is subject to Allee effects and infection. We first establish the existence, uniqueness, positivity and uniform ultimate boundedness of the solutions for the proposed system in the positive octant. For three subsystems, we investigate the existence of their respective trivial and positive equilibria and determine the conditions for some bifurcations (Hopf bifurcation, Bogdanov–Takens bifurcation of codimension-2 and saddle-node bifurcation) to occur. We find that the Allee effect, nonmonotonic functional response and intra-class competition in susceptible preys enable the S–I and S–P subsystems to have richer dynamics. For example, the S–I subsystem can have up to three positive equilibria, the S–P subsystem with nonmonotonic functional response can have two positive equilibria while it is impossible in monotonic situation, and high intra-class competition in susceptible preys may lead to the extinction of the predator population, etc. We show that the strong Allee effect can create a separatrix curve (or surface), leading to multistability. Then, we study the uniform persistence of the full system and identify an interior periodic orbit by applying Poincaré map and bifurcation theory. Our analysis reveals that the introduction of the infection or predation may act as a biological control to save the population from extinction and the interaction between these two factors yields a diverse array of biologically relevant behaviors. Finally, some numerical simulations are performed to support and supplement our analytical findings. [ABSTRACT FROM AUTHOR]

In this paper, a predator–prey model with age structure, Beddington–DeAngelis functional response and time delays is considered. Using a geometric method for studying transcendental equation with two delays, we conduct detailed analysis on the distribution of the roots for the characteristic equation of the model. Then, applying the integrated semigroup theory and the Hopf bifurcation theorem for an abstract Cauchy problem within a nondense domain, we proved the existence of Hopf bifurcation for the model. Stability switches can also occur, as the two time delays pass through a continuous curve in the parameter plane. To illustrate the theoretical results, numerical simulations are presented. [ABSTRACT FROM AUTHOR]

In this paper, a couple map lattice (CML) model is used to study the spatiotemporal dynamics and Turing patterns for a space-time discrete generalized toxic-phytoplankton-zooplankton system with self-diffusion and cross-diffusion. First, the existence and stability conditions for fixed points are obtained by using linear stability analysis. Second, the conditions for the occurrence of flip bifurcation, Neimark–Sacker bifurcation and Turing bifurcation are obtained by using the center manifold reduction theorem and bifurcation theory. The results show that there exist two nonlinear mechanisms, flip-Turing instability and Neimark–Sacker–Turing instability. Moreover, some numerical simulations are used to illustrate the theoretical results. Interestingly, rich dynamical behaviors, such as periodic points, periodic or quasi-periodic orbits, chaos and interesting patterns (plaques, curls, spirals, circles and other intermediate patterns) are found. The results obtained in the CML model contribute to comprehending the complex pattern formation of spatially extended discrete generalized toxic-phytoplankton-zooplankton system. [ABSTRACT FROM AUTHOR]

In this paper, time-delayed sampled-data feedback control technique is used to asymptotically stabilize a class of unstable delayed differential systems. Through the analysis for the distribution change of eigenvalues, an effective interval of the control parameter is obtained for a given sampling period. Here an indirect strategy is taken. Specifically, the system of continuous-time delayed feedback control is studied first by Hopf bifurcation theory. And then, the result and implicit function theorem are used to analyze the system of time-delayed sampled-data feedback control with a sufficiently small sampling period. Considering the practical criterion for the size of sampling period, the upper bound of sampling period is estimated. Finally, an application example, an unstable Mackey–Glass model, is asymptotically stabilized by introducing a blood transfusion item with time-delayed sampled-data feedback control. The blood transfusion speed and blood collection test period are derived from the main results. Some simulations and comparisons show the correctness and advantages of the main theoretical results. [ABSTRACT FROM AUTHOR]

The stability and the two-parameter bifurcation of a two-dimensional discrete Gierer–Meinhardt system are investigated in this paper. The analysis is carried out both theoretically and numerically. It is found that the model can exhibit codimension-two bifurcations (1 : 2 , 1 : 3 , and 1 : 4 strong resonances) for certain critical values at the positive fixed point. The normal forms are obtained by using a series of affine transformations and bifurcation theory. Numerical simulations including bifurcation diagrams, phase portraits and basins of attraction are conducted to validate the theoretical predictions, which can also display some interesting and complex dynamical behaviors. [ABSTRACT FROM AUTHOR]

In recent publications [Llibre, 2014; Llibre & Makhlouf, 2020], time-averaging method was applied to studying periodic orbits bifurcating from zero-Hopf critical points of two Rössler systems. It was shown that the averaging method is successful for a certain type of zero-Hopf critical points, but fails for some type of such critical points. In this paper, we apply normal form theory to reinvestigate the bifurcation and show that the method of normal forms is applicable for all types of zero-Hopf bifurcations, revealing why the time-averaging method fails for some type of zero-Hopf bifurcation. [ABSTRACT FROM AUTHOR]

Digital bifurcation analysis is a new algorithmic method for exploring how the behavior of a parameter-dependent discrete system varies with a change in its parameters and, in particular, for the identification of bifurcation points where such variation becomes dramatic. We have developed the method in an analogy with the traditional bifurcation theory and have successfully applied it to models taken from systems biology. In this paper, we report on the application of the digital bifurcation analysis for analyzing the stability of internet congestion control protocols by inspecting their attractor bifurcations. In contrast to the analytical methods, our approach allows fully automated analysis. We compared the robustness of the basic Random Early Drop (RED) approach with four substantially different extensions, namely gentle, adaptive, gradient descent, and integral feedback RED. The basic RED protocol is well known to exhibit unstable behavior when parameters are varied. In the case of adaptive and gradient descent RED protocol, the analysis showed significant improvements in stability, whereas in the results for gentle and integral feedback RED protocols the improvement was negligible. We performed a series of model simulations, the results of which were in accordance with our bifurcation analysis. Based on our results, we can recommend both adaptive and gradient descent RED to improve the robustness of the RED protocol. [ABSTRACT FROM AUTHOR]

Over the past few decades, interval arithmetic has been attracting widespread interest from the scientific community. With the expansion of computing power, scientific computing is encountering a noteworthy shift from floating-point arithmetic toward increased use of interval arithmetic. Notwithstanding the significant reliability of interval arithmetic, this paper presents a theoretical inconsistency in a simulation of dynamical systems using a well-known implementation of arithmetic interval. We have observed that two natural interval extensions present an empty intersection during a finite time range, which is contrary to the fundamental theorem of interval analysis. We have proposed a procedure to at least partially overcome this problem, based on the union of the two generated pseudo-orbits. This paper also shows a successful case of interval arithmetic application in the reduction of interval width size on the simulation of discrete map. The implications of our findings on the reliability of scientific computing using interval arithmetic have been properly addressed using two numerical examples. [ABSTRACT FROM AUTHOR]

Research on the output game behavior of oligopoly has greatly advanced in recent years. But many unknowns remain, particularly the influence of consumers' willingness to buy green products on the oligopoly output game. This paper constructs a triopoly output game model with multiple delays in the competition of green products. The influence of the parameters on the stability and complexity of the system is studied by analyzing the existence and local asymptotic stability of the equilibrium point. It is found that the system loses stability and increases complexity if delay parameters exceed a certain range. In the unstable or chaotic game market, the decisions of oligopoly will be counterproductive. It is also observed that the influence of weight and output adjustment speed on the firm itself is obviously stronger than the influence of other firms. In addition, it is important that weight and output adjustment speed cannot increase indefinitely, otherwise it will bring unnecessary losses to the firm. Finally, chaos control is realized by using the variable feedback control method. The research results of this paper can provide a reference for decision-making for the output of the game of oligopoly. [ABSTRACT FROM AUTHOR]

This paper considers the weakened Hilbert's 16th problem for symmetric planar perturbed polynomial Hamiltonian system. A Z13-equivariant planar vector field of degree 12 is deduced to find as many as possible limit cycles and their configuration patterns. By using bifurcation theory of planar dynamical system and the method of detection function, we have obtained that, under the thirteenth-order perturbation, the above Z13-equivariant planar perturbed Hamiltonian vector field of 12-degree has at least 117 limit cycles. Moreover, this paper also shows the configuration of compound eyes of the corresponding perturbed systems. [ABSTRACT FROM AUTHOR]

In this paper, we consider a single cell model of pre-Bötzinger complex, which is derived by adding an external tonic drive () to the model developed by Park and Rubin. Using fast-slow geometric decomposition and bifurcation analysis, we study firing activities of the system and try to reveal the mechanisms underlying the bursts related to the mean level of excitatory input () and the maximal conductance associated with the sodium (). Since a regular bursting requires at least two timescales, we consider the effects of timescale, especially of the slow timescale, on the bursting oscillations. Unlike the previous works, in this paper, we conduct our investigation by choosing different slow variables. We show how and affect bifurcations of the fast subsystem and how the bifurcations further determine firing activities of the full system with different slow variables. [ABSTRACT FROM AUTHOR]

In this paper, we extend the equivariant Hopf bifurcation theory for semilinear functional differential equations in general Banach spaces and then apply it to reaction–diffusion models with delay effect and homogeneous Dirichlet boundary condition on a general open domain with a smooth boundary. In the process we derive the criteria for the existence and directions of branches of bifurcating periodic solutions, avoiding the process of center manifold reduction. [ABSTRACT FROM AUTHOR]

Re-emergence of cholera threatens people's health globally. However, its periodic re-emerging outbreaks are still poorly understood. In this paper, we develop a simple ordinary differential equation (ODE) model to study the cholera outbreak cycles. Our model involves both direct (i.e. human-to-human) and indirect (i.e. environment-to-human) transmission routes, due to the multiple interactions between the human host, the pathogen, and the environment. In particular, we model the pathogen searching distance as a Poisson point process, and then formulate the host-pathogen encounter (HPE) rate. A thorough mathematical analysis is performed to investigate local and global dynamics of the model. Necessary and sufficient condition under which the backward bifurcation occurs is derived. Fold, Hopf, and Bogdanov–Takens bifurcations are studied with original model parameter values to reveal their relations with model behaviors. One- and two-dimensional bifurcation diagrams are provided to categorize model dynamics with respect to its parameter values. Analytical and numerical analyses show that our simple model is sufficient to exhibit complex epidemic patterns of cholera dynamics including bistability and annual and multiannual periodic outbreaks. Our result regarding the backward bifurcation and complex dynamics of cholera epidemics highlight the challenges in the prevention and control of the disease. [ABSTRACT FROM AUTHOR]

This paper proposes a novel three-dimensional autonomous chaotic system. Interestingly, when the system has infinitely many stable equilibria, it is found that the system also has infinitely many hidden chaotic attractors. We show that the period-doubling bifurcations are the routes to chaos. Moreover, the Hopf bifurcations at all equilibria are investigated and it is also found that all the Hopf bifurcations simultaneously occur. Furthermore, the approximate expressions and stabilities of bifurcating limit cycles are obtained by using normal form theory and bifurcation theory. [ABSTRACT FROM AUTHOR]