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In this paper, we study Poincaré, Hopf and homoclinic bifurcations of limit cycles for planar near-Hamiltonian systems. Our main results establish Hopf and homoclinic bifurcation theories by higher order Melnikov functions, obtaining conditions on upper bounds and lower bounds of the maximum number of limit cycles. As an application, we concern a cubic near-Hamiltonian system, and study Hopf and homoclinic bifurcations in detail, finding more limit cycles than [26]. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the normal form a class of partial functional differential equations, which takes account of delays in the diffusion terms as well as a wider scope of nonlinear terms. We first study the associated linear theory, mainly including the spectral properties of infinitesimal generator, formal adjoint and decomposition of phase space. Based on these results, the normal form theory for nonlinear equation is established, which can be used to study the local dynamics near the steady state for such equations. As an application, we consider the Hopf bifurcation problem of a scalar diffusive equation with delay not only involved in the diffusive term but also in reaction terms. The normal form, depending on the original coefficients, up to the third order term is calculated, which allows us to determine the direction of Hopf bifurcation and stability of bifurcated periodic solutions. The results are then applied to study the scalar population model with memory-based diffusion for modeling the movement of highly-developed animals. [ABSTRACT FROM AUTHOR]

In this paper, a reaction-diffusion population model with Dirichlet boundary condition and a directed movement oriented by a temporally distributed delay is proposed to describe the lasting memory of animals moving over space. The temporal kernel of the memory is taken as Gamma distribution function, among which there are two biologically meaningful cases: one is the weak kernel which implies that animals can immediately acquire knowledge and memory decays over time, the other is the strong kernel by which we assume that animals' memory undergoes learning and memory decay stages. It is shown that the population stabilizes to a positive steady state and aggregates in the interior of the territory when the delay kernel is the weak type. In the strong kernel case, oscillatory patterns can first arise via a Hopf bifurcation with a small memory delay and then vanish when the system undergoes a Hopf bifurcation with a large memory delay, which implies that stability switch occurs and spatial-temporal patterns emerge for intermediate value of delays. [ABSTRACT FROM AUTHOR]

This paper is concerned with a reaction-diffusion population model with nonlocal delayed effect and zero-Dirichlet boundary condition. Under the condition when the delayed feedback control is dominant, the normal form for spatially nonhomogeneous Hopf bifurcation from the sufficiently small positive equilibrium is computed by means of the normal form method and the center manifold theorem for partial functional differential equations. It is revealed that Hopf bifurcations appearing at the small positive equilibrium are forward and all bifurcating periodic solutions are locally orbitally asymptotically stable on the center manifold. To verify the validity of the obtained theoretical results, numerical simulations are also provided. [ABSTRACT FROM AUTHOR]

For decades, the network-organized reaction-diffusion models have been widely used to study ecological and epidemiological phenomena in discrete space. However, the high dimensionality of these nonlinear systems places a long-standing restriction to develop the normal forms of various bifurcations. In this paper, we take an important step to present a rigorous procedure for calculating the normal form associated with the Hopf bifurcation of the general network-organized reaction-diffusion systems, which is similar to but can be much more intricate than the corresponding procedure for the extensively explored PDE systems. To show the potential applications of our obtained theoretical results, we conduct the detailed Hopf bifurcation analysis for a multi-patch predator-prey system defined on any undirected connected underlying network and on the particular non-periodic one-dimensional lattice network. Remarkably, we reveal that the structure of the underlying network imposes a significant effect on the occurrence of the spatially nonhomogeneous Hopf bifurcations. [ABSTRACT FROM AUTHOR]

In this paper, we mainly investigate a Lotka-Volterra competition-diffusion-advection system with time delay, where the diffusion and advection rates of two competitors are different. By employing the Lyapunov-Schmidt reduction method, we obtain the existence of steady state solution. A weighted inner product has been introduced to study stability and Hopf bifurcation at the spatially nonhomogeneous steady-state. Our results imply that the infinitesimal generator associated with the linearized system have two pairs of purely imaginary eigenvalues, and time delay can make the spatially nonconstant positive steady state unstable for a reaction-diffusion-advection model. In addition, the bifurcation direction and stability of Hopf bifurcating periodic orbits was obtained by means of the center manifold reduction and the normal form theory. [ABSTRACT FROM AUTHOR]

Refuge provides an important mechanism for preserving many ecosystems. Prey refuges directly benefit prey but also indirectly benefit predators in the long term. In this paper, we consider the complex dynamics and bifurcations in Holling-Tanner model with generalist predator and prey refuge. It is shown that the model admits a nilpotent cusp or focus of codimension 3, a nilpotent elliptic singularity of codimension at least 4, and a weak focus with order at least 3 for different parameter values. As the parameters vary, the model can undergo three types degenerate Bogdanov-Takens bifurcations of codimension 3 (cusp, focus and elliptic cases), and degenerate Hopf bifurcation of codimension 3. The system can exhibit complex dynamics, such as multiple coexistent periodic orbits and homoclinic loops. Moreover, our results indicate that the constant prey refuge prevents prey extinction and causes global coexistence. A preeminent finding is that refuge can induce a stable, large-amplitude limit cycle enclosing one or three positive steady states. Numerical simulations are provided to illustrate and complement our theoretical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with equivariant normal forms of semilinear functional differential equations (FDEs) in general Banach spaces. The analysis is based on the theory previously developed for autonomous delay differential equations and on the existence of invariant manifolds. We show that in the neighborhood of trivial solutions, variables can be chosen so that the form of the reduced vector field relies not only on the information of the linearized system at the critical point but also on the inherent symmetry. We observe that the normal forms give critical information about dynamical properties, such as generic local branching spatiotemporal patterns of equilibria and periodic solutions. As an important application of equivariant normal forms, we not only establish equivariant Hopf bifurcation theorem for semilinear FDEs in general Banach spaces, but also in a natural way derive criteria for the existence, stability, and bifurcation direction of branches of bifurcating periodic solutions. We employ these general results to obtain the existence of infinite many small-amplitude wave solutions for a delayed Ginzburg-Landau equation on a two-dimensional disk with the homogeneous Dirichlet boundary condition. [ABSTRACT FROM AUTHOR]

This paper devotes to the study on the asymptotic behavior of an age-structured prey-predator model with distributed delay. The existence of positive stationary solutions of the considered model is first discussed and the nonlinear system is linearized around a positive stationary solution. Then the resulting linear system is formulated as an abstract non-densely defined Cauchy problem so that the well-posedness is obtained for the system. Following that the problems of locally asymptotic stability/instability, asynchronous exponential growth as well as Hopf bifurcations for the linearized system are investigated with the help of operator semigroups theory and spectral analysis under some conditions. Finally, some numerical simulations are provided to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the Holling-Tanner model with constant-yield prey harvesting. It is shown that the highest codimension of a nilpotent cusp is 4, and the model can undergo degenerate Bogdanov-Takens bifurcation of codimension 4. Moreover, when the model has a center-type equilibrium, we show that it is a weak focus with order at least 3 and at most 4, and the model can exhibit Hopf bifurcation of codimension 3. Some algebraic methods including resultant elimination and pseudo-division are used to solve the semi-algebraic varieties of normal form coefficients or focal values. Our results indicate that constant-yield prey harvesting can cause not only richer dynamics and bifurcations, but also the coextinction of both populations with some positive initial densities. Finally, numerical simulations, including the coexistence of limit cycle and homoclinic cycle, and three limit cycles, are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the bifurcations of local and global isolated periodic traveling waves in a single species population model described by a reaction-diffusion equation. Based on the singular point quantity algorithm of conjugate symmetric complex systems, we investigate Hopf bifurcation from all equilibrium points for the corresponding planar traveling wave system. We obtain all center conditions and construct one perturbed Hamiltonian system to study Poincaré bifurcation. Further, using the Chebyshev criterion, we develop a utilized approach to prove the existence of at most two limit cycles in a piecewise continuous parameter interval. Finally, the existence of double isolated periodic traveling waves for the model is established, and the results are illustrated by numerical simulation. It is shown that in a population model with density-dependent migrations and Allee effect, two large amplitude oscillations (isolated periodic traveling waves) can exist simultaneously. [ABSTRACT FROM AUTHOR]

In this paper, the existence, stability, and multiplicity of steady-state solutions and periodic solutions for a reaction-diffusion model with nonlocal delay effect and nonlinear boundary condition are investigated by using Lyapunov-Schmidt reduction. When the interior reaction term is weaker than the boundary reaction term, it is found that there is no Hopf bifurcation no matter how either of the interior reaction delay and the boundary reaction delay changes. When the interior reaction term is stronger than the boundary reaction term, it is the interior reaction delay instead of the boundary reaction delay that determines the existence of Hopf bifurcation. Moreover, the general results are illustrated by applications to models with either a single delay or bistable boundary condition. [ABSTRACT FROM AUTHOR]

This paper provides the first Liapunov coefficient formula of 3D Lotka-Volterra equations. This formula gives applications to stability of positive equilibrium and to detecting sub/super criticality of Hopf bifurcation. For 3D competitive Lotka-Volterra equations, combining this formula with the Poincaré-Bendixson theorem, we obtain criteria on multiplicity of limit cycles among Zeeman's classes 27-31, and present a series of examples to admit at least two limit cycles, which are rigorously proved by the first Liapunov coefficient formula, rather than by symbolic computation using Maple. A new Hopf bifurcation that all 2 × 2 principal minors of the community matrix are positive is found, and numerical simulation reveals its global limit cycle bifurcations are plenty. [ABSTRACT FROM AUTHOR]

In this paper, we study a three-stage tick population model with three development delays. Using the total delay τ as the bifurcation parameter, we conduct local and global Hopf bifurcation analysis. Especially, we examine the onset and termination of Hopf bifurcations of periodic solutions from the unique positive equilibrium. We locate all of the stability switches for the equilibrium and demonstrate that the global Hopf bifurcation branches are bounded. This result implies the system undergoes oscillatory behavior only with bounded delays. The key step in the proof is the exclusion of 2 τ -periodic solutions due to the negative Lozinskiĭ measure of the corresponding compound matrix. Numerical simulations are provided to show the complicated patterns of the tick population when the development delays are altered by the temperature and other environmental factors. [ABSTRACT FROM AUTHOR]

Spruce budworm is a major defoliator of forests in North America and its periodic outbreak can cause severe economic growth loss. In 2008, Vaidya and Wu proposed a delayed spruce budworm population model and studied the impact of physiological structure on outbreak control (Vaidya and Wu, 2008 [27]). In this paper, we revisit this model and study its global dynamics that has not been analyzed previously. By carefully dealing with the major difficulty caused jointly by the time delay and the nonlinearity, we give a rather complete stability analysis for the model, including global asymptotic stability, bistability, semi-stability and Hopf bifurcation. Moreover, we show that the time delay can induce interesting dynamics including multitype bistability and long transients. [ABSTRACT FROM AUTHOR]

In this paper, we study the well-known Bazykin's model with Holling II functional response and predator competition. A detailed bifurcation analysis, depending on all four parameters, reveals a rich bifurcation structure, including supercritical and subcritical Bogdanov-Takens bifurcation, degenerate Hopf bifurcation of codimension at most 2, and a focus type degenerate Bogdanov-Takens bifurcation of codimension 3, originating from a nilpotent focus of codimension 3 which acts as the organizing center for the bifurcation set. Moreover, some sufficient conditions to guarantee the global asymptotical stability of the semi-trivial equilibrium or the unique positive equilibrium are also given. Our analysis indicates that we can classify the long-time dynamics of the model with a threshold value c 0 for the natural mortality rate c of predators, in detail, the following are true. (i) When c ≥ c 0 , the prey will persist and predators will eventually go extinct for all positive initial populations. (ii) When c < c 0 , the prey and predators will coexist, for all positive initial populations, in the form of multiple positive equilibria or multiple periodic orbits. Our results can be seen as a complement to the work by Bazykin et al. [2–5] , Hainzl [22,23] , Kuznetsov [30]. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamics of a general reaction-diffusion-advection equation with nonlocal delay effect and Dirichlet boundary condition. The existence and stability of positive spatially nonhomogeneous steady state solution are shown. By analyzing the distribution of eigenvalues of the infinitesimal generator associated with the linearized equation, the existence of Hopf bifurcation is proved. We introduce the weighted space to overcome the hurdle from advection term. We also show that the effect of adding a term advection along environmental gradients to Hopf bifurcation values for a Logistic equation with nonlocal delay. [ABSTRACT FROM AUTHOR]

In this paper we study a host-generalist parasitoid model with Holling II functional response where the generalist parasitoids are introduced to control the invasion of the hosts. It is shown that the model can undergo a sequence of bifurcations including cusp, focus and elliptic types degenerate Bogdanov-Takens bifurcations of codimension three, and a degenerate Hopf bifurcation of codimension at most two as the parameters vary, and the model exhibits rich dynamics such as the existence of multiple coexistent steady states, multiple coexistent periodic orbits, homoclinic orbits, etc. Moreover, there exists a critical value for the carrying capacity of generalist parasitoids such that: (i) when the carrying capacity of the generalist parasitoids is smaller than the critical value, the invading hosts can always persist despite of the predation by the generalist parasitoids, i.e., the generalist parasitoids cannot control the invasion of hosts; (ii) when the carrying capacity of the generalist parasitoids is larger than the critical value, the invading hosts either tend to extinction or persist in the form of multiple coexistent steady states or multiple coexistent periodic orbits depending on the initial populations, i.e., whether the invasion can be stopped and reversed by the generalist parasitoids depends on the initial populations; (iii) in both cases, the generalist parasitoids always persist. Numerical simulations are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we take advantage of the averaging theory to investigate a torus bifurcation in two-parameter families of 2 D nonautonomous differential equations. Our strategy consists in looking for generic conditions on the averaged functions that ensure the existence of a curve in the parameter space characterized by a Neimark-Sacker bifurcation in the corresponding Poincaré map. A Neimark-Sacker bifurcation for planar maps consists in the birth of an invariant closed curve from a fixed point, as the fixed point changes stability. In addition, we apply our results to study a torus bifurcation in a family of 3 D vector fields. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the number of small amplitude limit cycles in arbitrary polynomial systems. It is found that almost all the results for the number of small amplitude limit cycles are obtained by calculating Lyapunov constants and determining the order of the corresponding Hopf bifurcation. It is well known that the difficulty in calculating the Lyapunov constants increases with the increasing of the degree of polynomial systems. So, it is necessary and valuable for us to achieve some general results about the number of small amplitude limit cycles in arbitrary polynomial systems with degree , which is denoted by . In this paper, by applying the method developed by C. Christopher and N. Lloyd in 1995, and M. Han and J. Li in 2012, we first obtain the lower bounds for , and then prove that if . Finally, we obtain that grows as least as rapidly as for all large (it is proved by M. Han, J. Li, Lower bounds for the Hilbert number of polynomial systems, J. Differential Equations 252 (2012) 3278–3304 that the number of all limit cycles in arbitrary polynomial systems with degree grows as least as rapidly as ). [Copyright &y& Elsevier]

Abstract: In this paper, we investigate a model for gene expression, unlike the models mathematically analyzed previously we have both transcriptional and translational time delays. The stability and Hopf bifurcation of the equilibrium point are investigated. Different to previous papers, a multiple time scale (MTS) technique is employed to calculate the normal form on the center manifold of system of delay differential equations, which is much easier to implement in practice than the conventional method, center manifold reduction. Our results show that when time delay is small the equilibrium is stable, when it is at its critical value Hopf bifurcation happens and while for very large value of time delay the oscillation sustains, which has been confirmed by the published data and proved mathematically by using the global continuity of the Hopf bifurcation in this paper. [Copyright &y& Elsevier]

In this paper, we study a susceptible-infectious-recovered (SIRS) epidemic model with a generalized nonmonotone and saturated incidence rate k I 2 S 1 + β I + α I 2 , in which the infection function first increases to a maximum when a new infectious disease emerges, then decreases due to psychological effect, and eventually tends to a saturation level due to crowding effect. It is shown that there are a weak focus of multiplicity at most two and a cusp of codimension at most two for various parameter values, and the model undergoes saddle-node bifurcation, Bogdanov-Takens bifurcation of codimension two, Hopf bifurcation, and degenerate Hopf bifurcation of codimension two as the parameters vary. It is shown that there exists a critical value α = α 0 for the psychological effect, and two critical values k = k 0 , k 1 (k 0 < k 1) for the infection rate such that: (i) when α > α 0 , or α ≤ α 0 and k ≤ k 0 , the disease will die out for all positive initial populations; (ii) when α = α 0 and k 0 < k ≤ k 1 , the disease will die out for almost all positive initial populations; (iii) when α = α 0 and k > k 1 , the disease will persist in the form of a positive coexistent steady state for some positive initial populations; and (iv) when α < α 0 and k > k 0 , the disease will persist in the form of multiple positive periodic coexistent oscillations and coexistent steady states for some positive initial populations. Numerical simulations, including the existence of one or two limit cycles and data-fitting of the influenza data in Mainland China, are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study a family of models with delays describing the process of angiogenesis, that is a physiological process involving the growth of new blood vessels from pre-existing ones. This family includes the well-known models of tumour angiogenesis proposed by Hahnfeldt et al. and dʼOnofrio–Gandolfi and is based on the Gompertz type of the tumour growth. As a consequence we start our analysis from the influence of delay onto the Gompertz model dynamics. The family of models considered in this paper depends on two time delays and a parameter which reflects how strongly the vessels dynamics depends on the ratio between tumour and vessels volume. We focus on the analysis of the model in three cases: one of the delays is equal to 0 or both delays are equal, depending on the parameter α. We study the stability switches, the Hopf bifurcation and the stability of arising periodic orbits for different , especially for and which reflects the Hahnfeldt et al. and the dʼOnofrio–Gandolfi models. For comparison we use also the value . [Copyright &y& Elsevier]

Abstract: The aim of this paper is to outline a formal framework for the analytical analysis of the Hopf bifurcations in the delay differential equations with two independent time delays. Some results for the differential–difference equations with two delays, when the both of the coefficients of linearized equation are negative were obtained in [X. Li, S. Ruan, J. Wei, Stability and bifurcation in delay-differential equations with two delays, J. Math. Anal. Appl. 236 (1999) 254–280]. In the paper we present some remarks on the case studied in [X. Li, S. Ruan, J. Wei, Stability and bifurcation in delay-differential equations with two delays, J. Math. Anal. Appl. 236 (1999) 254–280] and also two other cases, namely when the coefficients of linearized equation have different signs and when coefficients are both positive. [Copyright &y& Elsevier]

This paper is aimed at the nonlinear flutter of a cantilevered plate with Hertzian contact in axial flow. The contact effect is modeled as a nonlinear spring force with both square and cubic nonlinearity. The fluid force is considered as the sum of two parts, one is the reactive fluid force due to plate motion and the other is the resistive fluid force independent on plate motion. The reactive fluid force is derived by solving the bound and wake vorticity with the help of Glauert’s expansions, and the resistive force is evaluated in terms of drag coefficient. The governing nonlinear partial differential equation of the system is discretized in space and time domains by using the Galerkin method. Results show that the plate loses its stability by flutter and then undergoes limit cycle motions due to the contact nonlinearity after instability. The present fluid model is reliable and shows good agreement with other theories archived. A heuristic analysis scheme based on the equivalent linearization method is developed for the analysis of bifurcations and limit cycles. The Hopf bifurcation is either supercritical or subcritical, which is closely dependent on the contact location. For some special cases the bifurcations are, interestingly, both supercritical and subcritical. When the plate experiences limit cycles, with the increasing dynamic pressure there firstly appear the lock-in motions; and then the quasi-periodic motions show up as a breaking of limit cycle by inclusion of a secondary significant frequency with an irrational value of 1 4 π of the dominant limit cycle frequency. Finally the plate undergoes dynamic buckling characterized by quasi-periodic divergence when the dynamic pressure is relatively large. [ABSTRACT FROM AUTHOR]

In this paper, we study dynamics in delayed micro-electromechanical nonlinear coupling system, with particular attention focused on Hopf and Hopf-pitchfork bifurcations. Based on the distribution of eigenvalues, we prove that a sequence of Hopf and Hopf-pitchfork bifurcations occur at the trivial equilibrium as the delay increases and obtain the critical values of two types of bifurcations. Next, by applying the multiple time scales method, the normal forms near the Hopf and Hopf-pitchfork bifurcations critical points are derived. Finally, bifurcation analysis and numerical simulations are presented to demonstrate the application of the theoretical results. We show the regions near above bifurcation critical points in which the micro-electromechanical nonlinear coupling system exists stable fixed point or stable periodic solution. Detailed numerical analysis using MATLAB extends the local bifurcation analysis to a global picture, and stable windows are observed as we change control parameters. Namely, the stable fixed point and stable periodic solution can exist in large regions of unfolding parameters as the unfolding parameters increase away from the critical value. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a reaction–diffusion–advection model with time delay effect. The stability/instability of the spatially nonhomogeneous positive steady state and the associated Hopf bifurcation are investigated when the given parameter of the model is near the principle eigenvalue of an elliptic operator. Our results imply that time delay can make the spatially nonhomogeneous positive steady state unstable for a reaction–diffusion–advection model, and the model can exhibit oscillatory pattern through Hopf bifurcation. The effect of advection on Hopf bifurcation values is also considered, and our results suggest that Hopf bifurcation is more likely to occur when the advection rate increases. [ABSTRACT FROM AUTHOR]

In this paper we establish bifurcation theory of limit cycles for planar C k smooth autonomous differential systems, with k ∈ N . The key point is to study the smoothness of bifurcation functions which are basic and important tool on the study of Hopf bifurcation at a fine focus or a center, and of Poincaré bifurcation in a period annulus. We especially study the smoothness of the first order Melnikov function in degenerate Hopf bifurcation at an elementary center. As we know, the smoothness problem was solved for analytic and C ∞ differential systems, but it was not tackled for finitely smooth differential systems. Here, we present their optimal regularity of these bifurcation functions and their asymptotic expressions in the finite smooth case. [ABSTRACT FROM AUTHOR]

The stability and Hopf bifurcation of the positive steady state to a general scalar reaction–diffusion equation with distributed delay and Dirichlet boundary condition are investigated in this paper. The time delay follows a Gamma distribution function. Through analyzing the corresponding eigenvalue problems, we rigorously show that Hopf bifurcations will occur when the shape parameter n ≥ 1 , and the steady state is always stable when n = 0 . By computing normal form on the center manifold, the direction of Hopf bifurcation and the stability of the periodic orbits can also be determined under a general setting. Our results show that the number of critical values of delay for Hopf bifurcation is finite and increasing in n , which is significantly different from the discrete delay case, and the first Hopf bifurcation value is decreasing in n . Examples from population biology and numerical simulations are used to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a predator-prey model with predator-stage structured and diffusion is concerned. We deal with the system by endowing it with the homogeneous Neumann boundary conditions. We first give the a priori estimates of positive solutions for the reduced reaction diffusion system. Secondly, we discuss the effects of predator cannibalism and prey growth on the stability of nonnegative constant steady states of the model in detail. Thirdly, we investigate the nonexistence and existence of nonconstant positive solutions. Finally, we discuss the Hopf bifurcation created by diffusion. [ABSTRACT FROM AUTHOR]

• The infected of COVID-19 showed multiple distinct peaks in 2020 and 2021 in Japan. • People switch between active and restrained states differing in the infection rate. • Transition rate to active state increases with the number currently active people. • Backward transition rate increases with the abundance of infected people. • The model showed a transient or sustained oscillation and various bifurcations. The coronavirus (SARS-CoV-2) exhibited waves of infection in 2020 and 2021 in Japan. The number of infected had multiple distinct peaks at intervals of several months. One possible process causing these waves of infection is people switching their activities in response to the prevalence of infection. In this paper, we present a simple model for the coupling of social and epidemiological dynamics. The assumptions are as follows. Each person switches between active and restrained states. Active people move more often to crowded areas, interact with each other, and suffer a higher rate of infection than people in the restrained state. The rate of transition from restrained to active states is enhanced by the fraction of currently active people (conformity), whereas the rate of backward transition is enhanced by the abundance of infected people (risk avoidance). The model may show transient or sustained oscillations, initial-condition dependence, and various bifurcations. The infection is maintained at a low level if the recovery rate is between the maximum and minimum levels of the force of infection. In addition, waves of infection may emerge instead of converging to the stationary abundance of infected people if both conformity and risk avoidance of people are strong. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear analysis is performed on a closed-loop system of articulated heavy vehicles with driver steering control. The nonlinearity arises from the nonlinear cubic tire force model. An integration method is employed to derive an analytical periodic solution of the system in the neighbourhood of the critical speed. The results show that excellent accuracy can be achieved for the calculation of periodic solutions arising from Hopf bifurcation of the vehicle motion. A criterion is obtained for detecting the Bautin bifurcation which separates branches of supercritical and subcritical Hopf bifurcations. The integration method is compared to the incremental harmonic balance method in both supercritical and subcritical scenarios. [ABSTRACT FROM AUTHOR]

This paper investigates the problem for limit cycle bifurcations of system x ˙ = y F ( x , y ) + ε p ( x , y ) , y ˙ = − x F ( x , y ) + ε q ( x , y ) , where F ( x , y ) consists of multiple circles and p ( x , y ) , q ( x , y ) are polynomials of degree n . The upper bound for the maximal number of limit cycles emerging from the period annulus surrounding the origin is provided in terms of n and the involved multiplicities of circles by using the first order Melnikov function. Furthermore, Hopf bifurcation for a cubic system of this type is discussed. [ABSTRACT FROM AUTHOR]

The 3-dimensional competitive Lotka–Volterra (LV) systems have been studied for more than two decades, and particular attention has been focused on bifurcation of limit cycles. For such a system, Zeeman (1993) identified 33 stable equivalence classes on a carrying simplex, among which only classes 26–31 may have limit cycles. It has been shown that all these 6 classes may possess two limit cycles, and the existence of three limit cycles was claimed in some of these classes. Recently, Gyllenberg and Yan (2009) studied the existence of four limit cycles, three of them are small-amplitude limit cycles due to Hopf bifurcation and one additional limit cycle, enclosing all the three small-amplitude limit cycles, is due to the existence of a heteroclinic cycle, and proposed a new conjecture including: (i) There exists a 3-d competitive LV system with at least 5 limit cycles. (ii) In the case of a heteroclinic cycle on the boundary of the carrying simplex of a 3-d competitive LV system, the vanishing of the first four focus values (the vanishing of the zero-order focus value means that there is a pair of purely imaginary eigenvalues at the positive equilibrium) does not imply that the heteroclinic cycle is neutrally stable, and hence it does not imply that the positive equilibrium is a center. (iii) In the case of a heteroclinic cycle on the boundary of the carrying simplex of a 3-d competitive LV system, the vanishing of the first three focus values and that the heteroclinic cycle is neutrally stable do not imply the vanishing of the third-order focus value, and hence they do not imply that the positive equilibrium is a center. In this paper, we will present two examples belonging to class 27 and another two examples belonging to class 26, which exhibit at least four small-amplitude limit cycles in the vicinity of the positive equilibrium due to Hopf bifurcations, and prove that the items (ii) and (iii) in the conjecture are true. Moreover, showing the existence of four small-amplitude limit cycles is a necessary step towards proving item (i) of the conjecture. [ABSTRACT FROM AUTHOR]

The dynamics of a diffusive Lotka–Volterra type model for two species with nonlocal delay effect and Dirichlet boundary conditions is investigated in this paper. The existence and multiplicity of spatially nonhomogeneous steady-state solutions are obtained by means of Lyapunov–Schmidt reduction. The stability of spatially nonhomogeneous steady-state solutions and the existence of Hopf bifurcation with the changes of the time delay are obtained by analyzing the distribution of eigenvalues of the infinitesimal generator associated with the linearized system. By the normal form theory and the center manifold reduction, the stability and bifurcation direction of Hopf bifurcating periodic orbits are derived. Finally, our theoretical results are illustrated by a model with homogeneous kernels and one-dimensional spatial domain. [ABSTRACT FROM AUTHOR]

A nonlocal delayed reaction–diffusion equation with Dirichlet boundary condition is considered in this paper. It is shown that a positive spatially nonhomogeneous equilibrium bifurcates from the trivial equilibrium. The stability/instability of the bifurcated positive equilibrium and associated Hopf bifurcation are investigated, providing us with a complete picture of the dynamics. [ABSTRACT FROM AUTHOR]

In this paper, the existence, stability, and multiplicity of spatially nonhomogeneous steady-state solution and periodic solutions for a reaction–diffusion model with nonlocal delay effect and Dirichlet boundary condition are investigated by using Lyapunov–Schmidt reduction. Moreover, we illustrate our general results by applications to models with a single delay and one-dimensional spatial domain. [ABSTRACT FROM AUTHOR]

In this paper, we consider a model containing two modes for HIV-1 infection and spread, one is the diffusion-limited cell-free virus transmission and the other is the direct cell-to-cell transfer of viral particles. We show that the basic reproduction number is underestimated in the existing models that consider only the cell-free virus transmission, or the cell-to-cell infection, ignoring the other. Assuming logistic growth for target cells, we find that if the basic reproduction number is greater than one, the infection can persist and the Hopf bifurcation can occur from the positive equilibrium within certain parameter ranges. [ABSTRACT FROM AUTHOR]

In this paper, we discussed all the possible codimension-one and partial codimension-two bifurcations existing in a symmetrical ring network, using the symmetric bifurcation theory of delay differential equations coupled with the representation theory of Lie groups, and dynamical analysis methods. We have not only figured out the pattern of each bifurcation, but also predicted the directions and stability analysis of the bifurcated solutions according to structure of the system. Numerical simulations were then given to verify our theoretical analysis and investigate the complexity of codimension-two mode intersections. Comments were also presented to highlight some future work. [ABSTRACT FROM AUTHOR]

This paper deals with a delayed reaction–diffusion three-species Lotka–Volterra model with interval biological parameters and harvesting. Sufficient conditions for the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analyzing the associated characteristic equation. Furthermore, formulas for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by applying the normal form method and center manifold theorem. Then an optimal control problem has been considered. Finally, numerical simulation results are presented to validate the theoretical analysis. Numerical evidence shows that the presence of harvesting can impact the existence of species and over harvesting can result in the extinction of the prey or the predator which is in line with reality. [ABSTRACT FROM AUTHOR]

Abstract: Normal form theory is very important and useful in simplifying the forms of equations restricted on the center manifolds in studying nonlinear dynamical problems. In this paper, using the center manifold theorem associated with the integrated semigroup theory, we develop a normal form theory for semilinear Cauchy problems in which the linear operator is not densely defined and is not a Hille–Yosida operator and present procedures to compute the Taylor expansion and normal form of the reduced system restricted on the center manifold. We then apply the main results and computation procedures to determine the direction of the Hopf bifurcation and stability of the bifurcating periodic solutions in a structured evolutionary epidemiological model of influenza A drift and an age structured population model. [Copyright &y& Elsevier]

Abstract: It is acknowledged that coral reefs are globally threatened. P.J. Mumby et al. [10] constructed a mathematical model with ordinary differential equations to investigate the dynamics of coral reefs. In this paper, we first provide a detailed global analysis of the coral reef ODE model in [10]. Next we incorporate the inherent time delay to obtain a mathematical model with delay differential equations. We consider the grazing intensity and the time delay as focused parameters and perform local stability analysis for the coral reef DDE model. If the time delay is sufficiently small, the stability results remain the same. However, if the time delay is large enough, macroalgae only state and coral only state are both unstable, while they are both stable in the ODE model. Meanwhile, if the grazing intensity and the time delay are endowed some suitable values, the DDE model possesses a nontrivial periodic solution, whereas the ODE model has no nontrivial periodic solutions for any grazing rate. We study the existence and property of the Hopf bifurcation points and the corresponding stability switching directions. [Copyright &y& Elsevier]

Abstract: In this paper we consider a class of higher dimensional differential systems in which have a two dimensional center manifold at the origin with a pair of pure imaginary eigenvalues. First we characterize the existence of either analytic or inverse Jacobian multipliers of the systems around the origin, which is either a center or a focus on the center manifold. Later we study the cyclicity of the system at the origin through Hopf bifurcation by using the vanishing multiplicity of the inverse Jacobian multiplier. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the dynamical behavior of a virus infection model with delayed humoral immunity. By using suitable Lyapunov functional and the LaSalleʼs invariance principle, we establish the global stabilities of the two boundary equilibria. If , the uninfected equilibrium is globally asymptotically stable; if , the infected equilibrium without immunity is globally asymptotically stable. When , we obtain the sufficient conditions to the local stability of the infected equilibrium with immunity . The time delay can change the stability of and lead to the existence of Hopf bifurcations. The stabilities of bifurcating periodic solutions is also studied. We check our theorems with numerical simulations in the end. [Copyright &y& Elsevier]

Abstract: This paper investigates both homoclinic bifurcation and Hopf bifurcation which occur concurrently in a class of planar perturbed discontinuous systems of Filippov type. Firstly, based on a geometrical interpretation and a new analysis of the so-called successive function, sufficient conditions are proposed for the existence and stability of homoclinic orbit of unperturbed systems. Then, with the discussion about Poincaré map, bifurcation analyses of homoclinic orbit and parabolic–parabolic (PP) type pseudo-focus are presented. It is shown that two limit cycles can appear from the two different kinds of bifurcation in planar Filippov systems. [Copyright &y& Elsevier]

Abstract: We consider a network of three identical neurons incorporating distributed and discrete signal transmission delays. The model for such a network is a system of coupled nonlinear delay differential equations. It is established that two cases of a single Hopf bifurcation may occur at the trivial equilibrium of the system, as a consequence of the symmetry of the network. These single Hopf bifurcations are the simple and the double root. The present paper looks at the simple root case, and addresses the issue of absolute stability of the trivial equilibrium and stability switching, leading up to calculation of the critical delay and formulation of a Hopf bifurcation theorem. [Copyright &y& Elsevier]

In this paper, we study the parkinson oscillation mechanism in a computational model by bifurcation analysis and numerical simulation. Oscillatory activities can be induced by abnormal coupling weights and delays. The bidirectional Hopf bifurcation phenomena are found in simulations, which can uniformly explain the oscillation mechanism in this model. The Hopf1 represents the transition between the low firing rate stable state (SS) and oscillation state (OS), the Hopf2 represents the transition between the high firing rate stable state (HSS) and the OS, the mechanisms of them are different. The Hopf1 and Hopf2 bifurcations both show that when the state transfers from the stable region to the oscillation region, oscillatory activities originate from the beta frequency band or the gamma frequency band. We find that the changing trends of the frequency (DF) and oscillation amplitude (OSAM) are contrary in many cases. The effect of the delay in inhibitory pathways is greater than that of in excitatory pathways, and appropriate delays improve the discharge activation level (DAL) of the system. In all, we infer that oscillations can be induced by the follow factors: 1. Improvement of the DAL of the globus pallidus externa (GPe); 2. Reduce the DAL of the GPe from the HSS or the discharge saturation state; 3. The GPe can also resonate with the subthalamic nucleus (STN). [ABSTRACT FROM AUTHOR]

Abstract: This paper proposes a nonstandard finite difference (NSFD) method for a class of delay differential equations. We prove that, for any step size (), this numerical method could intrinsically preserve the qualitative behavior of the dynamical system, including the local stability of equilibrium, the existence and the direction of Hopf bifurcation and the stability of bifurcating periodic solution. Finally, to illustrate the analytic results, the NSFD method is used for a model of the survival of red blood cells in animals. [Copyright &y& Elsevier]

Abstract: A medical experiment published in Nature has shown that humanized mice receiving the vectored immunoprophylaxis can be fully protected from HIV infection. In this paper, a mathematical model is proposed to investigate the viral dynamics under the effect of antibodies in the experiment. It is shown that the introduction of vectored immunoprophylaxis can induce the backward bifurcation and the ignorance of antibodies'' loss due to their involvement with virus may result in the loss of backward bifurcation. By numerical simulations, it is found that the model also exhibits some other complicated dynamical behaviors. A subcritical Hopf bifurcation, a fold bifurcation of equilibria and a limit point bifurcation of limit cycles are detected, which induce five typical patterns of dynamical behaviors including the bistable phenomenon. [Copyright &y& Elsevier]

Abstract: This paper is concerned with an autocatalysis model subject to no-flux boundary conditions. The existence of Hopf bifurcation are firstly obtained. Then by the center manifold theory and the normal form method, the direction of Hopf bifurcation and the stability of spatially homogeneous and inhomogeneous periodic solutions are established. On the other hand, the steady-state bifurcations from simple and double eigenvalues are intensively studied. The techniques of space decomposition and implicit function theorem are adopted to deal with the case of double eigenvalues. Finally, some numerical simulations are shown to verify the analytical results. [Copyright &y& Elsevier]