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In this paper, we provide a rigorous description of the birth of canard limit cycles in slow-fast systems in R 3 through the folded saddle-node of type II and the singular Hopf bifurcation. In particular, we prove – in the analytic case only – that for all 0 < ϵ ≪ 1 there is a family of periodic orbits, born in the (singular) Hopf bifurcation and extending to O (1) cycles that follow the strong canard of the folded saddle-node. Our results can be seen as an extension of the canard explosion in R 2 , but in contrast to the planar case, the family of periodic orbits in R 3 is not explosive. For this reason, we have chosen to call the phenomena in R 3 , the "dud canard". The main difficulty of the proof lies in connecting the Hopf cycles with the canard cycles, since these are described in different scalings. As in R 2 , we use blowup to overcome this, but we also have to compensate for the lack of uniformity near the Hopf bifurcation, due to its singular nature; it is a zero-Hopf bifurcation in the limit ϵ = 0. In the present paper, we do so by imposing analyticity of the vector-field. This allows us to prove existence of an invariant slow manifold, that is not normally hyperbolic. [ABSTRACT FROM AUTHOR]

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 present a memory-based diffusion predator-prey model that incorporates spatial heterogeneity and is subject to homogeneous Dirichlet boundary conditions. Prey species lack memory or cognitive abilities, exhibiting only random diffusion. In contrast, predators utilize memory-based self-diffusion. For the proposed model, we establish the existence and explicit expression of a spatially non-constant positive steady-state. Furthermore, we demonstrate that memory-based diffusion and the averaged memory period can lead to richer dynamics. Specifically, when the memory-based diffusion coefficient is not dominant, the averaged memory period has no impact on the non-constant steady-state. However, when the memory-based diffusion coefficient takes precedence, the averaged memory period can destabilize the non-constant steady-state, resulting in Hopf bifurcation and the emergence of spatially non-homogeneous periodic solutions. [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]

In this paper, we consider a diffusive predator-prey system with fear response delay, where a benefit from the ani-predation response in addition to the cost is taken into account. As a step toward understanding the underlying mechanism of the spatiotemporal pattern formations by rigorous mathematical analysis, we are interested in how the joint effects of time delay, the fear effect strength, the carrying capacity, as well as the diffusion rates could affect the complex spatiotemporal pattern formations of the system. In particular, Turing instability of the Hopf bifurcating periodic solutions is investigated in details. To that end, a general formula is derived to determine Turing instability of the Hopf bifurcating periodic solutions for general 2-component delayed-diffusive system. This extends our earlier general results for non-delayed diffusive system to its delayed-diffusive counterparts. Our general formula tends to be new and can be applied to a variety of 2-component delayed-diffusive systems. [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]

In this paper, we study a diffusion-advection Lotka-Volterra competition model with stage structure in a spatially heterogeneous environment. The existence and local asymptotic stability of spatially non-homogeneous semi-trivial steady-state solutions and spatially non-homogeneous positive steady-state solutions are obtained by the implicit function theorem and spectral analysis. We show that three scenarios can occur: if random diffusion rates of two species are sufficiently large, both species go extinct; if random diffusion rates of two species are relatively small, two competing species coexist; if one species has a large random diffusion rate and the other has a small random diffusion rate, the species with a large random diffusion rate are driven to extinction. An interesting finding is that a large delay does not lead to Hopf bifurcation at the spatially non-homogeneous steady-state solution, but makes this steady-state solution approach zero. We numerically demonstrate the effects of spatial heterogeneity on spatiotemporal dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we are devoted to studying a Holling type-II predator-prey system with cross diffusion. First of all, by a series of estimates, we establish the global existence of generalized solutions under some proper assumptions. By analyzing the distribution of eigenvalues, we investigate the local and global stability of the constant solution and also consider the steady state bifurcation and Hopf bifurcation near the constant steady state in details. In addition, the nonexistence of non-constant steady states is also investigated when diffusion rate d is large enough. Finally, sufficient conditions ensuring the existence of non-constant steady states are obtained by using Leray-Schauder degree theory. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a spatial model with memory delay of the prey, Allee effect and maturation delay with delay-dependent coefficients of predators. We first explore the model without delays and diffusions, and find that it can undergo a saddle-node bifurcation when the intensity of Allee effect is at the tipping point. Then for the scenario of stability of the coexistence steady state without delays, we obtain the crossing curves on the delays plane. The model can undergo Hopf bifurcation when delays pass through these crossing curves from a stable region to an unstable one. We further calculate the normal form of Hopf bifurcation and hence obtain the direction of Hopf bifurcation and the stability of the bifurcation periodic solutions. It is shown that the model can possess multiple stability switches and a stable spatially heterogeneous periodic solution with mode-4 as delays vary. [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]

To incorporate spatial memory and nonlocal effect of animal movements, we propose and investigate the spatiotemporal dynamics of the single population model with memory-based diffusion and nonlocal reaction. We first study the stability of a positive equilibrium and the steady state bifurcation induced by diffusion and nonlocality. We then investigate the impact of the averaged memory period on stability and bifurcation, and show that the combination of the averaged memory period and the diffusion can lead to the occurrence of Turing-Hopf and double Hopf bifurcations. The paper originally derives the normal form theory for Turing-Hopf bifurcation in the general reaction-diffusion equation with memory-based diffusion and nonlocal reaction. This novel algorithm can be widely used to classify the spatiotemporal dynamics near the Turing-Hopf bifurcation point. Finally, we apply the obtained results to a model proposed by Britton and numerically illustrate the spatiotemporal patterns induced by Hopf, Turing-Hopf and double Hopf bifurcations. Stable spatially homogeneous/nonhomogeneous periodic solutions, homogeneous/nonhomogeneous steady states and the transition from one of these solutions to another are provided in this paper. We additionally acquire the coexistence of two stable spatially nonhomogeneous steady states or two spatially nonhomogeneous periodic solutions near the Turing-Hopf bifurcation point. • Propose the single population model with memory-based diffusion and nonlocal reaction. • Find the codimension-two Turing-Hopf and double Hopf bifurcation in the scalar reaction-diffusion equation. • Develop the normal form of Turing-Hopf bifurcation in the equation with memory-based diffusion and nonlocal term. • Find the coexistence of two stable spatially inhomogeneous steady states or two spatially inhomogeneous periodic solutions. • Find the transition from one of unstable solutions to another stable one. [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]

Highlights • The constant excitation induced by flight maneuver is included in the rotor model. • The HB-AFT method is formulated to solve sub-harmonic resonance problems. • The bifurcation of the system affected by the constant excitation is investigated. • Different bifurcation types are identified. Abstract This paper focuses on the mechanism of a complex bifurcation behavior caused by flight maneuvers in an aircraft rub-impact rotor system with Duffing type nonlinearity. The maneuver load during flight maneuvers may induce a rub-impact phenomenon, accompanied by complex nonlinear behaviors such as periodic, sub-harmonic and quasi-periodic motions, but the bifurcation mechanism is not so clear. In this study, the harmonic balance method combined with an alternating frequency/time domain procedure (HB-AFT method) is formulated and used to derive the approximate periodic solutions of the system. In conjunction with the arc-length continuation, the solution branches for both periodic 1-T motion (synchronous oscillation) and periodic 2-T motion (sub-harmonic oscillation) are traced. Then with the aid of the Floquet theory, the stabilities of the obtained periodic solutions are examined. The changes in the number or the stability of the solutions lead to the identification of the bifurcation points, which can be classified qualitatively into three types, i.e. Neimark-Sacker bifurcation point (NSBP), quasi-periodic Hopf bifurcation point (QPHBP) and saddle-node bifurcation point (SNBP). In addition, the constant excitation significantly affects the instability as well as the bifurcation of the rotor system. In the case of a smaller constant excitation, the rotation region with respect to the quasi-periodic motions may disappear, accompanied with a PBP (pitchfork bifurcation point) instead of the two NSBPs and one QPHBP. The bifurcation analysis in this paper provides deep insight into the mechanism of the complex nonlinear phenomenon induced by the constant excitation. The results obtained will also contribute to a better understanding of the nonlinear dynamic behaviors of aircraft rotor systems during flight maneuvers. [ABSTRACT FROM AUTHOR]

Abstract There has been a substantial amount of well mixing epidemic models devoted to characterizing the observed complex phenomena (such as bistability, hysteresis, oscillations, etc.) during the transmission of many infectious diseases. A comprehensive explanation of these phenomena by epidemic models on complex networks is still lacking. In this paper we study epidemic dynamics in an adaptive network proposed by Gross et al., where the susceptibles are able to avoid contact with the infectious by rewiring their network connections. Such rewiring of the local connections changes the topology of the network, and inevitably has a profound effect on the transmission of the disease, which in turn influences the rewiring process. We rigorously prove that the adaptive epidemic model investigated in this paper exhibits degenerate Hopf bifurcation, homoclinic bifurcation and Bogdanov–Takens bifurcation. Our study shows that adaptive behaviors during an epidemic may induce complex dynamics of disease transmission, including bistability, transient and sustained oscillations, which contrast sharply to the dynamics of classical network models. Our results yield deeper insights into the interplay between topology of networks and the dynamics of disease transmission on networks. [ABSTRACT FROM AUTHOR]

Abstract The goal of this paper is to develop the Equivariant Degree based method for studying relative periodic solutions in the setiings with lack of smoothness and/or genericity. In this paper, we consider an equivariant Hopf bifurcation of relative periodic solutions from relative equilibria in systems of functional differential equations respecting Γ × S 1 -spatial symmetries. The existence of branches of relative periodic solutions together with their symmetric classification is established using the equivariant twisted Γ × S 1 -degree with one free parameter. As a case study, we consider a delay differential model of coupled identical passively mode-locked semiconductor lasers with the dihedral symmetry group Γ = D 8 ; and, a system of hysterestic electro-mechanical oscillators coupled in the same symmetric fashion. [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]

The transition between strong and weak Allee effects in prey provides a simple regime shift in ecology. In this paper, we study the interplay between the functional response of Holling type IV and both strong and weak Allee effects. The model investigated here presents complex dynamics and high codimension bifurcations. In particular, nilpotent cusp singularity of order 3 and degenerate Hopf bifurcation of codimension 3 are completely analyzed. Remarkably it is the first time that three limit cycles are discovered in predator-prey models with multiplicative Allee effects. Moreover, a new unfolding of nilpotent saddle of codimension 3 with a fixed invariant line is discovered and fully developed, and the existence of codimension 2 heteroclinic bifurcation is proven. Our work extends the existing results of predator-prey systems with Allee effects. The bifurcation analysis and diagram allow us to give biological interpretations of predator-prey interactions. [ABSTRACT FROM AUTHOR]

Spatial memory has been considered significant in animal movement modeling. In this paper, we formulate a two-species interaction model by incorporating both random walk and spatial memory-based walk in their movement. The spatial memory-based walk, described by a chemotactic-like term, is derived by a modified Fick's law involving a directed movement toward the gradient of the density distribution function at a past time. For the proposed model, local stability and bifurcations are studied at constant steady states. Unlike a classical reaction-diffusion equation, we show that the accumulation points of eigenvalues for the model will locate at a vertical line in the complex plane, which will make the model generate spatially inhomogeneous time-periodic patterns through Hopf bifurcation. As illustrations, we apply these results to competition and cooperative models with memory-based diffusion. For the competition model, it turns out that the outcomes are far more complicated than those of classic Lotka-Volterra reaction-diffusion models, due to the consideration of memory-based diffusion. In particular, the existence of periodic oscillations is proved under weak competition. Similar conclusions hold for the cooperative model. [ABSTRACT FROM AUTHOR]

In (Euzébio et al., 2016 [10] ; Chen and Tang, 2020 [8]), the bifurcation diagram and all global phase portraits of a degenerate planar piecewise linear differential system x ˙ = F (x) − y , y ˙ = g (x) − α with three zones were given completely for the non-extreme case. In this paper we deal with the system for the extreme case and find new nonlinear phenomena of bifurcation for this planar piecewise linear system, i.e., a generalized degenerate Hopf bifurcation occurs for points at infinity. Moreover, the bifurcation diagram and all global phase portraits in the Poincaré disc are obtained, presenting scabbard bifurcation curves, grazing bifurcation curves for limit cycles, generalized supercritical (or subcritical) Hopf bifurcation curve for points at infinity, generalized degenerate Hopf bifurcation value for points at infinity and double limit cycle bifurcation curve. [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]

The self-synchronization of rotors in a nonlinear system has both academic and practical importance because of its applicability as a source of excitation for vibratory machines. Although the characteristics of such self-synchronization can be clarified by analyzing the periodic solutions of the equations of motion, only self-synchronization corresponding to the stable periodic solution is realized in actual systems. The stability of the solution is thus one of the most important characteristics of self-synchronization. In this paper, to stabilize periodic solutions suitable for application to vibratory machines, the effects of the damping of the rigid body and the moment of inertia of the rotors on stability are examined using a system that consists of two unbalanced rotors installed in a rigid body. High-precision analysis conducted using the shooting method shows that it is possible to stabilize the instabilities caused by period-doubling and Hopf bifurcations while keeping other characteristics, such as the synchronous frequency and the rigid body amplitude, mostly unchanged. The findings of this study can serve as a guide to extending the range of stable periodic solutions and are thus expected to be useful in the design of systems that utilize the self-synchronization of rotors. [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, the dynamics of a general predator-prey system with diffusion and stage structures are investigated. Under some assumptions, the stability of positive steady state and the existence of local Hopf bifurcation are obtained by analyzing the distribution of eigenvalues. Moreover, the global existence of positive periodic solution is established by using the global Hopf bifurcation result of Wu. Finally, an example and some numerical simulations are carried out for illustrating the theoretical results. [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]

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]

In this paper, we study bifurcations of small-amplitude limit cycles of Liénard systems of the form x ˙ = y − F (x) , y ˙ = − g (x) , where g (x) is a cubic polynomial, and F (x) is a smooth or piecewise smooth polynomial of degree n. By using involutions, we obtain sharp upper bounds of the number of small-amplitude limit cycles produced around a singular point for some systems of this type. [ABSTRACT FROM AUTHOR]

A newly-developed medium–low-speed maglev test vehicle was used to obtain the vertical dynamic response signals of the vehicle-line coupled system when the test vehicle was standing still on four different types of lines through field tests. It was found that abnormal low-frequency vertical vibration occurred on the car body, so that an excellent ride index of the test vehicle cannot be ensured. As the stability of the vehicle at standstill is the prerequisite to the stability of the vehicle running at a certain speed, it is necessary to investigate the causes underlying the abnormal vibration and propose corrective measures through theoretical studies in combination with the measured results. Therefore, this paper established a coupled vehicle-track system dynamics model considering active feedback control, and investigated the stability and bifurcation behavior of the coupled system through nonlinear dynamics theory and numerical simulation. Field test results show that: the vertical ride index of the test vehicle was rated as "Unqualified" when it was standing still; the vertical vibration eigenfrequency of the car body is basically the same (between 2.10 and 2.20 Hz) when the test vehicle was standing still on the four different lines, indicating that the abnormal vibration was basically independent of the line type; the vertical dynamic response of each measurement point on the electromagnet, F-rail and bridge was normal. The theoretical and simulation studies show that: the numerical simulation of the theoretical model can effectively reproduce the abnormal vibration; the malfunction in the nonlinear characteristics of the air spring vertical damping (i.e. the damping malfunction) is the main reason for this vibration, and the abnormal vibration corresponded to the supercritical Hopf bifurcation dynamics behavior in nonlinear dynamics; after the air spring was restored to normal, the equilibrium point of the coupled system was stable under the influence of different initial disturbances. This study may provide an important reference for reducing the abnormal low-frequency vertical vibration of the test vehicle at standstill. [ABSTRACT FROM AUTHOR]

This paper investigates a conceptual pitch–plunge airfoil in hypersonic flows with both structural and aerodynamic nonlinearity, meanwhile the effects of irregular fluctuations in the flow and external random loads modeled as a Gaussian white noise are considered. The dynamic responses of the airfoil model especially the bifurcation behaviors are analyzed via the harmonic balance method for the deterministic case. We found the airfoil system undergo both subcritical and supercritical Hopf bifurcations. Subsequently, the effects of stochasticity on the dynamical behaviors of the hypersonic airfoil system are explored in the regimes of subcritical and supercritical Hopf bifurcations depending on the system parameters. Several interesting phenomena are triggered under random perturbations such as intermittency and stochastic transition between the low-amplitude oscillation state and the undesirable high-amplitude oscillation state. These dynamics are further characterized via the probability density function and time history. This work will provide new insights into the safety and reliability design of hypersonic aircraft. • The airfoil models in hypersonic flow with stochastic perturbations are introduced. • The effects of the structural parameters on Hopf bifurcation are analyzed theoretically. • The influence mechanisms of random fluctuations on both subcritical and supercritical bifurcations are explored. • The stochastic perturbations can induce intermittency and stochastic transition behaviors. [ABSTRACT FROM AUTHOR]

This paper proposes a planar delay differential equation for cancer virotherapy. The model simulates the situation in which an oncolytic virus is injected for the second time, and the immune system suppresses the viral infection with a time delay. Our purpose is to provide theoretical conditions so that the therapy can be continued successfully. With the help of the characteristic equation, we examine the singularities and their local stability. Hopf bifurcation is also investigated around the endemic singularity. It is shown that there is a sequence of Hopf bifurcations, but the Hopf cycles do not persist continuously between the two sequential bifurcations. Finally, we see that virotherapy can be conducted successfully by controlling the delay parameter. [ABSTRACT FROM AUTHOR]

The majority of methods for calculating the dynamic response of nonlinear aeroelastic systems considers only steady-state periodic behaviour. An exception is the Multiple Time Scales method, which can estimate both the transient and steady-state solutions of such systems; nevertheless, this approach is only accurate close to the Hopf bifurcation. This paper proposes a novel combined approach whereby the transient response is obtained from the Multiple Time Scales method and the asymptotic periodic behaviour is corrected using the Harmonic Balance method. This consistent and efficient framework mutually empowers both techniques and accounts for large parameter variations around the critical condition. The effect of cubic aero-structural nonlinearities on the dynamic response of a generic aeroelastic system is then investigated. Both the Multiple Time Scales and Harmonic Balance methods are adopted and perfect agreement of the explicit results is demonstrated, albeit near the system instability. In contrast, the proposed combined solution is valid for a wider range of perturbations, is analytical and has negligible computational cost while retaining accuracy. The role of key parameters and terms on the core mechanism of the dynamic behaviour is rigorously identified and discussed, from both physical and mathematical points of view. Galloping is finally considered as the simplest but complete application to a fundamental yet practical problem, featuring full conceptual complexity while exploiting the solid synthesis capability of the newly proposed analytical approach. Excellent agreement was found in all cases with results from the numerical integration of the nonlinear equations of motion in the time domain. [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]

Predicting critical transitions and post-transition dynamics of complex systems is a unique challenge. In this paper, a novel approach is introduced to forecast Hopf bifurcations and the post-bifurcation dynamics of nonlinear fluid-structural systems. The forecasting method is model-less and uses measurements of the system response collected only in the pre-bifurcation regime. To demonstrate the method, it is applied to a cantilever high aspect ratio wing exposed to gust loads as perturbations. To generate surrogate measurements required for forecasting bifurcations in this large-dimensional complex system, a nonlinear strain-based finite element formulation coupled with unsteady aerodynamics is used to model the fluid–structure interaction. Results show that the method successfully forecasts the linear flutter speed and the amplitude of the limit cycle oscillations that occur in the post-bifurcation regime. The procedure is shown to be time efficient, model-less, and to require only few measurements, which makes the proposed forecasting method a unique tool for nonlinear analysis of complex systems. [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]

This paper revisits the problem of an unbalanced motor attached to a fixed frame by means of a nonlinear spring and a linear damper. The excitation provided by the motor is, in general, nonideal, which means it is affected by the vibratory response. Since the system behaviour is highly dependent on the order of magnitude of the motor characteristic slope, the case of large slope is considered herein. Some Perturbation Methods are applied to the system of equations, which allows transforming the original 4D system into a much simpler 2D system. The fixed points of this reduced system and their stability are carefully studied. We find the existence of a Hopf bifurcation which, to the authors’ knowledge, has not been addressed before in the literature. These analytical results are supported by numerical simulations. We also compare our approach and results with those published by other authors. [ABSTRACT FROM AUTHOR]

This paper focuses on the study of codimension one and two Hopf bifurcations and the pertinent Lyapunov stability coefficients for a general reaction–diffusion system. We display algebraic expressions for the first and second Lyapunov coefficients for the infinite dimensional system subject to Neumann boundary conditions. As an application, a special subset of a three-dimensional Lotka–Volterra dynamical system with diffusions subject to Neumann boundary conditions is analyzed. The main goal is to perform a detailed local stability analysis for the proposed predator–prey model to show the existence of multiple spatially homogeneous and non-homogeneous periodic orbits, due to the occurrence of codimension one Hopf bifurcation. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of a spruce budworm model with diffusion and physiological structures are investigated. The stability of steady state and the existence of Hopf bifurcation near positive steady state are investigated by analyzing the distribution of eigenvalues. The properties of Hopf bifurcation are determined by the normal form theory and center manifold reduction for partial functional differential equations. And global existence of periodic solutions is established by using the global Hopf bifurcation result of Wu. Finally, some numerical simulations are carried out to illustrate the analytical results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to provide an efficient frequency-domain method for bifurcation analysis of nonlinear dynamical systems. The proposed method consists in directly tracking the bifurcation points when a system parameter such as the excitation or nonlinearity level is varied. To this end, a so-called extended system comprising the equation of motion and an additional equation characterizing the bifurcation of interest is solved by means of the Harmonic Balance Method coupled with an arc-length continuation technique. In particular, an original extended system for the detection and tracking of Neimark-Sacker (secondary Hopf) bifurcations is introduced. By applying the methodology to a nonlinear energy sink and to a rotor-stator rubbing system, it is shown that the bifurcation tracking can be used to efficiently compute the boundaries of stability and/or dynamical regimes, i.e., safe operating zones. [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]

Many biophysical models have the property that some variables in the model evolve much faster than others. A common step in the analysis of such systems is to simplify the model by assuming that the fastest variables equilibrate instantaneously, an approach that is known as quasi-steady state reduction (QSSR). QSSR is intuitively satisfying but is not always mathematically justified, with problems known to arise, for instance, in some cases in which the full model has oscillatory solutions; in this case, the simplified version of the model may have significantly different dynamics to the full model. This paper focusses on the effect of QSSR on models in which oscillatory solutions arise via one or more Hopf bifurcations. We first illustrate the problems that can arise by applying QSSR to a selection of well-known models. We then categorize Hopf bifurcations according to whether they involve fast variables, slow variables or a mixture of both, and show that Hopf bifurcations that involve only slow variables are not affected by QSSR, Hopf bifurcations that involve fast and slow variables (i.e., singular Hopf bifurcations) are generically preserved under QSSR so long as a fast variable is kept in the simplified system, and Hopf bifurcations that primarily involve fast variables may be eliminated by QSSR. Finally, we present some guidelines for the application of QSSR if one wishes to use the method while minimising the risk of inadvertently destroying essential features of the original model. [ABSTRACT FROM AUTHOR]