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Purpose: The present paper investigates the bursting dynamics and its mechanism of the general hybrid Rayleigh-van der Pol-Duffing oscillator with two external periodic excitations. The study aims to avoid or reasonably utilize the vibrations induced by self-excited system in engineering. Methods: This paper mainly uses the fast–slow analysis method. First, we analyze the stability and bifurcation structure of the equilibrium point of the system with a single external excitation. Through the bifurcation set, we discover that the system has fold bifurcation and Hopf bifurcation. The direction of the Hopf bifurcation is obtained by calculating the first Lyapunov coefficient. When two external excitations exist, by employing de Moivre's equation, the original system transforms into a new form with a single slow variable. The classical machinery of fast subsystem analysis is used to investigate the bursting oscillations. Besides, the transformed phase diagrams are used to reveal the bifurcation mechanism. Results: Two bursting oscillations, namely symmetric fold/supHopf bursting and symmetric fold/fold bursting, are observed in the system with one periodic excitation. With two external excitations, the "turnover-of-fold-hystersis-induced" bursting, "symmetric periodic 2-fold" bursting, "symmetric periodic 6-fold" bursting, "symmetric periodic 10-fold" bursting and "symmetric periodic 14-fold" bursting are found and analyzed in detail. At the same time, the results show that with the increase of amplitudes of the external excitations, qualitative changes occur. Conclusions: With the addition of the second periodic excitation, the original bursting patterns induced by a single periodic excitation change completely. The phenomena will impact the vibration in the actual engineering. [ABSTRACT FROM AUTHOR]

In this paper, a system containing switching manifolds is constructed based on a non-linear conveyor belt model by introducing a periodic external excitation term. When the frequency of the external excitation is much smaller than the intrinsic frequency of the system, a series of bursting phenomena can be observed. Using the Filippov convex method, bifurcation mechanisms on the non-smooth boundary are discussed. Particularly, the explicit expression of the sliding region is obtained. Based on that, different bursting dynamics, especially the switching patterns dominated by sliding phenomena, are revealed by considering two cases of stiffness and excitation amplitude. It is shown that there exist different types of equilibrium points, leading to multiple modes of bifurcation, as stiffness changes. However, there always exist a pair of subcritical Hopf bifurcation points, which lie on the unstable equilibrium curve. Varying the excitation amplitude, different types of bursting oscillation modes may occur. In addition, both the transition of the type of equilibrium and the intrinsic structure of the system have effects on the local structure of the trajectory. [ABSTRACT FROM AUTHOR]

A general time-delay reaction–diffusion–advection system with the Dirichlet boundary condition and spatial heterogeneity is investigated in this paper. By using the implicit function theorem, we obtain the existence and asymptotic expression of the spatially non-homogeneous positive steady-state solution. This is the steady-state bifurcation from zero equilibrium. Via analyzing the corresponding characteristic equation, the stability of the spatially non-homogeneous positive steady-state solution and the occurrence of Hopf bifurcation at the positive steady-state solution are obtained, and the spatially non-homogeneous periodic solution is derived from Hopf bifurcation, this is the secondary bifurcation behavior of the system. Utilizing the normal form method and center manifold theory, we prove that the direction of Hopf bifurcation is supercritical and the bifurcating spatially non-homogeneous periodic solution is stable. Furthermore, We show that there exist two sequences Hopf bifurcation values and the orders of two sequences Hopf bifurcation values are given. Moreover, theoretical and numerical results are applied to competition and cooperation systems, respectively. Finally, the effect of the advection rate and spatial heterogeneity are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we derive the algorithm for calculating the normal form of the double Hopf bifurcation that appears in a memory-based diffusion system via taking memory-based diffusion coefficient and the memory delay as the perturbation parameters. Using the obtained theoretical results, we study the dynamical classification near the double Hopf bifurcation point in a predator-prey system with Holling type II functional response. We show the existence of different kinds of stable spatially inhomogeneous periodic solutions, the transition from one kind to the other as well as the coexistence of two types of periodic solutions with different spatial profiles by varying the memory-based diffusion coefficient and the memory delay. [ABSTRACT FROM AUTHOR]

The coronavirus disease 2019 (COVID-19) remains serious around the world and causes huge deaths and economic losses. Understanding the transmission dynamics of diseases and providing effective control strategies play important roles in the prevention of epidemic diseases. In this paper, to investigate the effect of delays on the transmission of COVID-19, we propose a delayed SEIR model to describe COVID-19 virus transmission, where two delays indicating the incubation and recovery periods are introduced. For this system, we prove its solutions are nonnegative and ultimately bounded with the nonnegative initial conditions. Furthermore, we calculate the disease-free and endemic equilibrium points and analyze the asymptotical stability and the existence of Hopf bifurcations at these equilibrium points. Then, by taking the weighted sum of the opposite number of recovered individuals at the terminal time, the number of exposed and infected individuals during the time horizon, and the system cost of control measures as the cost function, we present a delay optimal control problem, where two controls represent the social contact and the pharmaceutical intervention. Necessary optimality conditions of this optimal control problem are exploited to characterize the optimal control strategies. Finally, numerical simulations are performed to verify the theoretical analysis of the stability and Hopf bifurcations at the equilibrium points and to illustrate the effectiveness of the obtained optimal strategies in controlling the COVID-19 epidemic. [ABSTRACT FROM AUTHOR]

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

This paper is devoted to a mathematical model with diffusion and cross-diffusion to describe the interaction between vegetation and soil water. First, the existence of Hopf bifurcation and cross-diffusion-driven Turing instability are discussed. Then, based on the nonlinear analysis, we obtain the exact parameters range for stationary patterns and show the dynamical behavior near Turing bifurcation point. It is found that the model has the properties of gap, strip and spot patterns. Moreover, the small water-uptake ability of vegetation roots promotes the growth of vegetation and the transitions of vegetation pattern. But with the continuous increase of the water-uptake ability of vegetation roots, the local vegetation biomass density increases and the isolation between vegetation patches also increases, which may induce the emergence of desertification. In addition, our results reveal that the water consumption rate induces the transitions of vegetation pattern and prohibits the increase of vegetation biomass density. [ABSTRACT FROM AUTHOR]

Purpose: This paper mainly studies the double Hopf bifurcation for coupled van der Pol–Rayleigh system with time delay. It aims to see how time delay and coupling factors affect the system, then avoid unstable or harmful vibrations and look for steady state in engineering. Methods: This paper mainly uses the normal form and center manifold theory and the Runge–Kutta method. First, we analyze the stability switching of the coupled van der Pol–Rayleigh system with time delay and the dynamical behavior caused by the double Hopf bifurcation. Secondly, we derive the normal form of double Hopf bifurcation by the normal form and center manifold theory. Then the dynamical phenomena of the system are investigated. Finally, we perform numerical simulations to verify the theoretical analysis by the Runge–Kutta method. Results: The stability switching region of the system with respect to the delay is obtained. And the parameter regions of amplitude death, stable periodic motion and quasi-periodic motion are obtained. And as parameter values change, the steady state of the periodic solution follows. Conclusions: In the field of engineering, it is more practical to consider systems with time delay. So we consider system with time delay, obtain the stability switching region about the time delay and discuss the dynamics of each bifurcation region. When doing some system design, it is necessary to avoid the occurrence of bifurcation behavior, because it will lead to changes in system stability. We can adjust the system parameters to avoid bifurcation according to stability region. Or some engineering problem requires the avoidance of harmful vibrations and the pursuit of steady state. With the change of time delay and coupling factors, we can find the amplitude death region and the region of stable periodic motion and quasi-periodic motion, thus avoiding harmful vibration in practical engineering by adjusting the parameters. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the periodic pattern formations with spatial multi-peaks in a classic diffusive Holling-Tanner predator-prey model with nonlocal intraspecific prey competition. The main innovation is that a spatial dependently kernel is considered in the nonlocal effect, which mathematically complicates the linear stability analysis. We first generate the existences of Hopf, Turing, Turing-Hopf and double-Hopf bifurcations, and determine the stability of the positive equilibrium. It turns out that the stable parameter region for the positive equilibrium decreases with α increasing, which implies that the parameter region of pattern formation for such kernel is smaller than the spatial average case. For double-Hopf bifurcation, we calculate the normal form up to the third-order term restricted on the center manifold, which is expressed by the original parameters of the system. Via analyzing the equivalent amplitude equations, the system exhibits stable spatially nonhomogeneous periodic patterns, the bistability of such periodic solutions, as well as unstable spatially nonhomogeneous quasi-periodic solutions, all of them possess multiple spatial peaks. Interestingly, some possible strange attractors are found numerically near the double-Hopf singularity. Biologically, the emerging spatio-temporal patterns imply that such nonlocal intraspecific competition can promote the coexistence of the prey and predator species in the form of more complex periodic states. [ABSTRACT FROM AUTHOR]

Purpose: The bifurcating solution generated by Neimark-Sacker bifurcation of map exhibits a typical quasi-periodic oscillation behavior, which is beneficial for the active use of bifurcation in bifurcation control field. In this paper, an impulsive state feedback control method is proposed to create the quasi-periodic oscillation with certain desired characteristic in the vibro-impact system. Method: The linear control gains determined by using the explicit criteria of Neimark-Sacker bifurcation without directly using eigenvalues are designed to guarantee the existence of quasi-periodic oscillation. On the other hand, the non-linear control gains, which are determined by using center manifold reduction theory and normal form approach, ensure that the created quasi-periodic oscillation is stable. The amplitude of the trajectory section on the torus of quasi-periodic oscillation is analytically derived to achieve the control of the amplitude of invariant torus section by selecting appropriate non-linear control gains. Results and conclusions: Numerical experiments are provided to show the effectiveness of the proposed control method in a two-degree-of-freedom vibro-impact system. The results of quantitative analysis show that a stable quasi-periodic oscillation in the vibro-impact system can be created at a preassigned bifurcation point by desiging linear and nonlinear control gains, and the effect of the nonlinear control gains on the amplitude is either a positive relationship or a negative correlation to the amplitude. [ABSTRACT FROM AUTHOR]

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

At present, there are few studies on the delayed kernel function of hyper-strong kernel. This paper attempts to analyze the stability and bifurcation of neural networks with distributed delayed hyper-strong kernels. Firstly, considering the average delay as a bifurcation parameter, the study discusses the characteristic equations of delayed kernels with weak kernel, strong kernel and hyper-strong kernel to provide sufficient conditions for the stability and generation of Hopf bifurcation. Secondly, it applies the normal theory and the center manifold theory to derive the formulas for determining the stability and direction of the bifurcating periodic solution. Finally, it verifies the correctness of the calculation results by numerical simulation with an example. [ABSTRACT FROM AUTHOR]

This paper studies an Internet congestion control system with two time delays, which are accessed by a single resource and considers both discrete and distributed delays of the system. By designing a new hybrid controller containing negative feedback control and time delay feedback control to control the system, and taking discrete time delay variables as bifurcation parameters, the local stability and Hopf bifurcation of the system are studied. The results show that the Hopf bifurcation can be effectively delayed or avoided by adjusting the value of the feedback control parameter β . The global asymptotically stable dynamic characteristics of the system are ideal, which has important functional significance for optimizing network congestion control. Finally, a large number of simulation examples verify the correctness of the conclusions. [ABSTRACT FROM AUTHOR]

This paper studies a two-age structured SIRS epidemic model with logistic growth of susceptible population and two-time delays. We simultaneously introduce two-time delays, i.e., the immunity and incubation periods, into this dynamic system and investigate their impact on different dynamic behaviors for the model. By means of the C 0 -semigroup theory, the model is transformed into a non-densely defined abstract Cauchy problem, and the condition of the existence and uniqueness of the endemic equilibrium is obtained. Following the spectral analysis, the characteristic equation technique, and the Hopf bifurcation theorem, we show that different combinations of the two delays perform a vital role in the instability/stability as well as the Hopf bifurcation results of equilibrium solutions. We numerically provide some graphical representations to check the main theoretical results and show the rich dynamics by varying the two delay parameters. [ABSTRACT FROM AUTHOR]

In this paper, a fractional model of a special structure of bidirectional associative memory (BAM) neural networks called tri-diagonal BAM neural networks (TdBAMNNs) is considered. The Hopf bifurcation analysis is made for the proposed fractional system in the presence of leakage and communication delays. The feasibility of the obtained theoretical results is verified by numerical simulations. [ABSTRACT FROM AUTHOR]

This paper is devoted to the high codimension bifurcations of a classical predator–prey system with Allee effects and generalized Holling type III functional response p (x) = m x 2 a x 2 + b x + 1 where b > - 2 a . We show that the maximal orders of nilpotent saddle, cusp singularity and weak focus are all three. The unfoldings of a cusp singularity and of a nilpotent saddle of order 3 with a fixed invariant line are developed. The dependence of codimension of degenerate Hopf bifurcation on b is thoroughly investigated. It is proven that there exist a homoclinic loop of order 2 and a heteroclinic loop of order 2 for - 2 a < b < 0 , and three limit cycles for b > 0 . Together with existing work for Holling type I, II and IV functional responses, our results complement the analysis of the classical predator–prey systems with Allee effects and four types of Holling functional responses. Furthermore, simple formulas are derived to characterize the order of nilpotent saddle, through which the existence and order of the heteroclinic loop can be easily obtained for a general class of predator–prey systems with any smooth functional response. [ABSTRACT FROM AUTHOR]

This paper is devoted to the existence, multiplicity, stability, and Hopf bifurcation of steady-state solutions of a diffusive Lotka–Volterra type model for two species with nonlocal delay effect and nonlinear boundary condition. It is found that there is no Hopf bifurcation when the interior reaction term is weaker than the boundary reaction term, and that the interior reaction delay determines the existence of Hopf bifurcation only when the interior reaction term is stronger than the boundary reaction term. This observation helps us to understand the nonlinear balance between the interior reaction and boundary flux in nonlinear boundary problems. Moreover, the general results are illustrated by applications to a model with homogeneous kernels. [ABSTRACT FROM AUTHOR]

The tea green leafhopper Empoasca onukii is one kind of insect pest threatening the tea production, and the mite Anystis baccarum has been used as an agent for pest control. In this paper, we introduce a generalist predator-prey model to study the dynamics for informing biological control. There have been some bifurcation studies of the generalist predator-prey model in the last few years. Except for the bifurcations include saddle-node bifurcation of codimension 1 and 2, Hopf bifurcations, and Bogdanov-Takens bifurcation of codimension 2 and 3, we also present the bifurcations of nilpotent singularities of elliptic and focus type of codimension 3. We find that the nilpotent singularities are associated with a cubic Liénard system, and the nilpotent bifurcations are three-parameter bifurcations of a codimension 4 nilpotent focus. Furthermore, we show that the nilpotent focus serves as an organizing center to connect all the codimension 3 bifurcations in the system. We also present the bifurcation diagrams to unfold the nilpotent singularities of codimension 3. One interesting observation is that we show numerically the existence of three limit cycles in the system. [ABSTRACT FROM AUTHOR]

In this paper, we consider a single species population model over patches with delay and nonlocal interactions, for which no symmetry for the dispersion (connection) matrix is assumed. We show that there exists a positive equilibrium when the dispersal rate is large. We also discuss the stability/instability of this positive equilibrium, establish the threshold dynamics and explore the associated Hopf bifurcation. Moreover, we demonstrate our theoretical results by a nonlocal logistic population model and by the Nicholson's blowflies model. [ABSTRACT FROM AUTHOR]

Using a semidiscretization method, we derive in this paper a discrete slow-fast predator-prey system with ratio-dependent functional response. First of all, a detailed study for the local stability of fixed points of the system is obtained by invoking an important lemma. In addition, by utilizing the center manifold theorem and the bifurcation theory some sufficient conditions are obtained for the transcritical bifurcation and Neimark-Sacker bifurcation of this system to occur. Finally, with the use of Matlab software, numerical simulations are carried out to illustrate the corresponding theoretical results and reveal some new dynamics of the system. Our results clearly demonstrate that the system is very sensitive to its fast time scale parameter variable. [ABSTRACT FROM AUTHOR]

This paper is concerned with a delayed reaction–diffusion equation on a unit disk. By means of the singularity theory and Lyapunov–Schmidt reduction, we not only derive universal conclusions about the existence of inhomogeneous steady-state solutions and the equivariant Hopf bifurcation theorems, but also obtain some more extraordinary properties of bifurcating solutions, which are produced by the radial symmetry through abstract methods based on the Lie group representation theory. Meanwhile, we illustrate our results by an application to a population model with a time delay. Furthermore, the methods established in this paper are applicable to specific delayed reaction–diffusion models with other symmetries. [ABSTRACT FROM AUTHOR]

This paper reports a new 3D sub-quadratic Lorenz-like system and proves the existence of two pairs of heteroclinic orbits to two pairs of nontrivial equilibria and the origin, which are completely different from the existing ones to the unstable origin and a pair of stable nontrivial equilibria in the published literature. This motivates one to further explore it and dig out its other hidden dynamics: Hopf bifurcation, invariant algebraic surface, ultimate boundedness, singularly degenerate heteroclinic cycle and so on. Particularly, numerical simulation illustrates that the Lorenz-like chaotic attractors coexist with one saddle in the origin and two stable nontrivial equilibria, which are created through the broken infinitely many singularly degenerate heteroclinic cycles and explosions of normally hyperbolic stable foci E z. [ABSTRACT FROM AUTHOR]

Neuronal networks control motor output through multi-rhythmic electrical activities with different frequencies and amplitudes of different neurons. For example, the pre-Bötzinger complex related to respiratory function exhibits sigh (slow frequency and large amplitude) and eupnea (fast frequency and small amplitude) rhythms. In the present paper, rhythm transitions modulated by several physiological factors are simulated in a coupling model composed of sigh and eupnea compartments, and the roles of ionic currents and the underlying bifurcation mechanism of the rhythm transitions are acquired. The transition from sigh- and eupnea-rhythm to the eupnea rhythm induced by the blockage of calcium or hyperpolarization-activated channels is related to supercritical Hopf bifurcation of the sigh compartment, and to the sigh rhythm caused by the blockage of persistent sodium channel or change of extracellular potassium concentration is associated with subcritical Hopf/fold limit cycle bifurcation of the eupnea compartment. The parameter regions of ionic currents and dynamical mechanism of rhythm transitions in the pre-Bötzinger complex are helpful for the modulation to the respiratory rhythms. [ABSTRACT FROM AUTHOR]

In this paper, based on the classical HR neuron model, a double time-delay flux neuron model with magnetron memristor is proposed in this paper. The stability, the existence of Hopf bifurcation, direction of bifurcation and bifurcation cycle solution of the model are studied using Routh Hurwitz judgment method and central manifold theorem. As a result, we prove that there exists a bifurcation periodic solution in the specific time-delay range of the model, and the change law of spike discharge behavior induced by increasing double time-delay is clarified, which is helpful to explain the abnormal discharge behavior of brain or nerve center caused by electromagnetic radiation. The time-series diagram and phase diagram of the model under different time lag are obtained by numerical simulation with MATLAB software. The simulation result shows that under appropriate conditions, the time-delay is less than a critical value, and the system is asymptotically stable. When the time-delay exceeds the critical value, Hopf bifurcation occurs at the equilibrium point of the system. [ABSTRACT FROM AUTHOR]

This paper introduces a three-stable locally active memristor whose internal state equation is composed of sign functions related to internal state and voltage. The memristor has three asymptotically stable equilibrium points and a locally active region. The non-volatility and local activity of the memristor are verified by the POP (power-off plot) and the DC V - I plot, respectively. Furthermore, the proposed memristor has a petal-like continuous DC V - I curve, and thus it also is a new type of 4-lobe Chua corsage memristor. Biasing the memristor at its locally active region and connecting it in series with an inductor, a second-order oscillator is developed in this paper. The required inductance and the oscillation frequency are determined by analyzing the admittance function Y(iw,V) of memristor's small-signal equivalent circuit. Based on Hopf bifurcation theory and the pole-zero diagram of the composite admittance function Y c (s , Q) , the dynamics of the resultant oscillator circuit are analyzed in detail. In addition, a third-order chaotic oscillation circuit is obtained by adding a capacitor to the second-order oscillation circuit. Its dynamical characteristics are analyzed through Lyapunov exponent spectrum, bifurcation diagram and dynamic diagram. Finally, simulation results based on Multisim are given to verify the correctness of the numerical simulation. [ABSTRACT FROM AUTHOR]

In this paper, a six-neuron incommensurate fractional order BAM neural network model with multi-delays is considered. We demonstrate that the equilibrium point of the system loses its stability and Hopf bifurcation emerges when the delay passes through a critical value. And the relationship between the critical delay of Hopf bifurcation and size of fractional orders is found. Finally, some numerical simulations are given to verify the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics of a general nonlocal delayed reaction–diffusion–advection vector-disease model with spatial heterogeneity. The existence of the nonconstant positive steady state is identified by means of the Rabinowitz's bifurcation theory. Further, the stability of the positive steady state and the Hopf bifurcation are analyzed in terms of the eigenvalue distribution associated with the steady state. Moreover, the direction of the Hopf bifurcation and the stability of the bifurcated periodic solution are obtained by the center manifold reduction and normal form theory with the weighted inner product. It is found that in spatial heterogeneity, the large diffusion can change the stability of the infected host population due to the combined effects of the nonlocal dispersal and time delay. [ABSTRACT FROM AUTHOR]

Aiming at the cause of limit cycles and the pathway of chaotic oscillation by doubly fed induction generator (DFIG) as its internal parameters change, this paper proposed a method to identify limit cycles and a numerical method to identify chaotic pathways. First, a simplified four-dimensional nonlinear dynamic model of the DFIG is developed. A simple criterion for the Hopf bifurcation parameter of the DFIG is given: the Routh–Hurwitz stability criterion. In addition, the relationship between the system at critical damping and the Hopf bifurcation is obtained. Second, the stability interval of the DFIG linearized system is verified using a time-domain approach when the system has recovered from a slight disturbance. Finally, periodic, quasi-periodic, and chaotic oscillations are predicted to occur based on the largest Lyapunov exponent and power spectrum to further determine the system's parameters. The path and threshold of the tuning parameter inducing the DFIG into chaos are obtained to provide a theoretical basis for revealing the mechanism inducing chaotic oscillations. The numerical simulation results demonstrate the effectiveness and feasibility of the proposed method. [ABSTRACT FROM AUTHOR]

Purpose: In order to reduce the hazards caused by the nonlinear torsional vibration of the shaft system of the direct-drive permanent magnet wind turbine, the dynamic characteristics of the controlled shaft system torsional vibration were analyzed. Methods: First, the shaft system of the direct-drive permanent magnet wind turbine is equivalent to a two-mass model. Based on considering the electromechanical coupling and the nonlinear change of the main shaft damping, the torsional vibration dynamic model of the shaft system is established by the Lagrange–Maxwell equation. Second, with the main shaft damping as the bifurcation parameter, the Routh–Hurwitz criterion is used to calculate the Hopf bifurcation critical point and stability range of the shafting system. Finally, a nonlinear state feedback controller is designed, and the controlled shafting system paradigm is obtained by the direct method. Results: The torsional vibration characteristics of shafting and the influence of supercritical Hopf bifurcation on shafting are discussed. In order to suppress the unstable torsional vibration of the shafting system and analyze the influence of the controller's linear gain and nonlinear gain on the torsional vibration of the shafting system, the numerical simulation verifies the correctness of the theoretical analysis. Conclusions: This paper can provide a theoretical basis for the stable operation of the direct-drive permanent magnet wind turbine. [ABSTRACT FROM AUTHOR]

This paper carries out an analysis of the global properties of solutions of an in-host model of hepatitis C for general values of its parameters. A previously unknown stable steady state on the boundary of the positive orthant is exhibited. It is proved that the model exhibits Hopf bifurcations and hence periodic solutions. A general parametrization of positive steady states is given and it is determined when the number of steady states is odd or even, according to the value of a certain basic reproductive ratio. This implies, in particular, that when this reproductive ratio is greater than one there always exists at least one positive steady state. A positive steady state which bifurcates from an infection-free state when the reproductive ratio passes through one is always stable, i.e. no backward bifurcation occurs in this model. The results obtained are compared with those known for related models of viral infections. [ABSTRACT FROM AUTHOR]

The dynamics of a general reaction–diffusion–advection two species model with nonlocal delay effect and Dirichlet boundary condition is investigated in this paper. The existence and stability of the positive spatially nonhomogeneous steady state solution are studied. Then by regarding the time delay τ as the bifurcation parameter, we show that Hopf bifurcation occurs near the steady state solution at the critical values τ n (n = 0 , 1 , 2 , ...) . Moreover, the Hopf bifurcation is forward and the bifurcated periodic solutions are stable on the center manifold. The general results are applied to a Lotka–Volterra competition–diffusion–advection model with nonlocal delay. [ABSTRACT FROM AUTHOR]

This paper reports on the dynamical analysis, field programmable gate array (FPGA) implementation of autonomous Josephson junction (JJ) jerk oscillator with cosine interference term (AJJJOCIT) and investigation of its collective behavior in a network. The AJJJOCIT derived from a resistive capacitive shunted JJ model with cosine interference term has two or no equilibrium points as a function of direct current (DC). One of the equilibrium points is unconditionally unstable and the other equilibrium point has a Hopf bifurcation where its expression depends on DC and coherence parameters. One-scroll self-excited chaotic attractor, one-scroll chaotic hidden attractor, steady state attractors, bistable periodic attractors, limit cycle and coexistence between periodic and one-scroll chaotic self-excited (or hidden) attractors are revealed in the AJJJOCIT during the numerical analysis. Moreover, the FPGA of AJJJOCIT is implemented and the FPGA results are qualitatively the same as those obtained during the numerical analysis. Finally, the collective dynamics of the AJJJOCIT are studied using a single-layer matrix of the AJJJOCIT. It is demonstrated that chimera states exist in the system and when increasing coupling strength, a completely synchronized network is revealed. [ABSTRACT FROM AUTHOR]

This paper aims to report complex bursting patterns in a parametrically and externally excited Jerk circuit system. Typically, the common compound bursting with two clusters in each period can be observed in this system. By the fast-slow analysis, we find that the active states related to this type of bursting are created by supercritical Hopf bifurcations. Interestingly, with the increase of parametric excitation amplitude, the common bursting pattern evolves into the ones that display complex structures. We show that the parametric excitation amplitude plays a vital role in influencing the dynamical behaviors, which is apparent from the two-parameters analysis. Then, a rich variety of dynamical phenomena including period-doubling and inverse period-doubling cascades in orbits of period 2, 4, and 8 can be observed with the increase of parametric excitation amplitude, thereby leading to complex bursting patterns with multiple-periodically active states. Besides, as a result of the period-doubling cascades, chaotic attractors are eventually discovered, which can be corroborated by the maximal Lyapunov exponent. In particular, periodic windows with period-doubling and inverse period-doubling cascades also emerge in the chaotic region when the parametric excitation amplitude increases to higher values. These give rise to the complex bursting patterns with chaotically active states. Based on this, two types of bursting patterns with complex structures, i.e., bursting patterns with multiple-periodically active states and bursting patterns with chaotically active states are revealed. Our results enrich the routes to complex bursting and deepen the understanding of bursting patterns with complex structures. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamical behaviors of a basic delayed physical model for aerosol–cloud–precipitation system, which was proposed by Koren and Feingold. The rain production delay τ is crucial for this system. The main work of the present paper is to study the effect of cloud and rain interaction on climate by the mathematical methods, such as double Hopf bifurcation analysis. First, we prove the uniqueness of the equilibrium of interest, i.e., the positive equilibrium, which has never been reported. Next, we use DDE-BIFTOOL to plot the bifurcation diagrams with respect to two bifurcation parameters, i.e., a and τ , and then find its double Hopf bifurcation points. Then, we study their unfolding and classification by the methods of multiple scales and normal form. Finally, we verify the results by numerical simulations. We find some complicated dynamic behaviors of the system via analytical method, such as stable equilibrium, stable periodic, and quasi-periodic solutions. For the stable equilibrium, the numerical simulations agree well with the analytical results. The stable equilibrium implies a balance between cloud formation and depletion (drizzle), and stable periodic solution is related to cycles of formation of thicker clouds that are later consumed by stronger rain (the moderately drizzle). These complex dynamical phenomena can be very helpful for the researchers to understand the mechanism of aerosol–cloud–precipitation system. And these results can also be used to predict rainfall and climate change, thereby reducing the impact of extreme weather, and its subsequent natural disasters and accidents on economic and social development. [ABSTRACT FROM AUTHOR]

In the brain, feedforward inhibition is a fundamental regulator of balancing neural excitation and inhibition, and its dysfunction may cause epilepsy. In this paper, we investigate the influence of feedforward inhibition on absence seizures and its dynamical mechanisms based on a neural mass model. On the one hand, pyramidal neurons (PY), fast kinetic interneurons ( I fast ) and slow kinetic interneurons ( I slow ) form two feedforward inhibition pathways, which are the PY– I fast – I slow pathway and the PY– I slow – I fast pathway, respectively. When changing the synaptic strength in different pathways, the system shows simple oscillation state, spike and wave discharge (SWD), and multi-spike and wave discharge (m-SWD). On the other hand, the one-parameter bifurcation analysis reveals that the system develops fold bifurcations, Hopf bifurcations, fold bifurcations on limit cycle, and period doubling bifurcations when state transitions occur. In particular, when the system is in the bistable region, the dynamic state of this region is closely related to the stable limit cycle and stable fixed point. Therefore, feedforward inhibition pathways are indeed involved in the regulation of absence seizures, excitatory connections in both feedforward inhibition pathways are more effective than the inhibitory connections in the control of absence seizures. More interestingly, the feedforward inhibition pathway PY– I slow – I fast has a stronger regulation of absence seizures than PY– I fast – I slow . These results provide a theoretical basis for a more detailed understanding of the underlying mechanisms of neurological disorders. [ABSTRACT FROM AUTHOR]

Photovoltaic (PV) system is the cleanest form of electricity generation, and it is the only form with no effect on the environment at all. However, some environmental challenges persist, which must be overcome before solar energy may be used to represent a source of truly clean energy. This paper aims to study the stability and dynamic behavior of a grid-connected environmentally friendly photovoltaic energy system using the bifurcation theory. This theory introduces a systematic method for stability analysis of dynamic systems, under changes in the system parameters. To produce bifurcation diagrams based on the bifurcation theory, a parameter is constantly changed in each step, using MATLAB and AUTO, and eigenvalues are monitored simultaneously. Considering how the eigenvalues approach the system's imaginary axis in accordance with the changes in the targeted parameter, the occurred saddle-node and Hopf bifurcations of the grid-connected PV system are extracted. Using the obtained bifurcations, the system's dynamic stability limits against changes in controlled (controller coefficients) and systematic parameters (such as the Thevenin impedance network) are found. [ABSTRACT FROM AUTHOR]

Periodic traveling waves (PTWs, wavetrains) have been extensively used to understand spatiotemporal oscillations observed in a number of field datasets. The aim of this paper is to investigate the existence, period functions and stability of PTWs with large speed in general two component reaction-diffusion systems. We first show the existence of small-amplitude (or large-amplitude) PTWs arising from Hopf bifurcation around a weak focus of order m (or from bifurcation of periodic orbits around a m-fold limit cycle). Second, we give the dominant terms of their period functions, which rarely appear in the existing references. Third, we establish the instability of PTWs in terms of the first-order derivative of their period functions. Our results indicate that the stability of small-amplitude PTWs depends on not only the real parts of the Lyapunov coefficients but also their imaginary parts. The invariant manifold theory developed by Fenichel is applied, along with bifurcation theory of planar dynamical systems. Finally, we apply the main results to a host-generalist parasitoid model for the biological control of a leafminer population. [ABSTRACT FROM AUTHOR]

In this paper, we consider a diffusive Lotka–Volterra predator–prey system with spatial heterogeneity and two delays. We first show that there exists a nonconstant positive steady state when the diffusion rates of prey and predator are large. Then, we obtain the stability of the steady state and show the existence of a Hopf bifurcation. Moreover, some numerical simulations are provided to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we show the existence of a Hopf bifurcation in a delayed single population model with patch structure. The effect of the dispersal rate on the Hopf bifurcation is considered. Especially, if each patch is favorable for the species, we show that when the dispersal rate tends to zero, the limit of the Hopf bifurcation value is the minimum of the "local" Hopf bifurcation values over all patches. On the other hand, when the dispersal rate tends to infinity, the Hopf bifurcation value tends to that of the "average" model. [ABSTRACT FROM AUTHOR]

This paper considers a chemical reaction-diffusion model for studying pattern formation with the Sel'kov–Schnakenberg model. Firstly, the stability conditions of the positive equilibrium and the existing conditions of the Hopf bifurcation are established for the local system. Then, Turing instability (diffusion-driven), which causes the spatial pattern is investigated and the existing condition of the Turing bifurcation is obtained. In addition, the dynamic behaviors near the Turing bifurcation are also studied by employing the method of weakly nonlinear analysis. The theoretical analysis shows that spatio-temporal patterns change from the spot, mixed (spot-stripe) to stripe with the variation of parameters, which can be verified by a series of numerical simulations. These numerical simulations give a visual representation of the evolution of spatial patterns. Our results not only explain the evolution process of reactant concentration, but also reveal the mechanism of spatio-temporal patterns formation. [ABSTRACT FROM AUTHOR]

In this paper, we study the global existence of a bifurcating periodic solution for a two zooplankton-phytoplankton model with two delays. First, we demonstrate that the bifurcating periodic solution exists when one delay increases and the other delay remains unchanged. Second, we give simulation to describe the bifurcating periodic solution when one delay increases. Our work answers the question in Sect. 5 (Shi and Yu in Chaos Solitons Fractals 100:62–73, 2017). [ABSTRACT FROM AUTHOR]

This paper focuses on the effect of leakage delays on the dynamics of nonlinear delay systems based on a neuron system with discrete and distributed delay. By introducing a virtual neuron, we can eliminate the impact of distributed delay and transform the original neuron system with mixed delays to an equivalent new system involving only discrete delay. This paper extends the existing works on neural networks to the case of leakage delays. We choose the leakage delay as the bifurcation parameter and analyze the role of leakage delays on the stability and Hopf bifurcation. Sufficient conditions for ensuring the neuron system to be stable, and undergoing the Hopf bifurcation are derived. Finally, by using the software package DDE-BIFTOOL, we provide the simulation results to substantiate our theoretical analysis, and give the relationship between the onset of Hopf bifurcation and the system parameters by drawing the bifurcation curves. [ABSTRACT FROM AUTHOR]

In the Antarctic, the whale population had been reduced dramatically due to the unregulated whaling. It was expected that Antarctic krill, the main prey of whales, would grow significantly as a consequence and exploratory krill fishing was practiced in some areas. However, it was found that there has been a substantial decline in abundance of krill since the end of whaling, which is the phenomenon of krill paradox. In this paper, to study the krill–whale interaction we revisit a harvested predator–prey model with Holling I functional response. We find that the model admits at most two positive equilibria. When the two positive equilibria are located in the region { (N , P) | 0 ≤ N < 2 N c , P ≥ 0 } , the model exhibits degenerate Bogdanov–Takens bifurcation with codimension up to 3 and Hopf bifurcation with codimension up to 2 by rigorous bifurcation analysis. When the two positive equilibria are located in the region { (N , P) | N > 2 N c , P ≥ 0 } , the model has no complex bifurcation phenomenon. When there is one positive equilibrium on each side of N = 2 N c , the model undergoes Hopf bifurcation with codimension up to 2. Moreover, numerical simulation reveals that the model not only can exhibit the krill paradox phenomenon but also has three limit cycles, with the outmost one crosses the line N = 2 N c under some specific parameter conditions. [ABSTRACT FROM AUTHOR]

In this paper, a stage structured predator-prey model with Holling type II functional response is formulated, considering the juvenile prey as the favorite food for the generalist predator. Further, the existence of predators is ensured by the sufficient amount of alternative food available in the habitat. The proportional harvesting of the adult prey is incorporated with the assumption that only the adult prey are of economic worth. The existence and local stability of the distinct equilibrium points of the system are investigated. The bifurcation from origin to predator free equilibrium state is obtained for the bifurcation parameter-effort on harvesting. The occurrence of Hopf bifurcation about the interior equilibrium state is established for arbitrary model parameters and the supercritical nature of this bifurcation is proved by fixing these parametric values. An algebraic equation is included to this modified model to analyze the economic benefits resulted from the harvesting of adult prey. The singularity-induced bifurcation (SIB) about the coexisting equilibrium state of differential-algebraic system is deduced along the parameter v at v = 0 , v being the profit/loss due to harvesting. The state feedback controller is recommended to eliminate the SIB about the coexisting equilibrium state for the differential algebraic system. Adopting an appropriate feedback control would ensure the stability of co-existence interior equilibrium state along with economic profit from harvesting. Numerical examples are used to elaborate the analytical results obtained. [ABSTRACT FROM AUTHOR]

Little seems to be considered about the globally exponentially asymptotical stability of parabolic type equilibria and the existence of heteroclinic orbits in the Lorenz-like system with high-order nonlinear terms. To achieve this target, by adding the nonlinear terms yz and x 2 y to the second equation of the system, this paper introduces the new 3D cubic Lorenz-like system: x ˙ = a (y - x) , y ˙ = b 1 y + b 2 y z + b 3 x z + b 4 x 2 y , z ˙ = - c z + y 2 , which does not belong to the generalized Lorenz systems family. In addition to giving rise to generic and degenerate pitchfork bifurcation, Hopf bifurcation, hidden Lorenz-like attractors, singularly degenerate heteroclinic cycles with nearby chaotic attractors, etc., one still rigorously proves that not only the parabolic type equilibria S x = { (x , x , x 2 c) | x ∈ R , c ≠ 0 } are globally exponentially asymptotically stable, but also there exists a pair of symmetrical heteroclinic orbits with respect to the z-axis, as most other Lorenz-like systems. This study may offer new insights into revealing some other novel dynamic characteristics of the Lorenz-like system family. [ABSTRACT FROM AUTHOR]

Seagrass meadows always provide a habitat for many kinds of fish. As the environment changes, some kinds of fish persist if they can evolve a positive growth rate to avoid extinction and others go to extinct in the habitat. To reveal the roles of environment and ability to evolve on the predator population persistence, in this paper, we formulate an age structured model by adopting the seagrass residence index (SRI) as the environment variable. By uniform persistence and stability analysis, we obtain the detailed dynamic behaviors of the age structured model which are mainly determined by the two thresholds η 1 ∗ and η 2 ∗ for the conversion rate η . If η < η 1 ∗ , the density of the predator population will go to zero. If η > η 1 ∗ , there is a unique positive steady state E ∗ and the predator population will be uniformly persistent. If η 1 ∗ < η < η 2 ∗ , the unique positive steady state E ∗ is stable and the predator population can establish itself in the environment. If η > η 2 ∗ , the unique positive steady state E ∗ will lose its stability and Hopf bifurcation occurs when maturation age τ = 0 . If τ > 0 , maturation age τ will play a key role in the stability criteria for the positive steady state E ∗ , and the establishment of the predator population will become more difficult and complex. Numerical simulations are presented to verify the theoretical results and reveal the evolution of predator population adaptation to the habitat. [ABSTRACT FROM AUTHOR]

This paper introduces a novel autonomous chaotic jerk circuit with an antiparallel diodes pair whose mathematical model involves an inverse hyperbolic sine function in the form: f x = k - 2 x + 4 arcsin h mx where k (i.e. a constant excitation source) controls the symmetry of the model while m represents the slope of the inverse hyperbolic sine. The presence of the inverse hyperbolic sine is unusual provided that such types of circuits are connected to hyperbolic sine nonlinearity. The analysis of the model indicates that in case of a perfect symmetry ( k = 0.0 ), the system undergoes spontaneous symmetry breaking, period doubling scenario to chaos, symmetry recovering crisis, coexistence of multiple pairs of symmetric attractors, and coexisting symmetric bubbles of bifurcation. More complex and incoherent nonlinear dynamic patterns occur in the presence of symmetry perturbation ( k ≠ 0.0 ) including for instance non-symmetric Hopf bifurcations, coexisting point attractor and limit cycle, coexisting asymmetric bubbles of bifurcations, critical transitions, and coexisting (i.e. up to five) non-symmetric periodic and chaotic attractors. The space magnetization resulting from the presence of various coexisting attractors is examined and illustrated by using basins of attraction. The predictions of theoretical investigations are supported by laboratory experimental tests based on a prototypal electronic circuit mounted on a breadboard. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamical innovations occur in an upgraded chaotic Pan system. The local dynamics includes the stability analysis, types of attractor and Hopf bifurcation analysis. The proposed system undergoes periodic, chaotic and hyperchaotic orbits with the variation of bifurcation and control parameter. The hidden self-excited chaotic attractors are localized within the parametric boundaries; also, proposed system is initialized in the self-excited attractor zone where all the unstable equilibria exist. The hidden and self-existing attractor of hyperchaotic Pan system is revealed with the help of multidimensional transient dynamical system. Moreover, a complete set of conditions are derived that guarantee the existence of supercritical Hopf bifurcation in the upgraded chaotic Pan system. The Hopf bifurcation diagram is also investigated for numerical confirmation of analytical results. [ABSTRACT FROM AUTHOR]

In this present paper, we deal with a generalized Lengyel–Epstein model with the zero-flux boundary conditions. Firstly, we give an attraction region and the boundedness estimates of the solutions to the parabolic equations. Hereafter, one performs the local and global stability of the unique positive equilibrium. The first Lyapunov exponent technique and the normal form theory are employed to investigate the directions of the Hopf bifurcation, respectively. It is found that the supercritical or the subcritical may occur in the generalized Lengyel–Epstein model. Finally, we explore the steady states of the elliptic equations. The boundedness, the nonexistence, and the existence of the steady states are performed. Numerical experiments well verify the theoretical analysis. Relevant theoretical results illustrate that the diffusion rates of the substance can affect the dynamical behaviors of such a generalized Lengyel–Epstein model. [ABSTRACT FROM AUTHOR]

In this paper, a diffusive predator-prey system with a prey-taxis response subject to Neumann boundary conditions is considered. The stability, the Hopf bifurcation, the existence of nonconstant steady states, and the stability of the bifurcation solutions of the system are analyzed. It is proved that a high level of prey-taxis can stabilize the system, the stability of the positive equilibrium is changed when χ crosses χ 0 , and the Hopf bifurcation occurs for the small s. The system admits nonconstant positive solutions around (u ¯ , v ¯ , χ i) , the stability of bifurcating solutions are controlled by ∫ Ω Φ i 3 d x and ∫ Ω Φ i 4 d x . Finally, numerical simulation results are carried out to verify the theoretical findings. [ABSTRACT FROM AUTHOR]