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We are concerned in this paper with dynamic behaviors and an optimal control problem of a new delayed epidemic model. There are three major ingredients. The first one is the dynamic behaviors of the state system. The locally asymptotic stability of the disease-free equilibrium and the endemic equilibrium are investigated and the effect of time delay on stability is also discussed. It is also found that the Hopf bifurcation appears at a specific time delay. The second, which is the main new ingredient of this paper, is an optimal control problem. Applying vaccine strategy in the system, an optimal control problem is proposed to minimize the total number of infected individuals as much as possible, maximize the total number of the uninfected individuals, and reduce the total control cost. In view of Pontryagin's maximum principle, the specific characteristics of the optimal control policy are given. The third ingredient is the numerical simulations of the theoretical results. • The model considers time delay, competition and spread of disease. • It studies dynamics and an optimal control problem of a delayed model. • Examples and numerical simulations are given to verify the theoretical results. [ABSTRACT FROM AUTHOR]

This paper serves as a continuation and expansion of the previous achievement by B. Han and D. Jiang in their article (Han and Jiang, 2024). Our initial step involves transforming the more complex stochastic Holling-II model featuring a stronger kernel into a simplified yet degenerate stochastic system composed of five interconnected equations. For the deterministic component of this model, we delve into an extensive examination of the local asymptotic stability of its positive equilibrium state. Subsequently, for the stochastic model, we derive critical conditions that determine the thresholds for exponential extinction and persistence of both predator and prey populations. Importantly, our findings not only encompass scenarios where there are no stochastic disturbances but also shed light on how environmental noise impacts the population dynamics within the predator–prey system. Through the exploration of the homologous Fokker–Planck equations, we present approximate representations characterizing the probability density function of the stochastic predator–prey model. To substantiate these theoretical advancements, several illustrative examples are provided, offering numerical elucidations of our proposed results and emphasizing the profound effects of stochastic noises and some important parameters on the behaviors of the stochastic model. • A stochastic Holling II predator–prey model with general distributed delay is investigated. • The sufficient and necessary condition for the coexistence of predator and prey populations is derived. • The approximate expression of the density function of the stochastic predator–prey model is obtained. • Several examples with simulations are presented to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper it is exhibited the bifurcation scenario concerning a typical singularity of planar piecewise smooth vector fields in two zones. This singularity is characterized by a quadratic contact of one vector field and a quartic contact of another vector field at the same point of the switching manifold. By means of a three parameter perturbation, we observe the presence of bifurcations like: Hopf Bifurcation, Sliding Hopf Bifurcation, the birth and annihilation of chaotic scenarios, among others. The two and three dimensional bifurcation diagrams are included. • quartic–quadratic fold singularity of planar piecewise smooth vector fields. • normal form of the quartic–quadratic fold singularity. • 3-parameter bifurcation of a planar piecewise smooth vector field. • Planar bifurcation diagrams. • tridimensional bifurcation diagram. [ABSTRACT FROM AUTHOR]

This paper investigates the bifurcations of homoclinic orbits to hyperbolic saddle points in a simplified railway wheelset model with cubic and quintuple nonlinear terms. Using Melnikov's method, the sufficient conditions for the existence of the supercritical and the subcritical pitchfork bifurcations of homoclinic orbits are proven. To determine the integrability of the variational equations around homoclinic orbits in the meaning of differential Galois theory, the corresponding Fuchsian second-order differential equation for the normal variational equation and the Riemann P function are obtained. It is shown that the coefficients of the linear terms and the cubic coupling terms play a very significant role on influencing the existence of homoclinic orbits. While, the cubic coupling terms have little effect on the size of the left-hand and right-hand potential wells of homoclinic orbits. These results are beneficial to explore the key mechanism of hunting stability of a simplified railway wheelset model. • The sufficient conditions for pitchfork bifurcations of homoclinic orbits are proven. • The integrability of variational equations around homoclinic orbits is determined. • Linear and cubic coupling term coefficients affect the existence of homoclinic orbits. [ABSTRACT FROM AUTHOR]

• A Light-fueled liquid crystal elastomer self-fluttering aircraft is proposed. • The system has two motion patterns: stationary and self-fluttering. • Triggering conditions, frequency, and amplitude of self-fluttering are adjustable. • The amplitude and frequency of self-fluttering can be designed by system parameters. In response to external stimuli, active materials are able to alter their shape or motion. Nonetheless, considerable challenges remain in the effective realization and control of active machines. Conventional control methods, involving sensor-based and external device control as well as electronic control, are intricate and tough to put into practice. Conversely, self-oscillators built upon active materials exhibit remarkable properties, including steady motion and straightforward regulation. Drawing inspiration from flying organisms, this paper puts forward an innovative light-fueled self-fluttering aircraft with a liquid crystal elastomer (LCE) engine, capable of sustained flight in the presence of steady illumination. The governing equations of the self-fluttering aircraft are derived in accordance with the well-known dynamic LCE model. The system is capable of experiencing a supercritical Hopf bifurcation, transitioning from a stationary pattern to a self-fluttering pattern, and the energy compensation mechanism of the self-fluttering pattern is revealed as well. Moreover, the quantitative investigation covers how distinct system parameters influence the Hopf bifurcation conditions and the self-fluttering amplitude and frequency, as well as the average flight velocity. With the benefits of customizable size, electronics-free and dependence on ambient energy, the proposed system may generate fresh insights into the practical use of active materials-based self-oscillators in micro actuators, soft robotics, bionic devices, and energy harvesting. [ABSTRACT FROM AUTHOR]

In this paper, we delve into double Hopf bifurcation induced by memory-driven directed movement in a spatial predator–prey model with Allee effect and maturation delay of predators. We first adopt a novel technique to handle the associated characteristic equation and thus obtain the crossing curves as well as the double Hopf points. We then calculate explicit formulae of normal form regarding non-resonant double Hopf bifurcation. We thus divide the dynamics of the developed model into several categories near the double Hopf bifurcation points. Our numerical and theoretical results both demonstrate that the model can exhibit various complex phenomena when the parameters are near the double Hopf bifurcation points. For example, the transition from one stable spatially inhomogeneous periodic orbit with mode-5 to another with mode-4 and the coexistence of them can be observed. • A spatial model with delays in both reaction and diffusion terms is considered. • An innovative technique is given to obtain crossing curves and bifurcation points. • The algorithm to compute normal form of double Hopf bifurcation is generalized. • Compared with only random diffusion, memory-based movement causes complex dynamics. [ABSTRACT FROM AUTHOR]

Travel between patches and behavior changes of susceptible individuals are two crucial factors affecting the disease dynamics for any emerging infectious disease. Therefore studies of their impacts on disease dynamics are of great importance, especially to public health policy-makers. In this paper, we propose an SIR model that incorporates these two factors into consideration. For our two-patch SIR model with infection force mediated by (past) disease surveillance, we identify the basic reproduction number and show how it is related to travel modified basic reproduction numbers. We also carry out stability and Hopf bifurcation analysis and obtain conditions under which the disease persists in one patch or both patches. Numerical simulations are carried out to illustrate how travel between two patches and disease surveillance mediated infection force affect the disease dynamics and thus provide us some helpful insights on proper control of emerging infectious diseases. • Build a two-patch SIR model with infection force mediated by (past) disease surveillance. • Identify the overall basic reproduction number and related it with several travel modified basic reproduction numbers (See Table 1). • Find that the travel of susceptible individuals alone can cause an endemic even if the disease dies out in each isolated patch. • Study the effects of the past disease surveillance on the spread of the disease. [ABSTRACT FROM AUTHOR]

In this paper, the qualitative properties of the fixed points in the non-hyperbolic cases, codimension-one bifurcations and weak resonances of a Cournot-Bertrand duopoly mixed competition model are explored. The two firms adopt different decision variables and different objective functions, which are more consistent with the actual economic market situation. The qualitative properties of all the fixed points in the non-hyperbolic cases are investigated using the reduction principle and the center manifold theorem. After that, all the potential codimension-one bifurcations, including transcritical bifurcation, supercritical or subcritical flip bifurcation and Neimark–Sacker bifurcation are analyzed using the bifurcation theory and the center manifold theorem. The direction, stability, and even the explicit approximate expression are derived for each type of bifurcation. By perturbing the closed invariant curve caused by the Neimark–Sacker bifurcation, the 2 : 5 weak resonance associated with Arnold's tongue is theoretically proved, and the absence of 1 : 6 and 5 : 6 weak resonances is further analyzed. A large number of numerical simulations show complete consistency with all theoretical analyses. Moreover, the continuation method is used to conduct numerical bifurcation analyses, further verifying the correctness of theoretical analyses, and testing more codimension 2 bifurcations, such as fold-flip bifurcation, generalized flip bifurcation and 1 : 2 strong resonance. In addition, the economic implications of these bifurcations are also explained accordingly. [ABSTRACT FROM AUTHOR]

In this paper, we present a complete parametric analysis on a nonlinear predator–prey system with the generalized Holling type III functional response and a strong Allee effect. We apply the hierarchical parametric analysis to derive explicit conditions for the existence and stability of equilibrium solutions in a 5-dimensional parameter space. Specifically, a detailed study on Hopf bifurcation is given to show bifurcation of multiple limit cycles, and complex dynamics of multistable states including bistable, tristable and tetrastable phenomena. We also conduct simulations and provide a biological interpretation of the multistable states under various conditions. • A predator–prey generalized Holling type III functional response and a strong Allee effect is investigated. • The hierarchical parametric analysis is applied to study the existence and stability of equilibrium solutions in a 5-dimensional parameter space. • A detailed study on Hopf bifurcation is given to show bifurcation of multiple limit cycles. • Complex dynamics of multistable states including bistable, tristable and tetrastable phenomena are explored. [ABSTRACT FROM AUTHOR]

In this paper, we use a cortex-basal ganglia resonance network to explore the possible Hopf bifurcation mechanism of beta oscillation. Different from traditional viewpoints, this model contains a direct inhibitory projection from the subthalamic nucleus (STN) to the cortex excitatory nuclei (EXN). First, we obtain the oscillation critical conditions of the delay by Hopf bifurcation analysis. Then, we explore the effects of some key parameters on Hopf bifurcation points. Delay is an indispensable factor in this model, oscillations will occur only when delay increases to a certain extent. Interestingly, the amplitude of oscillation increases with an increase in delay, but the trend of frequency is opposite; and heterogeneous delays have different effects on amplitude and frequency. The inhibitory coupling weight in the pathway " S T N → E X N " can reduce the firing activation level of cortical circuit, and the increase of coupling weights in the closed loop of the STN and globus pallidus external (GPe) can disinhibition this effect. On the stable boundary curve, the quantitative relationship of coupling weights between the direct and indirect projection from the basal ganglia to cortex is monotonously increasing. An increase in some critical coupling weights in the model, the bidirectional Hopf bifurcations (supercritical and subcritical) occur alternately, which can uniformly explain the origin mechanism of oscillations. In the inhibition-excitation closed loops, the excitatory and inhibitory connection weights have different effects on oscillations, the quantitative relationship of which is monotonically decreasing under Hopf bifurcation critical conditions. In the oscillation region, we observe that the amplitude is the smallest near bifurcation points and gradually increases with the evolution of oscillations. Finally, we find that the direct inhibitory projection from the STN to the EXN is a key pathway for external stimuli to control cortical oscillations. All theoretical analysis results compares well to numerical simulations, we hope that theoretical mechanisms obtained in this paper can inspire experimental research in Parkinson's disease. • The projection from the STN to the EXN can inhibit cortical circuit. • The quantitative relationship of various projections from BG to cortex is increasing. • The supercritical and subcritical Hopf bifurcations can occur alternately. • The quantitative relationship of different weights in closed loops is decreasing. • The oscillation amplitude is the smallest near bifurcation points. [ABSTRACT FROM AUTHOR]

This paper focuses on a general Gause-Kolmogorov-Type predator–prey model with direct predator-taxis and indirect prey-taxis. The global existence and boundedness of solutions to the system are proved. Our approach is applicable to more general situation. Some measures of success for the constant coexistence steady state are rigorously calculated, such as decreasing the indirect prey-taxis sensitivity or the release rate of stimulus, and increasing the predator-taxis sensitivity or the decay rate of stimulus. By choosing the indirect prey-taxis coefficient as the bifurcation parameter, the existence of Hopf and double-Hopf bifurcations is established. We find that both indirect prey-taxis and direct predator-taxis can promote the complexity of the spatiotemporal pattern, e.g. spatially nonhomogeneous periodic patterns with different spatial frequency, spatially nonhomogeneous quasi-periodic patterns. Finally, we apply our theoretical analyses to a Rosenzweig–MacArthur model with indirect prey-taxis and direct predator-taxis, spatially homogeneous periodic patterns. • We bond the effects of direct and indirect taxis in a predator-prey system. • We prove the global well-posedness of a predator-prey system with direct and indirect taxis. • We determine several beneficial strategies of success about the indirect and direct taxis. • We establish double-Hopf bifurcations and pattern formations for a general predator-prey system. • In an application, complex nonhomgeneous periodic and quasi-periodic patterns emerge. [ABSTRACT FROM AUTHOR]

In this paper, based on the restoring force coupling method, a novel mathematical model is established to study the VC parameter vibration responses of the ball bearing-rotor system, and both the rotor radial and tilting motions have been considered. Firstly, the steady state analysis on the stiffness fluctuation is conducted to determine the equivalent linear resonant frequencies of the nonlinear ball bearing-rotor system. Secondly, both the two-way sweep-frequency and fixed frequency-analysis are adopted to study the different transient response mechanisms of the rotor VC parameter vibrations, Finally, the results show that not only the multiple resonant responses including the super-harmonic, sub-harmonic, Hopf bifurcation and chaos are generated for the rotor parametric vibration, but also the complex internal resonance phenomenon occurs when the rotor natural frequency ratio in different radial directions is close to 1:2. • A novel M-DOF model for the rotor VC vibration analysis is built. • The method for rotor equivalent linear resonance frequencies calculation is given. • The rotor transient response under different VC vibration mechanism is given. • The generation mechanism of the rotor multiple resonant responses is discussed. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical model describing the interaction of malignant glioma cells, macrophages and glioma specific CD8+T cells is discussed. The biologically feasible equilibria and corresponding local stability are deduced. The bifurcations and related dynamical behaviors of this model are further studied thoroughly. The existence of transcritical bifurcation and saddle–node bifurcation is derived based on Sotomayor's theorem and Hopf bifurcation is well discussed. The codimension 2 bifurcation such as Bogdanov–Takens bifurcation is investigated using the normal form theory and center manifold theorem in more detail. Finally, numerical simulations are obtained to validate our analytical findings by varying the parameters. • The dynamics of a mathematical model describing the interaction of malignantglioma cells, macrophages and glioma specific CD8+T cells are discussed. • The co-existence of equilibria of the model are obtained by numerical simulation. • The existence of codimension 1 and codimension 2 bifurcations is investigated. • Extensive numerical simulations are obtained to validate our analytical findings. [ABSTRACT FROM AUTHOR]

Malware spreading, characterized by differential equations, in networked systems has become a common topic of study in the scientific literature. In this paper, we focus primarily on the propagation dynamics of malware in cyber–physical systems (CPSs). Although some work prior has been done to understand how the malware dynamics evolves in complex networks, these studies have restricted their attention to stability, bifurcation and oscillation based on ordinary differential equation models. Here, we address the question "how does one detect tipping phenomena which result in a total collapse of CPSs if the spread of malware is related to both time and space?" To this end, we use a reaction–diffusion equation to model spatio-temporal dynamic evolutions of malware in CPSs and accurately predict the emergence of tipping points due to the Turing instability and Hopf bifurcation for the first time. In order to further reveal the mechanism of tipping, the specific conditions of the Turing instability and Hopf bifurcation are given. It is confirmed that the introduction of space factor and transmission delay causes CPSs to change from stable to unstable, and then the tipping occurs. Moreover, the delay-dependent stability is discussed and the stability and direction of Hopf bifurcation are explored. These studies provide a strategic guidance for the prediction and control of malicious virus in CPSs. Finally, some numerical simulation examples are provided to substantiate the theoretical results. • A novel reaction-diffusion malware spreading model with multiple delays is proposed. • The basic reproductive ratio is calculated and the Turing instability induced by tipping is studied. • The tipping point of time delay is given and the direction of Hopf bifurcation is determined. • The spatial-temporal evolution of infected nodes in CPSs can be predicted quantitatively. [ABSTRACT FROM AUTHOR]

In this paper, the influence of the fear effect and Leslie–Gower function on the dynamic behavior of the predator–prey model is considered. First, the well-posedness analysis of this model is demonstrated, and the existence and local stability of the equilibrium point are given. Then by using the bifurcation theory and selecting the appropriate bifurcation parameters, many types of bifurcation phenomena in the system are discovered, including transcritical bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation. Meanwhile, the numerical simulations on the basis of theoretical analysis are further carried out to intuitively show the influence of fear effect on population. As the degree of fear increases, the system will undergo multiple dynamic behaviors switching until the final extinction of prey population, while the predator population will survive due to the presence of substitute prey. In addition, the initial density of predator and prey can determine whether the solution of system tends to a coexisting steady state or a periodic oscillation. • A novel prey–predator model with the impact of fear is introduced. • The system can experience many types of bifurcation phenomena. • Numerical simulations and sensitivity analysis are performed. • Our results show that fear effect plays an important role on deriving the complex dynamics. [ABSTRACT FROM AUTHOR]

In this paper we study a three-dimensional tumor–immune system interaction model consisted of tumor cells, activated T cells, and immune checkpoint inhibitor anti-PD-1. Based on the uncontrollable character of tumor cells in the absence of immune response and treatment, the growth of tumor cells is assumed to be exponential. We discuss the distribution of equilibria qualitatively and study the stability of all possible equilibria with and without anti-PD-1 drug. When no drug is applied, the model has a tumor-free equilibrium and at most one tumorous equilibrium. Biologically, there exists a threshold d T 1 for the death rate d T of T cells: when d T ≥ d T 1 tumor cells will keep growing; when d T < d T 1 tumor cells may be eradicated for some positive initial values and keep growing for some other positive initial values. For the case with anti-PD-1 treatment, the model has at most five tumor-free equilibria and two interior equilibria. Our analysis indicates that there exists a threshold γ A 1 for the intravenous continuous injection γ A : when γ A ≤ γ A 1 the fate of tumor cells is the same as the case with no drug applied; when γ A > γ A 1 the model may exhibit bistable phenomena and periodic orbits. Furthermore, we establish the existence of local Hopf bifurcation around the interior equilibrium and determine the stability of the bifurcating periodic orbits. Our simulations show that the model exhibits a stable periodic orbit which implies the long term coexistence and balance of the tumor and immune system. • Study a tumor–immune system interaction model with an immune checkpoint inhibitor. • Discuss the number and stability of all possible equilibria. • Establish the existence of Hopf bifurcation and the stability of the periodic orbit. • Carry out numerical simulations to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we prove the equivalence of the multiple time scales (MTS) method and the center manifold reduction (CMR) method for deriving the normal form of Hopf bifurcation for delayed reaction–diffusion equations. First, we generalize the MTS method associated with functional differential equations to delayed reaction–diffusion equation for solving the normal form of Hopf bifurcation. This provides a user-friendly approach of analyzing the stability of time-periodic solutions, particularly, for the engineering researchers who are not familiar with CMR method. Second, we outline the CMR method associated with the computation of normal form of Hopf bifurcation for delayed reaction–diffusion equations. Third, we prove that the two methods can derive equivalent up to the third order normal form of Hopf bifurcation. Finally, different types of examples are presented, such as reaction–diffusion equations with Neumann boundary condition and Dirichlet boundary conditions, single delay and two delays, spatially homogeneous and spatially inhomogeneous periodic solutions. We obtain the equivalent Hopf bifurcation normal forms results derived by using the CMR and MTS methods for above examples, respectively. • We generalize MTS method to Hopf bifurcation of delayed reaction–diffusion equation. • We outline CMR method for Hopf bifurcation of delayed reaction–diffusion equations. • We prove the equivalence of the two methods up to the third order normal form. • Four typical examples are presented. [ABSTRACT FROM AUTHOR]

To analyze the effect of animal spirits on economic dynamics, we construct a simple Keynesian business-cycle model with animal spirits incorporated into it. Similar to the well-known Kaldor model, we assume that each firm possesses the investment function depending on demand, in which the animal spirits are connected with the degree of the propensity to invest. However, we would like to emphasize that, unlike the Kaldor model, even if the level of demand for each firm is the same, the degree of propensity to invest (i.e., the strength of animal spirits) depends on the change in demand. Thus, we assume that animal spirits in demand-upswing phases are stronger than those in demand-downswing phases. In each phase, the degree of dependence on demand is firm specific. Therefore, we adopt a statistical approach to connect such heterogeneous micro-investment behaviors with a macro-investment function. Through this approach, some kind of nonlinearity is naturally introduced into the macro-investment function. We demonstrate that the nonlinear macro-investment function yields persistent business cycles. We also demonstrate the occurrence of a generalized Hopf bifurcation. • We change of the title of the paper, that is, entitled a new title according to reviewer's comment. • We have made the text as easy to understand as possible including corrections of not only theoretical points but also grammatical and stylistic ones. • We also tried to reduce the number of pages. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the species interaction model with the ratio-dependent Holling III functional response and strong Allee effect. To explore nonhomogeneous solutions of the model, we consider the existence and non-existence of non-constant steady states and temporal bifurcation. Then we show the boundedness of the global positive solutions for the parabolic system and present the upper and lower bounds of positive solutions for the associated elliptic system. The non-existence and existence of the non-constant steady states of the elliptic system with the homogeneous Neumann boundary conditions are obtained by using the priori estimates, maximum principle and index theory. Furthermore, the existence and the direction of Hopf bifurcation are investigated via the stability analysis, center manifold theory and normal form reduction. Numerical simulations are carried out to verify our theoretical analysis and to illustrate that the ratio-dependent Holling III functional response and strong Allee effect have strong impact on dynamical behaviors of the species interaction systems. • Boundedness of parabolic and elliptic models is analyzed. • Nonconstant steady states are presented. • Hopf bifurcation with strong Allee effect is described. [ABSTRACT FROM AUTHOR]

In this paper, we consider the glycolysis system with symmetry and degeneration. Based on the detailed stability of constant steady state solution and symmetry of nonlinear glycolysis system, for the Hopf bifurcations, we obtain the existence, stability and bifurcation direction not only for the homogeneous and inhomogeneous periodic solutions, but also for the degenerate bifurcating periodic solutions. The approach we adopted is the Lyapunov–Schmidt reduction method instead of the center manifold theory. On the other hand, we derive the nonexistence of the steady state solutions for small input flux of substrate. Additionally, for the steady state bifurcations, main object is to discuss degenerate bifurcations by means of Lyapunov–Schmidt procedure and singularity theory because the classical Crandall–Rabinowitz theorem cannot be applied. Some numerical simulations are achieved to illustrate the analytical results, especially the degenerate bifurcation results. It is interesting to noticed that the degenerate bifurcations can lead to richer spatiotemporal patterns, including the meeting of two Hopf bifurcation branches and the coupling of two eigenfunction models. [ABSTRACT FROM AUTHOR]

In this paper, we systematically study Parkinson oscillation origin mechanism, oscillation amplitude and frequency characteristics in an improved cortex-basal ganglia (EXN-INN-STN-GPe) resonance model. Firstly, Hopf bifurcation critical conditions T 1 0 and T 2 0 for heterogeneous delays are derived, which are greatly affected by coupling weights. The delay is a key factor to affect the frequency of oscillation, many frequency bands (such as gamma oscillation, beta oscillation, alpha oscillation, etc.) appear with an increase in delay. According to T 1 0 and T 2 0 , the state of STN-GPe loop can be divided into four types: stable state, induced oscillation, spontaneous oscillation and compound oscillation, which represent different origin mechanism of oscillation. They have different amplitude characteristics, which increases with an increase in delay in most cases. The amplitude of the induced oscillation is smaller than the amplitude of the spontaneous oscillation in many cases. In the compound oscillation region, the EXN-STN antiphase oscillation mode can inhibit the oscillation amplitude in the STN-GPe. Induced oscillation may indicate that the oscillation origins in the cortex, which spreads to the STN-GPe loop with the evolution of disease. Self-feedback connections of EXN (W E E) and INN (W I I) play a very important role in affecting the amplitude of induced oscillation. The induced oscillation amplitude of the GPe is increased with an increase in W E E , but the trend is the opposite when it is affected by W I I. As an increase in coupling weight between the EXN and INN, the minimum of W E E and W I I required to induce oscillation are both increased. The theoretical analysis is in good agreement with the numerical simulation, the mechanism obtained may inspire experimental and new theoretical research in Parkinson's disease. • Hopf bifurcation critical conditions T 1 0 and T 2 0 for heterogeneous delays are obtained. • The delay is a key factor to affect the frequency of oscillation. • According to T 1 0 and T 2 0 , the state of STN-GPe loop can be divided into four types. • The amplitude of induced oscillation is smaller than that of spontaneous oscillation. • EXN-STN antiphase oscillation mode can inhibit oscillation amplitude in the STN-GPe. [ABSTRACT FROM AUTHOR]

In this paper we consider a 3D three-parameter unfolding close to the normal form of the triple-zero bifurcation exhibited by the Lorenz system. First we study analytically the double-zero degeneracy (a double-zero eigenvalue with geometric multiplicity two) and two Hopf bifurcations. We focus on the more complex case in which the double-zero degeneracy organizes several codimension-one singularities, namely transcritical, pitchfork, Hopf and heteroclinic bifurcations. The analysis of the normal form of a Hopf-transcritical bifurcation allows to obtain the expressions for the corresponding bifurcation curves. A degenerate double-zero bifurcation is also considered. The theoretical information obtained is very helpful to start a numerical study of the 3D system. Thus, the presence of degenerate heteroclinic and homoclinic orbits, T-point heteroclinic loops and chaotic attractors is detected. We find numerical evidence that, at least, four curves of codimension-two global bifurcations are related to the triple-zero degeneracy in the system analyzed. • The 3D system considered is invariant under the change (x, y, z) → (−x, −y, z). • A double-zero degeneracy (a double-zero eigenvalue with geometric multiplicity two) is studied. • Transcritical, pitchfork, Hopf and heteroclinic bifurcations are involved. • Degenerate global connections, T-point heteroclinic loops and chaotic attractors are detected. • Four degenerate global connections related to the triple-zero bifurcation have been found. [ABSTRACT FROM AUTHOR]

This paper presents a bi-directional N-type locally-active memristor (LAM) model, which has two symmetrical locally-active regions with respect to the origin. For this memristor, the locally-active regions coincide with the edge of chaos regimes, where complex dynamic behaviors may occur. By deriving the small-signal admittance of the memristor, it is found that the LAM may be pure resistive, inductive or capacitive in terms of biasing voltages. When the LAM operates at the edge of chaos and connects with an external inductor in series, a second-order periodic oscillator is established. In this case, it is possible to move the poles of the system from the left-half plane to the right-half plane, thereby generating periodic oscillations. Complex dynamics of the second-order memristor-based oscillator is analyzed via Hopf bifurcation analysis and based on the theories of local activity and edge of chaos. Using the parasitic capacitor of the LAM in the second-order oscillator, a third-order chaotic oscillator is designed, which has a negative real pole and a pair of complex conjugate poles with positive real parts, satisfying the Shilnikov criterion. It is found that both periodic oscillation and chaos appear only at the edge of chaos of the LAM, thus verifying the complexity theorem, no-complexity theorem and edge of chaos theorem. Finally, circuit experiments are performed, which verify the dynamics and the effectiveness of the LAM model, the periodic oscillator and the chaotic oscillator. • A novel bi-directional N-type locally-active memristor model is built. • This memristor shows capacitive, inductive and resistive properties with different operating voltages via small-signal method. • The memristor-based second-order periodic and third-order chaotic oscillators are designed and realized. • The complexity theorem, no-complexity theorem and edge of chaos theorem are verified. [ABSTRACT FROM AUTHOR]