Dynamical analysis of a Lotka-Volterra learning-process model

A Lotka-Volterra learning-process model was proposed by Monteiro and Notargiacomo in [{\it Commum. Nonlinear Sci. Numer. Simulat.} {\bf 47}(2017), 416-420] to approach learning process as an interplay between understanding and doubt. They studied the stability of the boundary equilibria and gave some numerical simulations but no further discussion for bifurcations. In this paper, we study the qualitative properties of the interior equilibria and a singular line segment completely. Moreover, we discuss their bifurcations such as transcritical, pitchfork, Hopf bifurcation on isolated equilibria and transcritical bifurcation without parameters on non-isolated equilibria. Finally, we also demonstrate these analytical theory by numerical simulations.     

Bogdanov‐Takens bifurcations of codimensions 2 and 3 in a Leslie‐Gower predator‐prey model with Michaelis‐Menten–type prey harvesting

The Bogdanov‐Takens bifurcations of a Leslie‐Gower predator‐prey model with Michaelis‐Menten–type prey harvesting were studied. In the paper “Diff. Equ. Dyn. Syst. 20(2012), 339‐366,” Gupta et al proved that the Leslie‐Gower predator‐prey model with Michaelis‐Menten–type prey harvesting has rich dynamics. Some equilibria of codimension 1 and their bifurcations were discussed. In this paper, we find that the model has an equilibrium of codimensions 2 and 3. We also prove analytically that the model undergoes Bogdanov‐Takens bifurcations (cusp cases) of codimensions 2 and 3. Hence, the model can have 2 limit cycles, coexistence of a stable homoclinic loop and an unstable limit cycle, supercritical and subcritical Hopf bifurcations, and homoclinic bifurcation of codimension 1 as the values of parameters vary. Moreover, several numerical simulations are conducted to illustrate the validity of our results.     

Stability and Hopf bifurcation analysis of an eco-epidemic model with a stage structure

In this paper, an eco-epidemiological model with a stage structure is considered. The asymptotical stability of the five equilibria, the existence of stability switches about positive equilibrium, is investigated. It is found that Hopf bifurcation occurs when the delay τ passes though a critical value. Some explicit formulae determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal form theory and center manifold theory. Some numerical simulations for justifying the theoretical analysis are also provided. Finally, biological explanations and main conclusions are given.     

Global and Bifurcation Analysis of an HIV Pathogenesis Model with Saturated Reverse Function

In this paper, an HIV dynamics model with the proliferation of CD4 T cells is proposed. The authors consider nonnegativity, boundedness, global asymptotic stability of the solutions and bifurcation properties of the steady states. It is proved that the virus is cleared from the host under some conditions if the basic reproduction number R_0 is less than unity. Meanwhile, the model exhibits the phenomenon of backward bifurcation. We also obtain one equilibrium is semi-stable by using center manifold theory. It is proved that the endemic equilibrium is globally asymptotically stable under some conditions if R_0 is greater than unity. It also is proved that the model undergoes Hopf bifurcation from the endemic equilibrium under some conditions. It is novelty that the model exhibits two famous bifurcations,backward bifurcation and Hopf bifurcation. The model is extended to incorporate the specific Cytotoxic T Lymphocytes(CTLs) immune response. Stabilities of equilibria and Hopf bifurcation are considered accordingly. In addition, some numerical simulations for justifying the theoretical analysis results are also given in paper.     

Two-parameter bifurcations in a network of two neurons with multiple delays

We consider a network of two coupled neurons with delayed feedback. We show that the connection topology of the network plays a fundamental role in classifying the rich dynamics and bifurcation phenomena. Regarding eigenvalues of the connection matrix as bifurcation parameters, we obtain codimension 1 bifurcations (including a fold bifurcation and a Hopf bifurcation) and codimension 2 bifurcations (including fold-Hopf bifurcations and Hopf-Hopf bifurcations). We also give concrete formulae for the normal form coefficients derived via the center manifold reduction that give detailed information about the bifurcation and stability of various bifurcated solutions. In particular, we obtain stable or unstable equilibria, periodic solutions, quasi-periodic solutions, and sphere-like surfaces of solutions. We also show how to evaluate critical normal form coefficients from the original system of delay-differential equations without computing the corresponding center manifolds.     

Local bifurcations of nonlinear viscoelastic panel in supersonic flow

The stability and bifurcation behaviors of a two-dimensional nonlinear viscoelastic panel in supersonic flow are investigated with analytical and numerical methods. One type of critical points for the bifurcation response equations is considered, which is characterized by a pair of purely imaginary eigenvalues and a pair of complex conjugate eigenvalues having negative real part. With the aid of computer language Maple and the normal form theory, Hopf bifurcation solution of the model is investigated. Finally, numerical simulations are shown, which agree with the theoretical analytical results.     

Local and global bifurcations in an SIRS epidemic model

This paper studies the existence and stability of the disease-free equilibrium and endemic equilibria for the SIRS epidemic model with the saturated incidence rate, considering the factor of population dynamics such as the disease-related, the natural mortality and the constant recruitment of population. Analytical techniques are used to show, for some parameter values, the periodic solutions can arise through the Hopf bifurcation, which is important to carry different strategies for the controlling disease. Then the codimension-two bifurcation, i.e. BT bifurcation, is investigated by using a global qualitative method and the curves of saddle-node bifurcation, Hopf bifurcation and homoclinic bifurcation are obtained at the degenerate equilibrium. Moreover, several numerical simulations are given to support the theoretical analysis.     

Complex bifurcations in the oscillatory reaction model

The mixing of different types of bifurcations, i.e. supercritical Andronov–Hopf (SAH), double loop (DL) and saddle-loop (SL) bifurcations in the vicinity of their total annihilation, is examined on the highly nonlinear six-variable model for the Bray–Liebhafsky (BL) oscillatory reaction under continuously well-stirred tank reactor (CSTR) conditions. For this kind of the reaction system where the law of mass conservation is additional constraint that must be satisfied and where because of that, some simple bifurcations cannot be formed independently to the others, the considered transformations of the bifurcations are particularly important. That is why as the control parameters for bifurcation analysis, the specific flow rate (j₀), as well as the inflow hydrogen peroxide concentration (h = [H₂O₂]_in), were used. The complex bifurcations obtained from numerical simulations are compared with some experimental results. It was shown that these complex bifurcations cannot be easily recognized in experimental investigations without knowing their evolution.     

Delay induced subcritical Hopf bifurcation in a diffusive predator-prey model with herd behavior and hyperbolic mortality

In this paper, we consider the dynamics of a delayed diffusive predator-prey model with herd behavior and hyperbolic mortality under Neumann boundary conditions. Firstly, by analyzing the characteristic equations in detail and taking the delay as a bifurcation parameter, the stability of the positive equilibria and the existence of Hopf bifurcations induced by delay are investigated. Then, applying the normal form theory and the center manifold argument for partial functional differential equations, the formula determining the properties of the Hopf bifurcation are obtained. Finally, some numerical simulations are also carried out and we obtain the unstable spatial periodic solutions, which are induced by the subcritical Hopf bifurcation.     

具阶段结构的多时滞SIR模型的稳定性分析

考虑了一类具有两个阶段结构的SIR模型,得到了解的正性和有界性,通过分析特征方程根的分布,以(?)_1和(?)_2为参数分析了平衡点的稳定性和局部Hopf分支存在性.进一步地,利用规范型和中心流型理论,给出了决定Hopf分支方向和分支周期解的稳定性的隐式算法.最后利用一些数值模拟来支持所得到的理论分析结果.     

Dynamics and bifurcations of a modified Leslie‐Gower–type model considering a Beddington‐DeAngelis functional response

In this paper, a planar system of ordinary differential equations is considered, which is a modified Leslie‐Gower model, considering a Beddington‐DeAngelis functional response. It generates a complex dynamics of the predator‐prey interactions according to the associated parameters. From the system obtained, we characterize all the equilibria and its local behavior, and the existence of a trapping set is proved. We describe different types of bifurcations (such as Hopf, Bogdanov‐Takens, and homoclinic bifurcation), and the existence of limit cycles is shown. Analytic proofs are provided for all results. Ecological implications and a set of numerical simulations supporting the mathematical results are also presented.     

Codimension one and two bifurcations of a discrete stage-structured population model with self-limitation

In this paper, the bifurcations of a discrete stage-structured population model with self-limitation between the two subgroups are investigated. We explore all possible codimension-one bifurcations associated with transcritical, flip (period doubling) and Neimark-Sacker bifurcations and discuss the stabilities of the fixed points in these non-hyperbolic cases. Meanwhile, we give the explicit approximate expression of the closed invariant curve which is caused by the Neimark-Sacker bifurcation. After that, through the theory of approximation by a flow, we explore the codimension two bifurcations associated with 1:3 strong resonance. We convert the nondegenerate condition of 1:3 resonance into a parametric polynomial, and determine its sign by the theory of complete discrimination system. We introduce new parameters and utilize some variable substitutions to obtain the bifurcation curves around 1:3 resonance, which are returned to the original variables and parameters to express for easy verification. By using a series of complicated approximate identity transformations and polar coordinate transformation, we explore 1:6 weak resonance. Moreover, we calculate the two boundaries of Arnold tongue which are caused by 1:6 weak resonance and defined as the resonance region. Numerical simulations and numerical bifurcation analyzes are made to demonstrate the effective of the theoretical analyzes and to present the relations between these bifurcations. Furthermore, our theoretical analyzes and numerical simulations are explained from the biological point of view.     

Codimension two bifurcations of fixed points in a class of vibratory systems with symmetrical rigid stops

A vibratory system having symmetrically placed rigid stops and subjected to periodic excitation is considered. Local codimension two bifurcations of the vibratory system with symmetrical rigid stops, associated with double Hopf bifurcation and interaction of Hopf and pitchfork bifurcation, are analyzed by using the center manifold theorem technique and normal form method of maps. Dynamic behavior of the system, near the points of codimension two bifurcations, is investigated by using qualitative analysis and numerical simulation. Hopf-flip bifurcation of fixed points in the vibratory system with a single stop are briefly analyzed by comparison with unfoldings analyses of Hopf-pitchfork bifurcation of the vibratory system with symmetrical rigid stops. Near the value of double Hopf bifurcation there exist period-one double-impact symmetrical motion and quasi-periodic impact motions. The quasi-periodic impact motions are represented by the closed circle and “tire-like” attractor in projected Poincaré sections. With change of system parameters, the quasi-periodic impact motions usually lead to chaos via “tire-like” torus doubling.     

Hopf bifurcations in a predator–prey system with a discrete delay and a distributed delay

A delayed Lotka–Volterra two-species predator–prey system with discrete hunting delay and distributed maturation delay for the predator population described by an integral with a strong delay kernel is considered. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the positive equilibrium is investigated and Hopf bifurcations are demonstrated. It is found that under suitable conditions on the parameters the positive equilibrium is asymptotically stable when the hunting delay is less than a certain critical value and unstable when the hunting delay is greater than this critical value. Meanwhile, according to the Hopf bifurcation theorem for functional differential equations (FDEs), we find that the system can also undergo a Hopf bifurcation of nonconstant periodic solution at the positive equilibrium when the hunting delay crosses through a sequence of critical values. In particular, by applying the normal form theory and the center manifold reduction for FDEs, an explicit algorithm determining the direction of Hopf bifurcations and the stability of bifurcating periodic solutions occurring through Hopf bifurcations is given. Finally, to verify our theoretical predictions, some numerical simulations are also included at the end of this paper.     

Synchronous dynamics of two coupled oscillators with inhibitory-to-inhibitory connection

We investigate the behaviour of a neural network model consisting of two coupled oscillators with delays and inhibitory-to-inhibitory connections. We consider the absolute synchronization and show that the connection topology of the network plays a fundamental role in classifying the rich dynamics and bifurcation phenomena. Regarding eigenvalues of the connection matrix as bifurcation parameters, we obtain codimension one bifurcations (including fold bifurcation and Hopf bifurcation) and codimension two bifurcation (including fold-Hopf bifurcations and Hopf–Hopf bifurcations). Based on the normal form theory and center manifold reduction, we obtain detailed information about the bifurcation direction and stability of various bifurcated equilibria as well as periodic solutions with some kinds of spatio-temporal patterns. Numerical simulation is also given to support the obtained results.     

Effects of Hard Limits on Bifurcation, Chaos and Stability

An SMIB model in the power systems,especially that concering the effects of hard limits onbifurcations,chaos and stability is studied.Parameter conditions for bifurcations and chaos in the absence ofhard limits are compared with those in the presence of hard limits.It has been proved that hard limits can affectsystem stability.We find that (1) hard limits can change unstable equilibrium into stable one;(2) hard limits canchange stability of limit cycles induced by Hopf bifurcation;(3) persistence of hard limits can stabilize divergenttrajectory to a stable equilibrium or limit cycle;(4) Hopf bifurcation occurs before SN bifurcation,so the systemcollapse can be controlled before Hopf bifurcation occurs.We also find that suitable limiting values of hard limitscan enlarge the feasibility region.These results are based on theoretical analysis and numerical simulations,such as condition for SNB and Hopf bifurcation,bifurcation diagram,trajectories,Lyapunov exponent,Floquetmultipliers,dimension of attractor and so on.     

Global dynamics of Nicholsonʼs blowflies equation revisited: Onset and termination of nonlinear oscillations

We revisit Nicholson?s blowflies model with natural death rate incorporated into the delay feedback. We consider the delay as a bifurcation parameter and examine the onset and termination of Hopf bifurcations of periodic solutions from a positive equilibrium. We show that the model has only a finite number of Hopf bifurcation values and we describe how branches of Hopf bifurcations are paired so the existence of periodic solutions with specific oscillation frequencies occurs only in bounded delay intervals. The bifurcation analysis and the Matlab package DDE-BIFTOOL developed by Engelborghs et al. guide some numerical simulations to identify ranges of parameters for coexisting multiple attractive periodic solutions.     

Controlling Hopf bifurcations of discrete-time systems in resonance

Resonance in Hopf bifurcation causes complicated bifurcation behaviors. To design with certain desired Hopf bifurcation characteristics in the resonance cases of discrete-time systems, a feedback control method is developed. The controller is designed with the aid of discrete-time washout filters. The control law is constructed according to the criticality and stability conditions of Hopf bifurcations as well as resonance constraints. The control gains associated with linear control terms insure the creation of a Hopf bifurcation in resonance cases and the control gains associated with nonlinear control terms determine the type and stability of bifurcated solutions. To derive the former, we propose the implicit criteria of eigenvalue assignment and transversality condition for creating the bifurcation in a desired parameter location. To derive the latter, the technique of the center manifold reduction, Iooss’s Hopf bifurcation theory and Wan’s Hopf bifurcation theory for resonance cases are employed. In numerical experiments, we show the Hopf circles and fixed points from the created Hopf bifurcations in the strong and weak resonance cases for a four-dimensional control system.     

Hopf bifurcations to quasi-periodic solutions for the two-dimensional plane Poiseuille flow

This paper studies various Hopf bifurcations in the two-dimensional plane Poiseuille problem. For several values of the wavenumber α, we obtain the branch of periodic flows which are born at the Hopf bifurcation of the laminar flow. It is known that, taking α ≈ 1, the branch of periodic solutions has several Hopf bifurcations to quasi-periodic orbits. For the first bifurcation, calculations from other authors seem to indicate that the bifurcating quasi-periodic flows are stable and subcritical with respect to the Reynolds number, Re. By improving the precision of previous works we find that the bifurcating flows are unstable and supercritical with respect to Re. We have also analysed the second Hopf bifurcation of periodic orbits for several α, to find again quasi-periodic solutions with increasing Re. In this case the bifurcated solutions are stable to superharmonic disturbances for Re up to another new Hopf bifurcation to a family of stable 3-tori. The proposed numerical scheme is based on a full numerical integration of the Navier-Stokes equations, together with a division by 3 of their total dimension, and the use of a pseudo-Newton method on suitable Poincaré sections. The most intensive part of the computations has been performed in parallel. We believe that this methodology can also be applied to similar problems.     

Bifurcation analysis of a Leslie-type predator–prey system with simplified Holling type IV functional response and strong Allee effect on prey

In this paper, a Leslie-type predator–prey system with simplified Holling type IV functional response and strong Allee effect on prey is proposed. The dissipativity of the system and the existence of all possible equilibria are investigated. The investigation emphasizes the exploring of bifurcation. It is shown that the system exists several non-hyperbolic positive equilibria, such as a weak focus of multiplicities one and two, (degenerate) saddle–nodes and Bogdanov–Takens singularities (cusp case) of codimensions 2 and 3. At these equilibria, it is proved that the system undergoes various kinds of bifurcations, such as saddle–node bifurcation, Hopf bifurcation, degenerate Hopf bifurcation and Bogdanov–Takens bifurcation of codimensions 2 and 3. With the parameters selected properly, there exhibits a limit cycle, a homoclinic loop, two limit cycles, a semistable limit cycle, or the simultaneous occurrence of a homoclinic loop and a limit cycle in the system. Moreover, it is also proved that the system has a cusp of codimension at least 4. Hence, there may exist three limit cycles generated from Hopf bifurcation of codimension 3. Numerical simulations are done to support the theoretical results.