Dynamics of a Discrete Two-species Competitive Model with Michaelies-Menten Type Harvesting in the First Species

In this paper, we use a semidiscretization method to derive a discrete two-species competitive model with Michaelis-Menten type harvesting in the first species. First, the existence and local stability of fixed points of the system are investigated by employing a key lemma. Subsequently, the transcritical bifurcation, period-doubling bifurcation and pitchfork bifurcation of the model are investigated by using the Center Manifold Theorem and bifurcation theory. Finally, numerical simulations are presented to illustrate corresponding theoretical results.     

Nonlinear analysis of a novel three-scroll chaotic system

This paper reports the nonlinear dynamics of a novel three-scroll chaotic system. The local stability of hyperbolic equilibrium and non-hyperbolic equilibrium are investigated by using center manifold theorem. Pitchfork bifurcation, degenerate pitchfork bifurcation and Hopf bifurcation are analyzed when the parameters are varied in the space of parameter. For a suitable choice of the parameters, the existence of singularly degenerate heteroclinic cycles and Hopf bifurcation without parameters are also investigated. Some numerical simulations are given to support the analytic results.     

Codimension two bifurcation in a delayed neural network with unidirectional coupling

In this paper, a delayed neural network model with unidirectional coupling is considered. Zero–Hopf bifurcation is studied by using the center manifold reduction and the normal form method for retarded functional differential equation. We get the versal unfolding of the norm form at the zero–Hopf singularity and show that the model can exhibit pitchfork, Hopf bifurcation, and double Hopf bifurcation is also found to occur in this model. Some numerical simulations are given to support the analytic results.     

Complex dynamics in Chen’s system

This paper characterizes some complex dynamics of Chen’s system. Some conditions of existence for pitchfork bifurcation and Hopf bifurcation are derived by using bifurcation theory and the center manifold theorem. Numerical simulation results not only show consistence with the theoretical analysis but also display some new and interesting dynamical behaviors including homoclinic bifurcation and the coexistence of two stable limit cycles and one chaotic attractor as well as some periodic solutions emerging from Hopf bifurcation but ending in homoclinic bifurcation, which are different from those reported in the literature before. All these show that Chen’s system has very rich nonlinear dynamics.     

Dynamics of two-cell systems with discrete delays

We consider the system of delay differential equations (DDE) representing the models containing two cells with time-delayed connections. We investigate global, local stability and the bifurcations of the trivial solution under some generic conditions on the Taylor coefficients of the DDE. Regarding eigenvalues of the connection matrix as bifurcation parameters, we obtain codimension one bifurcations (including pitchfork, transcritical and Hopf bifurcation) and Takens-Bogdanov bifurcation as a codimension two bifurcation. For application purposes, this is important since one can now identify the possible asymptotic dynamics of the DDE near the bifurcation points by computing quantities which depend explicitly on the Taylor coefficients of the original DDE. Finally, we show that the analytical results agree with numerical simulations.     

On catastrophe and cavitation for spherical cavity

This work deals with catastrophe of a spherical cavity and cavitation of a spherical cavity for Hooke material with 1/2 Poisson's ratio. A nonlinear problem, which is the Cauchy traction problem, is solved analytically. The governing equations are written on the deformed region or on the present configuration. And the conditions are described on moving boundary. A closed form solution is found. Furthermore, a bifurcation solution in closed form is given from the trivial homogeneous solution of a solid sphere. The results indicate that there is a tangent bifurcation on the displacement-load curve for a sphere with a cavity. On the tangent bifurcation point, the cavity grows up suddenly, which is a kind of catastrophe. And there is a pitchfork bifurcation on the displacement-load curve for a solid sphere. On the pitchfork bifurcation point, there is a cavitation in the solid sphere.     

Hopf‐pitchfork bifurcation in a two‐neuron system with discrete and distributed delays

Both discrete and distributed delays are considered in a two‐neuron system. We analyze the influence of interaction coefficient and time delay on the Hopf‐pitchfork bifurcation. First, we obtain the codimension‐2 unfolding with original parameters for Hopf‐pitchfork bifurcation by using the center manifold reduction and the normal form method. Next, through analyzing the unfolding structure, we give complete bifurcation diagrams and phase portraits, in which multistability and other dynamical behaviors of the original system are found, such as a stable periodic orbit, the coexistence of two stable nontrivial equilibria, and the coexistence of a stable periodic orbit and two stable equilibria. In addition, the obtained theoretical results are verified by numerical simulations. Finally, we perform the comparisons of the obtained results of Hopf‐pitchfork bifurcation with other Hopf‐fold bifurcation results in some biological neural systems and give the obtained mathematical results corresponding to the physical states of neurons. Copyright © 2015 JohnWiley & Sons, Ltd.     

Dynamic complexities in a parasitoid-host-parasitoid ecological model

Chaotic dynamics have been observed in a wide range of population models. In this study, the complex dynamics in a discrete-time ecological model of parasitoid-host-parasitoid are presented. The model shows that the superiority coefficient not only stabilizes the dynamics, but may strongly destabilize them as well. Many forms of complex dynamics were observed, including pitchfork bifurcation with quasi-periodicity, period-doubling cascade, chaotic crisis, chaotic bands with narrow or wide periodic window, intermittent chaos, and supertransient behavior. Furthermore, computation of the largest Lyapunov exponent demonstrated the chaotic dynamic behavior of the model.     

Poisson比为1/2材料中球形空穴突变和球形空穴萌生的分岔问题研究

研究Ｐｏｉｓｓｏｎ比为１／２的Ｈｏｏｋｅ材料中,空穴的突变和萌生现象·求解一个球对称几何非线性弹性力学的移动边界（ｍｏｖｉｎｇｂｏｕｎｄａｒｙ）问题,空穴为球形,远离空穴处为三向均匀拉伸应力状态,在当前构形上列控制方程;在当前构形边界上列边界条件·找到了这个自由边界问题的封闭解并得到空穴半径趋于零时的叉型分岔解·计算结果显示,在位移＿载荷曲线上存在一个切分岔型分岔点（或鞍结点型分岔点、极值型分岔点）,这个分岔点说明在外力作用下空穴会发生突变,即突然“长大”;当球腔半径趋于零时,这个切分岔转化为叉型分岔（或分枝型分岔）,这个叉型分岔可以解释实心球中的空穴萌生现象     

A Direct Method for Pitchfork Bifurcation Points

To overcome the difficulty caused by the singularity at the pitchfork bifurcation points, we introduce the homotopy parameter so that the problem of computing the pitchfork bifurcation points can be transferred to that of computing the fold points of degree 3 with respect to the homotopy parameter. An extended system for pitchfork bifurcation points is given. The regularity of the extended system is proved. Finally, the numerical examples show the effectiveness of our method.     

Complex dynamics in a two-dimensional noninvertible map

A two-dimensional noninvertible map is investigated. The conditions of existence for pitchfork bifurcation, flip bifurcation and Naimark–Sacker bifurcation are derived by using center manifold theorem and bifurcation theory. Chaotic behavior in the sense of Marotto’s definition of chaos is proven. And numerical simulations not only show the consistence with the theoretical analysis but also exhibit the complex dynamical behaviors, including period-34, period-5 orbits, quasi-period orbits, intermittency, boundary crisis as well as chaotic transient. The computation of Lyapunov exponents conforms the dynamical behaviors.     

Role of transmissible disease in an infected prey-dependent predator – prey system

The role of disease in ecological systems is a very important issue from both mathematical and ecological points of view. This paper deals with the qualitative analysis of a prey-dependent predator – prey system in which a disease is spreading among the prey species only. We have analysed the behaviour of the system around each equilibrium and obtained conditions for global stability of the system around an equilibrium by using suitable Lypunov functions. We have also worked out the region of parametric space under which the system enters a Hopf bifurcation and a transcritical bifurcation but does not experience either saddle-node bifurcations or pitchfork bifurcations around the disease-free equilibrium E ₂. Finally, we have given an example of a real ecological situation with experimental data simulations.     

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.     

Some new insights into a known Chen‐like system

In the paper entitled ‘A novel chaotic system and its topological horseshoe’ in [Nonlinear Analysis: Modelling and Control 18 (1) (2013) 66–77], proposed the 3D chaotic system, , and discussed some of its dynamics according to theoretical and numerical analysis of its parameters . The present work is devoted to giving some new insights into the system for b≥0. Combining theoretical analysis and numerical simulations, some new results are formulated. On the one hand, after some known errors, mainly the distribution of its equilibrium point which is pointed out, correct results are formulated. On the other hand, some of its more rich dynamical properties hiding and not found previously, such as the stability, fold bifurcation, pitchfork bifurcation, degenerated pitchfork bifurcation, and Hopf bifurcation of its isolated equilibria, the dynamics of non‐isolated equilibria, the singularly degenerate heteroclinic cycle, the heteroclinic orbit, and the dynamics at infinity are clearly revealed. Using these results, one can easily explain those interesting phenomena for invariant Lyapunov exponent spectrum and amplitude control that are presented in the known literature. What is more important, we probably demonstrate a new route to chaos. Copyright © 2015 John Wiley & Sons, Ltd.     

The dynamical behaviors of a Ivlev-type two-prey two-predator system with impulsive effect

In this paper, considering the strategy of integrated Pest Management (IPM), a class of two-prey two-predator system with the Ivlev-type functional response and impulsive effect at different fixed time is established. By using impulsive comparison theorem, Floquent theory and small amplitude perturbation skill, the sufficient conditions for the system to be extinct of prey and permanence are proved. Moreover, we give two sufficient conditions for the extinction of one of two prey and remaining three species are permanent. Numerical simulation shows that there exist complex dynamics for system, such as symmetry-breaking pitchfork bifurcation, periodic doubling bifurcation, chaos, periodic halving cascade. Lastly, a brief discussion is given.     

A Bayesian estimation of unfolded pitchfork bifurcation structure based upon experimental data

We consider the practical problem of estimating parameter valuesfor the unfolding of a pitchfork bifurcation from observed data.Although the existence of a bifurcation point may have beentheoretically determined, observed data from experiments ormeasurement must reflect the natural unfolding due to existingperturbations and imperfections. We demonstrate how a Bayesianapproach provides a natural formulation to this problem andhow incremental observations alter our best assessment of thetrue, unfolded, bifurcation structure, close to the prior estimate(the pitchfork). This method may be useful in practice to estimatethe location of bifurcation branches other than a particularone experimentally observed.     

Dynamics of an intraguild predation food web model with strong Allee effect in the basal prey

Since intraguild predation (IGP) is a ubiquitous and important community module in nature and Allee effect has strong impact on population dynamics, in this paper we propose a three-species IGP food web model consisted of the IG predator, IG prey and basal prey, in which the basal prey follows a logistic growth with strong Allee effect. We investigate the local and global dynamics of the model with emphasis on the impact of strong Allee effect. First, positivity and boundedness of solutions are studied. Then existence and stability of the boundary and interior equilibria are presented and the Hopf bifurcation curve at an interior equilibrium is given. The existence of a Hopf bifurcation curve indicates that if competition between the IG prey and IG predator for the basal resource lies below the curve then the interior equilibrium remains stable, while if it lies above the curve then the interior equilibrium loses its stability. In order to explore the impact of Allee effect, the parameter space is classified into sixteen different regions and, in each region, the number of interior equilibria is determined and the corresponding bifurcation diagrams on the Allee threshold are given. The extinction parameter regions of at least one species and the necessary coexistence parameter regions of all three species are provided. In addition, we explore possible dynamical patterns, i.e., the existence of multiple attractors. By theoretical analysis and numerical simulations, we show that the model can have one (i.e. extinction of all species), two (i.e. bi-stability) or three (i.e. tri-stability) attractors. It is also found by simulations that when there exists a unique stable interior equilibrium, the model may generate multiple attracting periodic orbits and the coexistence of all three species is enhanced as the competition between the IG prey and IG predator for the basal resource is close to the Hopf bifurcation curve from below. Our results indicate that the intraguild predation food web model exhibits rich and complex dynamic behaviors and strong Allee effect in the basal prey increases the extinction risk of not only the basal prey but also the IG prey or/and IG predator.     

Normal form for the symmetry-breaking bifurcation in the nonlinear Schrödinger equation

We derive and justify a normal form reduction of the nonlinear Schrödinger equation for a general pitchfork bifurcation of the symmetric bound state that occurs in a double-well symmetric potential. We prove persistence of normal form dynamics for both supercritical and subcritical pitchfork bifurcations in the time-dependent solutions of the nonlinear Schrödinger equation over long but finite time intervals.     

Homoclinic bifurcation with nonhyperbolic equilibria

The problem of homoclinic bifurcation is studied for a high dimensional system with nonhyperbolic equilibria. By constructing local coordinate systems near the unperturbed homoclinic orbit, Poincaré maps for the new system are established. Then the persistence of the homoclinic orbit and the bifurcation of the periodic orbit for the system accompanied with pitchfork bifurcation are obtained. Some known results are extended.