Bifurcation analysis of the Gierer–Meinhardt system with a saturation in the activator production

The reaction–diffusion Gierer–Meinhardt system with a saturation in the activator production is considered. Stability of the unique positive constant steady state solution is analysed, and associated Hopf bifurcations and steady state bifurcations are obtained. A global bifurcation diagram of non-trivial periodic orbits and steady state solutions with respect to key system parameters is obtained, which improves the understanding of dynamics of Gierer–Meinhardt system with a saturation in different parameter regimes.     

Bifurcation analysis of a predator–prey model with memory-based diffusion

In this paper, we investigate the predator–prey model equipped with Fickian diffusion and memory-based diffusion of predators. The stability and bifurcation analysis explores the impacts of the memory-based diffusion and the averaged memory period on the dynamics near the positive steady state. Specifically, when the memory-based diffusion coefficient is less than a critical value, we show that the stability of the positive steady state can be destabilized as the average memory period increases, which leads to the occurrence of Hopf bifurcations. Moreover, we also analyze the bifurcation properties using the central manifold theorem and normal form theory. This allows us to prove the existence of stable spatially inhomogeneous periodic solutions arising from Hopf bifurcation. In addition, the sufficient and necessary conditions for the occurrence of stability switches are also provided.     

Bifurcation analysis in a diffusive Logistic population model with two delayed density-dependent feedback terms

The present paper is concerned with a diffusive population model of Logistic type with an instantaneous density-dependent term and two delayed density-dependent terms and subject to the zero-Dirichlet boundary condition. By regarding the delay as the bifurcation parameter and analyzing in detail the associated eigenvalue problem, the local asymptotic stability and the existence of Hopf bifurcation for the sufficiently small positive steady state solution are shown. It is found that under the suitable condition, the positive steady state solution of the model will become ultimately unstable after a single stability switch (or change) at a certain critical value of delay through a Hopf bifurcation. However, under the other condition, the positive steady state solution of the model will become ultimately unstable after multiple stability switches at some certain critical values of delay through Hopf bifurcations. In addition, the direction of the above Hopf bifurcations and the stability of the bifurcating periodic solutions are analyzed by means of the center manifold theory and normal form method for partial functional differential equations. Finally, in order to illustrate the correction of the obtained theoretical results, some numerical simulations are also carried out.     

Hopf-flip bifurcations of vibratory systems with impacts

Two vibro-impact systems are considered. The period n single-impact motions and Poincaré maps of the vibro-impact systems are derived analytically. Stability and local bifurcations of single-impact periodic motions are analyzed by using the Poincaré maps. A center manifold theorem technique is applied to reduce the Poincaré map to a three-dimensional one, and the normal form map associated with Hopf-flip bifurcation is obtained. It is found that near the point of codim 2 bifurcation there exists not only Hopf bifurcation of period one single-impact motion, but also Hopf bifurcation of period two double-impact motion. Period doubling bifurcation of period one single-impact motion is commonly existent near the point of codim 2 bifurcation. However, no period doubling cascade emerges due to change of the type of period two fixed points and occurrence of Hopf bifurcation associated with period two fixed points. The results from simulation shows that there exists an interest torus doubling bifurcation occurring near the value of Hopf-flip bifurcation. The torus doubling bifurcation makes the quasi-periodic attractor associated with period one single-impact motion transit to the other quasi-periodic attractor represented by two attracting closed circles. The torus bifurcation is qualitatively different from the typical torus doubling bifurcation occurring in the vibro-impact systems.     

Resonance-sensitivity in dynamic Hopf bifurcations under fluid loading

Under steady fluid loading, elastic structures are liable to exhibit dynamic bifurcations to limit cycles: such unimodal instabilities are referred to as galloping while such multimodal instabilities are referred to as flutter. The trace of limit cycles energing from the critical equilibrium state can be either super-critical and stable, in analogy with a stable symmetric static bifurcation, or sub-critical and unstable, in analogy with an unstable symmetric static bifurcation. Galloping of a bluff body in a steady flow can be of the unstable type, and we might expect some form of imperfection sensitivity, although in contrast to static bifurcations, a Hopf bifurcation is actually topologically stable under the operation of a single control parameter: the form of the Hopf bifurcation cannot be rounded off or destroyed by imperfections as in the static case. However, since the dynamic instabilities are associated with a well defined and non-zero circular frequency we might expect the failure ‘load’ to be sensitive to resonant periodic forcing, and this is here shown to be the case, with a two-thirds power law sensitivity analogous to the static cusp.The conclusion is drawn that the concept of structural stability, vital as it is to good mathematical modelling, must be examined with care, particular attention being given to any restrictions on the class of allowable perturbations.     

一类高次自催化耦合反应扩散系统的分歧和斑图

考虑了一类由于自催化剂的耦合而发生的反应扩散系统的空间结构．利用线性化理论讨论了平衡态解的稳定性并且证明了在非耦合系统中空间非一致解出现分歧的必要条件．进一步,利用弱非线性理论讨论了分歧点并且给出了弱耦合系统的图灵分歧解的振幅方程及其性质．     

Steady-state,self-oscillating and chaotic behavior of a PID controlled nonlinear servomechanism by using Bogdanov–Takens and Andronov–Poincaré–Hopf bifurcations

This paper analyzes a controlled servomechanism with feedback and a cubic nonlinearity by means of the Bogdanov–Takens and Andronov–Poincaré–Hopf bifurcations, from which steady-state, self-oscillating and chaotic behaviors will be investigated using the center manifold theorem. The system controller is formed by a Proportional plus Integral plus Derivative action (PID) that allows to stabilize and drive to a prescribed set point a body connected to the shaft of a DC motor. The Bogdanov–Takens bifurcation is analyzed through the second Lyapunov stability method and the harmonic-balance method, whereas the first Lyapunov value is used for the Andronov–Poincaré–Hopf bifurcation. On the basis of the results deduced from the bifurcation analysis, we show a procedure to select the parameters of the PID controller so that an arbitrary steady-state position of the servomechanism can be reached even in presence of noise. We also show how chaotic behavior can be obtained by applying a harmonical external torque to the device in self-oscillating regime. The advantage of achieving chaotic behavior is that it can be used so that the system reaches a set point inside a strange attractor with a small control effort. The analytical calculations have been verified through detailed numerical simulations.     

HARMONIC AND SUBHARMONIC BIFURCATION IN THE BRUSSEL MODEL WITH PERIODIC FORCE

1.IntroductionABrusselatorisoneofthebestexaminedmodelchemicalreactionswhichconsistsoffourstepsItisshowninFig.1schematicallyandisrepreselltedbythefollowingsetofequationsffevedFebruary6,1995.*~workissupportedbytheNationalNaturalScienceFOundationmanYuan"TermsinChina.ThemodelweadoptistheoneduetoPrigogine,Lefever,andNicolis(Brusselator)t'.Fig.1.'TheschematicdiagramofBrusselmodel(AdditionalcirculararrowsrepreseDttheexistenceofautocatalysis.)Herexandystandfortheconcentrationsofreferencereacta…     

A Five-Dimensional System of Navier-Stokes Equation for a Two-Dimensional Incompressible Fluid on a Torus

A five-mode truncation of Navier-Stokes equation for a two-dimensional incompressible fluid on a torus is studied. Its stationary solutions and stability are presented, the existence of attractor and the global stability of the system are discussed. The whole process, which shows a chaos behavior approached through an involved sequence of bifurcations with the changing of Reynolds number, is simulated numerically. Based on numerical simulation results of bifurcation diagram, Lyapunov exponent spectrum, Poincare section, power spectrum and return map of the system are revealed.     

非线性动力系统中两鞍-结分岔点间非稳定曲线的确定

采用将伪弧长延拓法与Poincaré映射法相结合的方法,确定非自治动力系统中两鞍-结分岔点间非稳定曲线,并对采用一般延拓法时出现的奇异性进行了证明。该方法引入了一正则化方程,避免了在求解过程中出现的奇异问题,并给出了相应的迭代格式。在曲线的延拓过程中,由于存在两个延拓方向,为保证将曲线延拓出来,给出了一种确定切线方向的方法,该方法在分析非线性振动系统中的双稳态现象等问题是很有效的。     

Rich dynamical behaviours for predator–prey model with weak Allee effect

The weak Allee effect on the predator is introduced into the classic predator–prey model of Lotka–Volterra type. Global qualitative and bifurcation analyses are combined to determine the global dynamics of the model. It is shown that the weak Allee effect can bring rich and complicated dynamics to the previous simple model, such as the saddle–node bifurcation, subcritical and supercritical Hopf bifurcations, and Bogdanov–Takens bifurcations, implying that weak Allee effect can be one of the simple reasons for many complicated behaviours in the predator–prey communities.     

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.     

Period-doubling and Neimark–Sacker bifurcations in a larch budmoth population model

We investigate a discrete consumer-resource system based on a model originally proposed for studying the cyclic dynamics of the larch budmoth population in the Swiss Alps. It is shown that the moth population can persist indefinitely for all of the biologically feasible parameter values. Using intrinsic growth rate of the consumer population as a bifurcation parameter, we prove that the system can either undergo a period-doubling or a Neimark–Sacker bifurcation when the unique interior steady state loses its stability.     

Period doubling cascades of prey–predator model with nonlinear harvesting and control of over exploitation through taxation

The present study investigates a prey predator type model for conservation of ecological resources through taxation with nonlinear harvesting. The model uses the harvesting function as proposed by Agnew (1979) [1] which accounts for the handling time of the catch and also the competition between standard vessels being utilized for harvesting of resources. In this paper we consider a three dimensional dynamic effort prey–predator model with Holling type-II functional response. The conditions for uniform persistence of the model have been derived. The existence and stability of bifurcating periodic solution through Hopf bifurcation have been examined for a particular set of parameter value. Using numerical examples it is shown that the system admits periodic, quasi-periodic and chaotic solutions. It is observed that the system exhibits periodic doubling route to chaos with respect to tax. Many forms of complexities such as chaotic bands (including periodic windows, period-doubling bifurcations, period-halving bifurcations and attractor crisis) and chaotic attractors have been observed. Sensitivity analysis is carried out and it is observed that the solutions are highly dependent to the initial conditions. Pontryagin’s Maximum Principle has been used to obtain optimal tax policy to maximize the monetary social benefit as well as conservation of the ecosystem.     

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.     

Multiple bifurcations in a delayed predator–prey diffusion system with a functional response

The present paper is concerned with a delayed predator–prey diffusion system with a Beddington–DeAngelis functional response and homogeneous Neumann boundary conditions. If the positive constant steady state of the corresponding system without delay is stable, by choosing the delay as the bifurcation parameter, we can show that the increase of the delay can not only cause spatially homogeneous Hopf bifurcation at the positive constant steady state but also give rise to spatially heterogeneous ones. In particular, under appropriate conditions, we find that the system has a Bogdanov–Takens singularity at the positive constant steady state, whereas this singularity does not occur for the corresponding system without diffusion. In addition, by applying the normal form theory and center manifold theorem for partial functional differential equations, we give normal forms of Hopf bifurcation and Bogdanov–Takens bifurcation and the explicit formula for determining the properties of spatial Hopf bifurcations.     

多重时滞富营养化生态模型的稳定性与分支分析

本文研究了多重时滞富营养化生态模型的稳定性与分支问题.利用特征值方法,分别研究了具有单时滞和双时滞模型的线性稳定性.发现当模型中的时滞经过一系列临界值时,模型在平衡点附近经历了Hopf分支和Hopf-zero分支,并给出Hopf分支和Hopf-zero分支存在的充分条件.最后数值模拟验证了理论结果.     

Bifurcation analysis of a generalized Gause model with prey harvesting and a generalized Holling response function of type III

In this paper we study a generalized Gause model with prey harvesting and a generalized Holling response function of type III: . The goal of our study is to give the bifurcation diagram of the model. For this we need to study saddle-node bifurcations, Hopf bifurcation of codimension 1 and 2, heteroclinic bifurcation, and nilpotent saddle bifurcation of codimension 2 and 3. The nilpotent saddle of codimension 3 is the organizing center for the bifurcation diagram. The Hopf bifurcation is studied by means of a generalized Liénard system, and for b=0 we discuss the potential integrability of the system. The nilpotent point of multiplicity 3 occurs with an invariant line and can have a codimension up to 4. But because it occurs with an invariant line, the effective highest codimension is 3. We develop normal forms (in which the invariant line is preserved) for studying of the nilpotent saddle bifurcation. For b=0, the reversibility of the nilpotent saddle is discussed. We study the type of the heteroclinic loop and its cyclicity. The phase portraits of the bifurcations diagram (partially conjectured via the results obtained) allow us to give a biological interpretation of the behavior of the two species.     

Bifurcation analysis and control of a class of hybrid biological economic models

This paper systematically studies a hybrid predator–prey economic model, which is formulated by differential-difference-algebraic equations. It shows that this model exhibits two bifurcation phenomena at the intersampling instants. One is saddle–node bifurcation, and the other is singular induced bifurcation which indicates that economic profit may bring impulse at some critical value, i.e., rapid expansion of biological population in terms of ecological implications. On the other hand, for the sampling instants, the system undergoes Neimark–Sacker bifurcation at a critical value of economic profit, i.e., the increase of economic profit destabilizes the system and generates a unique closed invariant curve. Moreover, the state feedback controller is designed so that singular induced bifurcation and Neimark–Sacker bifurcation can be eliminated and the population can be driven to steady states by adjusting harvesting costs and the economic profit. At the same time, by using Matlab software, numerical simulations illustrate the effectiveness of the results obtained here.     

Existence and asymptotic behavior of solutions for a mathematical ecology model with herd behavior

In this work, we consider a prey-predator model with herd behavior under Neumann boundary conditions. For the system without diffusion, we establish a sufficient condition to guarantee the local asymptotic stability of all nontrivial equilibria and prove the existence of limit cycle of our proposed model. For the system with diffusion, we consider the long time behavior of the model including global attractor and local stability, and the Hopf and steady-state bifurcation analysis from the unique homogeneous positive steady state are carried out in detail. Furthermore, some numerical simulations to illustrate the theoretical analysis are performed to expand our theoretical results.