Bifurcation analysis for a regulated logistic growth model

In this paper, we consider a regulated logistic growth model. We first consider the linear stability and the existence of a Hopf bifurcation. We show that Hopf bifurcations occur as the delay τ passes through critical values. Then, using the normal form theory and center manifold reduction, we derive the explicit algorithm determining the direction of Hopf bifurcations and the stability of the bifurcating periodic solutions. Finally, numerical simulation results are given to support the theoretical predictions.     

Synchronous Dynamics and Bifurcation Analysis in Two Delay Coupled Oscillators with Recurrent Inhibitory Loops

In this study, the dynamics and low-codimension bifurcation of the two delay coupled oscillators with recurrent inhibitory loops are investigated. We discuss the absolute synchronization character of the coupled oscillators. Then the characteristic equation of the linear system is examined, and the possible low-codimension bifurcations of the coupled oscillator system are studied by regarding eigenvalues of the connection matrix as bifurcation parameter, and the bifurcation diagram in the γ–ρ plane is obtained. Applying normal form theory and the center manifold theorem, the stability and direction of the codimension bifurcations are determined. Moreover, the symmetric bifurcation theory and representation theory of Lie groups are used to investigate the spatio-temporal patterns of the periodic oscillations. Finally, numerical results are applied to illustrate the results obtained.     

On the Analysis of Stability,Bifurcation, Chaos and Chaos Control of Kopel Map

A dynamical system of two nonlinear difference equations introduced by Kopel is studied (Kopel, M., Simple and complex adjustment dynamics in Cournot Duopoly model, Chaos, Solitons and Fractals, 1996, 7 (12), 2031–2048.). In this study, we determine conditions for the stability of the fixed points and we study the bifurcations and chaos (Devany, R. L., Introduction to Chaotic Dynamical Systems. Addison–Wesley, New York, 1989.) by computing the maximum Laypunov exponents (Luhta, I. and Virtanen, I., Chaos, Solitons and Fractals, 1996, 7, 2083.). The OGY chaos control formula is used to control chaos in this system.     

On the period of the limit cycles appearing in one-parameter bifurcations

The generic isolated bifurcations for one-parameter families of smooth planar vector fields {X_μ} which give rise to periodic orbits are: the Andronov-Hopf bifurcation, the bifurcation from a semi-stable periodic orbit, the saddle-node loop bifurcation and the saddle loop bifurcation. In this paper we obtain the dominant term of the asymptotic behaviour of the period of the limit cycles appearing in each of these bifurcations in terms of μ when we are near the bifurcation. The method used to study the first two bifurcations is also used to solve the same problem in another two situations: a generalization of the Andronov-Hopf bifurcation to vector fields starting with a special monodromic jet; and the Hopf bifurcation at infinity for families of polynomial vector fields.     

Spherically symmetric internal layers for activator-inhibitor systems II. Stability and symmetry breaking bifurcations

A reaction-diffusion system of activator-inhibitor type is studied on an N-dimensional ball with the homogeneous Neumann boundary conditions. We analyze the stability property of the spherically symmetric solutions and their symmetry-breaking bifurcations into layer solutions which are not spherically symmetric.     

Bifurcations for a predator-prey system with two delays

In this paper, a predator-prey system with two delays is investigated. By choosing the sum τ of two delays as a bifurcation parameter, we show that Hopf bifurcations can occur as τ crosses some critical values. By deriving the equation describing the flow on the center manifold, we can determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. In addition, special attention is paid to the global continuation of local Hopf bifurcations. Using a global Hopf bifurcation result of [J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc. 350 (1998) 4799-4838], we may show the global existence of periodic solutions.     

Resonance and bifurcation in a discrete-time predator-prey system with Holling functional response

We perform a bifurcation analysis of a discrete predator-prey model with Holling functional response. We summarize stability conditions for the three kinds of fixed points of the map, further called F₁,F₂ and F₃ and collect complete information on this in a single scheme. In the case of F₂ we also compute the critical normal form coefficient of the flip bifurcation analytically. We further obtain new information about bifurcations of the cycles with periods 2, 3, 4, 5, 8 and 16 of the system by numerical computation of the corresponding curves of fixed points and codim-1 bifurcations, using the software package MatContM. Numerical computation of the critical normal form coefficients of the codim-2 bifurcations enables us to determine numerically the bifurcation scenario around these points as well as possible branch switching to curves of codim-1 points. Using parameter-dependent normal forms, we compute codim-1 bifurcation curves that emanate at codim-2 bifurcation points in order to compute the stability boundaries of cycles with periods 4, 5, 8 and 16.     

On the bifurcation analysis of a food web of four species

This paper is concerned with bifurcations of equilibria and the chaotic dynamics of a food web containing a bottom prey X, two competing predators Y and Z on X, and a super-predator W only on Y. Conditions for the existence of all equilibria and the stability properties of most equilibria are derived. A two-dimensional bifurcation diagram with the aid of a numerical method for identifying bifurcation curves is constructed to show the bifurcations of equilibria. We prove that the dynamical system possesses a line segment of degenerate steady states for the parameter values on a bifurcation line in the bifurcation diagram. Numerical simulations show that these degenerate steady states can help to switch the stabilities between two far away equilibria when the system crosses this bifurcation line. Some observations concerned with chaotic dynamics are also made via numerical simulations. Different routes to chaos are found in the system. Relevant calculations of Lyapunov exponents and power spectra are included to support the chaotic properties.     

Resonance bifurcation from homoclinic cycles

Differential equations that are equivariant under the action of a finite group can possess robust homoclinic cycles that can moreover be asymptotically stable. For differential equations in R⁴ there exists a classification of different robust homoclinic cycles for which moreover eigenvalue conditions for asymptotic stability are known. We study resonance bifurcations that destroy the asymptotic stability of robust ‘simple homoclinic cycles’ in four-dimensional differential equations. We establish that typically a periodic trajectory near the cycle is created, asymptotically stable in the supercritical case.     

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.     

Stability study for the MHD problem in perforated and parallel walls

We have studied the stability of a conducting fluid when it is continuously injected or ejected through a pair of parallel porous walls and escapes in both directions along the channel. The flow forms a stagnation point at the center and the effluence is restricted by a magnetic field perpendicular to the walls. A theoretical analysis of the steady state solutions of the MHD equations in the incompressible case is given as a function of three parameters R_e, R_m and M_A (R_e: Reynolds number, R_m: magnetic Reynolds number, M_A: Alfvenic Mach number), for some asymptotic limits. In the case of suction, critical values of the parameters are found for which there are bifurcations in the system. Such bifurcations are pitchfork type. In the case of injection, the flux results always stable.     

Asymptotic behavior of an SEIR epidemic model with quadratic treatment

Treatment is of great importance in fighting against infectious diseases. In this paper, we investigate a disease transmission model of SEIR type with quadratic treatment function. We consider r as the parameter which describes the societal effort to fight the infection. It is found that the system has four possible equilibria when r changes. Moreover, the existence and stability of equilibria of the model are discussed. Finally, the complex dynamics of the system are displayed by numerical simulations such as bi-stability, backward bifurcations.     

Analysis on existence of bifurcation solutions for a predator-prey model with herd behavior

This paper is concerned with a diffusive predator-prey model with herd behavior. The local and global stability of the unique homogeneous positive steady state U* is obtained. Treating the conversion or consumption rate γ as the bifurcation parameter, the steady-state bifurcations both from simple and double eigenvalues are studied near U*. The techniques include the Lyapunov function, the spectrum analysis of operators, the bifurcation theory, space decompositions and the implicit function theorem.     

一类捕食者-食饵模型的稳定性及Hopf分支

研究一类具有时滞和阶段结构的捕食模型的稳定性和Hopf分支.以滞量为参数,得到了系统正平衡点的稳定性和Hopf分支存在的充分条件.利用规范型和中心流形定理,给出了确定Hopf分支方向和分支周期解的稳定性的计算公式.     

The dynamics of a Bertrand duopoly with differentiated products: Synchronization,intermittency and global dynamics

We study the dynamics of a duopoly game à la Bertrand with horizontal product differentiation as proposed by Zhang et al. (2009) [35] by introducing opportune microeconomic foundations. The final model is described by a two-dimensional non-invertible discrete time dynamic system T. We show that synchronized dynamics occurs along the invariant diagonal being T symmetric; furthermore, we show that when considering the transverse stability, intermittency phenomena are exhibited. In addition, we discuss the transition from simple dynamics to complex dynamics and describe the structure of the attractor by using the critical lines technique. We also explain the global bifurcations causing a fractalization in the basin of attraction. Our results aim at demonstrating that an increase in either the degree of substitutability or complementarity between products of different varieties is a source of complexity in a duopoly with price competition.     

Stability diagram for 4D linear periodic systems with applications to homographic solutions

We consider a family of 4-dimensional Hamiltonian time-periodic linear systems depending on three parameters, λ₁, λ₂ and ε such that for ε=0 the system becomes autonomous. Using normal form techniques we study stability and bifurcations for ε>0 small enough. We pay special attention to the d'Alembert case. The results are applied to the study of the linear stability of homographic solutions of the planar three-body problem, for some homogeneous potential of degree −α, 0<α<2, including the Newtonian case.     

Buckling and lateral buckling interaction in thin-walled beam-column elements with mono-symmetric cross sections

Effects of axial forces on beam lateral buckling strength are investigated here in the case of elements with mono-symmetric cross sections. A unique compact closed-form is established for the interaction of lateral buckling moment with axial forces. This new equation is derived from a non-linear stability model. It includes first order bending distribution, load height level and effect of mono-symmetry terms (Wagner’s coefficient and shear point position). Compared to the so-called three-factors (C₁–C₃) formula commonly employed in beam lateral buckling stability, another factor C₄ is added in presence of axial loads. Pre-buckling deflection effects are considered in the study and the case of doubly-symmetric cross sections is easily recovered. The proposed solutions are validated and compared to finite element simulations where 3D beam elements including warping are used. The agreement of the proposed solutions with bifurcations observed on the non-linear equilibrium paths is good. Dimensionless interaction curves are dressed for the beam lateral buckling strength and the applied axial load, where the flexural-torsional buckling axial force is a taken as reference.     

Hopf bifurcations in dynamical systems

The onset of instability in autonomous dynamical systems (ADS) of ordinary differential equations is investigated. Binary, ternary and quaternary ADS are taken into account. The stability frontier of the spectrum is analyzed. Conditions necessary and sufficient for the occurring of Hopf, Hopf–Steady, Double-Hopf and unsteady aperiodic bifurcations—in closed form—and conditions guaranteeing the absence of unsteady bifurcations via symmetrizability, are obtained. The continuous triopoly Cournot game of mathematical economy is taken into account and it is shown that the ternary ADS governing the Nash equilibrium stability, is symmetrizable. The onset of Hopf bifurcations in rotatory thermal hydrodynamics is studied and the Hopf bifurcation number (threshold that the Taylor number crosses at the onset of Hopf bifurcations) is obtained.

     

有保护区的海洋渔业资源离散动力学模型分析

假设海洋渔业资源分属于保护区和非保护区两个区域,本文建立一个渔业资源储量-捕捞力度模型,用聚合方法得到一个简化的离散动力系统,从而分析正不动点的存在性、稳定性以及关于保护区面积比例的局部分叉,运用中心流形定理分析平衡点的局部稳定性,并用数值模拟验证不动点的局部分叉.最后,用全局分析方法分析保护区面积比例变化对可行吸引域的结构和大小的影响,从而揭示保护区对渔业资源可持续利用的影响.     

The dynamics of Bertrand model with bounded rationality

The paper considers a Bertrand model with bounded rational. A duopoly game is modelled by two nonlinear difference equations. By using the theory of bifurcations of dynamical systems, the existence and stability for the equilibria of this system are obtained. Numerical simulations used to show bifurcations diagrams, phase portraits for various parameters and sensitive dependence on initial conditions. We observe that an increase of the speed of adjustment of bounded rational player may change the stability of Nash equilibrium point and cause bifurcation and chaos to occur. The analysis and results in this paper are interesting in mathematics and economics.