Hopf bifurcation in symmetric configuration of predator-prey-mutualist systems

In this paper we apply the equivariant degree method to study Hopf bifurcations in a system of differential equations describing a symmetric predator-prey-mutualist model with diffusive migration between interacting communities. A topological classification (according to symmetry types), of symmetric Hopf bifurcation in configurations of populations with D₈, D₁₂, A₄ and S₄ symmetries, is presented with estimation on minimal number of bifurcating branches of periodic solutions.     

Taylor–Couette problem and related topics

The O(2)×S¹-equivariant degree is applied to study the Hopf bifurcation for the Taylor–Couette problem. Isotypical crossing numbers and equivariant bifurcation invariants are evaluated and are applied to classify the equivariant Hopf bifurcation for the Taylor–Couette flow.     

Global Hopf bifurcation for differential equations with state-dependent delay

We develop a global Hopf bifurcation theory for a system of functional differential equations with state-dependent delay. The theory is based on an application of the homotopy invariance of S¹-equivariant degree using the formal linearization of the system at a stationary state. Our results show that under a set of mild conditions the information about the characteristic equation of the formal linearization with frozen delay can be utilized to detect the local Hopf bifurcation and to describe the global continuation of periodic solutions for such a system with state-dependent delay.     

Spatially nonhomogeneous equilibrium in a reaction-diffusion system with distributed delay

We consider a reaction-diffusion system with general time-delayed growth rate and kernel functions. The existence and stability of the positive spatially nonhomogeneous steady-state solution are obtained. Moreover, taking minimal time delay τ as the bifurcation parameter, Hopf bifurcation near the steady-state solution is proved to occur at a critical value τ=τ₀. Especially, the Hopf bifurcation is forward and the bifurcated periodic solutions are stable on the center manifold. The general results are applied to competitive and cooperative systems with weak or strong kernel function respectively.     

Bifurcations of limit cycles in a Z 4-equivariant quintic planar vector field

In this paper, a Z4-equivariant quintic planar vector field is studied. The Hopf bifurcation method and polycycle bifurcation method are combined to study the limit cycles bifurcated from the compounded cycle with 4 hyperbolic saddle points. It is found that this special quintic planar polynomial system has at least four large limit cycles which surround all singular points. By applying the double homoclinic loops bifurcation method and Hopf bifurcation method, we conclude that 28 limit cycles with two different configurations exist in this special planar polynomial system. The results acquired in this paper are useful for studying the weakened 16th Hilbert＇s Problem.     

Numerical Hopf bifurcation analysis in nonlinear ordinary and partial differential systems from chemical reactor theory

A code has been developed which will automatically locate and analyze points of Hopf bifurcation in autonomous ordinary differential systems. The code first locates critical value(s) v_c of a user-specified parameter v (the bifurcation parameter) such that a stationary (equilibrium) solution x_*(v) loses linear stability by virtue of a complex conjugate pair of eigenvalues. The code computes x_*(v) during the location of v_c. Then the code computes the various coefficients in a local approximation to the family of periodic solutions which arise, a process which involves computation of second and third partial derivatives by numerical differencing of the user-supplied Jacobian matrix. The current version of the code, called BIFOR2, is fully described in Hassard, Kazarinoff, and Wan, Theory and Applications of Hopf Bifurcation, Cambridge U.P., 1981. In this paper we demonstrate the code in applications to two systems drawn from chemical reactor theory. The first application is to a 4th order ordinary differential system modeling a coupled tank reactor. The second application is to a partial differential system modeling a catalyst particle system. These represent the first applications of the code to chemically reacting systems other than the Brusselator. The second application demonstrates how collocation methods may be used in conjunction with BIFOR2 to perform Hopf bifurcation analysis of partial differential systems.     

Analysis and control of the bifurcation of Hodgkin–Huxley model

The Hodgkin–Huxley (HH) equations are parameterized by a number of parameters and show a variety of qualitatively different behaviors depending on the various parameters. This paper finds the bifurcation would occur when the leakage conductance g_l is lower than a special value. The Hopf bifurcation of HH model is controlled by applying a simple and unified state-feedback method and the bifurcation point is moved to an unreachable physiological point at the same time, so in this way an absolute bifurcation control is achieved. The simulation results demonstrate the validity of such theoretic analysis and control method. This new method could be a great help to the design of new closed-loop electrical stimulation systems for patients suffering from different nerve system dysfunctions.     

LIMIT CYCLE PROBLEM OF QUADRATIC SYSTEM OF TYPE （ Ⅲ ）m=0, （ Ⅲ ）

To continue the discussion in （Ⅰ ） and （ Ⅱ ）,and finish the study of the limit cycle problem for quadratic system （ Ⅲ ）m=0 in this paper. Since there is at most one limit cycle that may be created from critical point O by Hopf bifurcation,the number of limit cycles depends on the different situations of separatrix cycle to be formed around O. If it is a homoclinic cycle passing through saddle S1 on 1 ＋ax-y = 0,which has the same stability with the limit cycle created by Hopf bifurcation,then the uniqueness of limit cycles in such cases can be proved. If it is a homoclinic cycle passing through saddle N on x= 0,which has the different stability from the limit cycle created by Hopf bifurcation,then it will be a case of two limit cycles. For the case when the separatrix cycle is a heteroclinic cycle passing through two saddles at infinity,the discussion of the paper shows that the number of limit cycles will change from one to two depending on the different values of parameters of 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.     

Dynamics of a competitive Lotka-Volterra system with three delays

In this paper, a competitive Lotka-Volterra system with three delays is investigated. By choosing the sum τ of three delays as a bifurcation parameter, we show that in the above system, Hopf bifurcation at the positive equilibrium can occur as τ crosses some critical values. And we obtain the formulae determining direction of Hopf bifurcation and stability of the bifurcating periodic solutions by using the normal form theory and center manifold theorem. Finally, numerical simulations supporting our theoretical results are also included.     

Dynamic Hopf bifurcations generated by nonlinear terms

We consider the so-called delayed loss of stability phenomenon for singularly perturbed systems of differential equations in case that the associated autonomous system with a scalar parameter undergoes the Hopf bifurcation at the zero equilibrium point. It is assumed that the linearization of the associated system is independent of the parameter and the next terms in the expansion of the right-hand parts at zero are positive homogeneous of order α>1. Simple formulas are presented to estimate the asymptotic delay for the delayed loss of stability phenomenon. More precisely, we suggest sufficient conditions which ensure that zeros of a simple function ψ defined by the positive homogeneous nonlinear terms are the Hopf bifurcation points of the associated system, the sign of ψ at other points determines stability of the zero equilibrium, and the asymptotic delay equals the distance between the bifurcation point and a zero of some primitive of ψ.     

Bifurcation for a functional yield chemostat when one competitor produces a toxin

A model, with general yield functions: Fi(S), i=1,2, of competition in the chemostat of two competitors for a single nutrient when one of the competitors produces toxin against its opponent is studied in this paper. The conditions in terms of the relevant parameters for the Hopf bifurcation of the three-dimensional system have been proved, which implies the existence of limit cycles in the 3-D system.     

Hopf bifurcation in numerical approximation of the sunflower equation

In this paper we consider the numerical solution of the sunflower equation. We prove that if the sunflower equation has a Hopf bifurcation point ata =a ₀, then the numerical solution with the Euler-method of the equation has a Hopf bifurcation point ata _h =a ₀ +O(h).     

Exploring action potential initiation in neurons exposed to DC electric fields through dynamical analysis of conductance-based model

Noninvasive direct current (DC) electric stimulation of central nervous system is today a promising therapeutic option to alleviate the symptoms of a number of neurological disorders. Despite widespread use of this noninvasive brain modulation technique, a generalizable explanation of its biophysical basis has not been described which seriously restricts its application and development. This paper investigated the dynamical behaviors of Hodgkin’s three classes of neurons exposed to DC electric field based on a conductance-based neuron model. With phase plane and bifurcation analysis, the different responses of each class of neuron to the same stimulation are shown to derive from distinct spike initiating dynamics. Under the effects of negative DC electric field, class 1 neuron generates repetitive spike through a saddle-node on invariant circle (SNIC) bifurcation, while it ceases this repetitive behavior through a Hopf bifurcation; Class 2 neuron generates repetitive spike through a Hopf bifurcation, meanwhile it ceases this repetitive behavior also by a Hopf bifurcation; Class 3 neuron can generate single spike through a quasi-separatrix-crossing (QSC) at first, then it generates repetitive spike through a Hopf bifurcation, while it ceases this repetitive behavior through a SNIC bifurcation. Furthermore, three classes of neurons’ spiking frequency f–electric field E (f–E) curves all have parabolic shape. Our results highlight the effects of external DC electric field on neuronal activity from the biophysical modeling point of view. It can contribute to the application and development of noninvasive DC brain modulation technique.     

Stable oscillations in a predator-prey model with time lag

The Lotka-Volterra model with carrying capacity at the prey and time delay in the equation concerning the predator is considered. The time delay is taken into consideration by an integral with the weight function a exp(?at). It is shown that under certain conditions imposed upon the parameters of the system a supercritical Hopf bifurcation takes place at a certain value a₀, of a and the bifurcating closed paths are orbitally asymptotically stable for values of a below a₀.     

Bifurcation analysis for a three-species predator-prey system with two delays

In this paper, a three-species predator-prey system with two delays is investigated. By choosing the sum τ of two delays as a bifurcation parameter, we first show that Hopf bifurcation at the positive equilibrium of the system can occur as τ crosses some critical values. Second, we obtain the formulae determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions by using the normal form theory and center manifold theorem. Finally, numerical simulations supporting our theoretical results are also included.     

On the number of limit cycles of a Z4-equivariant quintic polynomial system

In this paper we study the number of limit cycles of a near-Hamiltonian system under Z₄-equivariant quintic perturbations. Using the methods of Hopf and heteroclinic bifurcation theory, we found that the perturbed system can have 13 limit cycles.     

Stability and Hopf bifurcation analysis in a three-level food chain system with delay

A class of three level food chain system is studied. With the theory of delay equations and Hopf bifurcation, the conditions of the positive equilibrium undergoing Hopf bifurcation is given regarding τ as the parameter. The stability and direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold argument, and numerical simulations are performed to illustrate the analytical results.     

Global Dynamics of a Tuberculosis Epidemic Model and the Influence of Backward Bifurcation

In this paper, we propose and analyze a tuberculosis (TB) model with exogenous re-infection. We assume that treated individual may be again infected by infectious individual. The model exhibits two bifurcations viz. transcritical bifurcation when the basic reproductive number R ₀?=?1 and backward bifurcation where the disease transmission rate β plays as control parameter. The persistent of the model and, the local and global stability criteria of disease-free and endemic equilibria are discussed. By carrying out bifurcation analysis, it is shown that the model exhibits the bistability and undergoes the Hopf bifurcation when immunological memory is everlasting i.e. when σ?=?0. Lastly, some simulations are given to verify our analytical results.