Twistless Tori Near Low-Order Resonances

In this paper, we investigate the behavior of the twist near low-order resonances of a periodic orbit or an equilibrium of a Hamiltonian system with two degrees of freedom. Namely, we analyze the case where the Hamiltonian has multiple eigenvalues (the Hamiltonian Hopf bifurcation) or a zero eigenvalue near the equilibrium and the case where the system has a periodic orbit whose multipliers are equal to 1 (the saddle-center bifurcation) or −1 (the period-doubling bifurcation). We show that the twist does not vanish at least in a small neighborhood of the period-doubling bifurcation. For the saddle-center bifurcation and the resonances of the equilibrium under consideration, we prove the existence of a “twistless” torus for sufficiently small values of the bifurcation parameter. An explicit dependence of the energy corresponding to the twistless torus on the bifurcation parameter is derived. Bibliography: 6 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 135–144.     

Double Hopf bifurcation for van der Pol-Duffing oscillator with parametric delay feedback control

The stability and bifurcation of a van der Pol-Duffing oscillator with the delay feedback are investigated, in which the strength of feedback control is a nonlinear function of delay. A geometrical method in conjunction with an analytical method is developed to identify the critical values for stability switches and Hopf bifurcations. The Hopf bifurcation curves and multi-stable regions are obtained as two parameters vary. Some weak resonant and non-resonant double Hopf bifurcation phenomena are observed due to the vanishing of the real parts of two pairs of characteristic roots on the margins of the “death island” regions simultaneously. By applying the center manifold theory, the normal forms near the double Hopf bifurcation points, as well as classifications of local dynamics are analyzed. Furthermore, some quasi-periodic and chaotic motions are verified in both theoretical and numerical ways.     

Stability and Hopf bifurcation for a regulated logistic growth model with discrete and distributed delays

In this paper, we investigate the stability and Hopf bifurcation of a new regulated logistic growth with discrete and distributed delays. By choosing the discrete delay τ as a bifurcation parameter, we prove that the system is locally asymptotically stable in a range of the delay and Hopf bifurcation occurs as τ crosses a critical value. Furthermore, explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by normal form theorem and center manifold argument. Finally, an illustrative example is also given to support the theoretical results.     

BIFURCATIONS OF DEGENERATE SINGULAR POINT OF A PREDATOR-PREY SYSTEM WITH HOLLING-IV FUNCTIONAL RESPONSE

A predator-prey system with Holling-IV functional response is investigated. It is shown that the system has a positive equilibrium?which is a cusp of co-dimension 2 under certain conditions. When the parameters vary in a small neighborhood of the values of parameters, the model undergoes the Bogdanov-Takens bifurcation. Different kinds of bifurcation phenomena are exhibited, which include the saddle~node bifurcation, the Hopf bifurcation and the homo-clinic bifurcation. Some computer simulations are presented to illustrate the conclusions.     

Bifurcations of Asymptotically Autonomous Differential Equations

This article deals with bifurcation phenomena of asymptotically autonomous differential equations. Under the assumption that the underlying autonomous system admits a bifurcation of pitchfork, saddle node, transcritical or Hopf type, nonautonomous bifurcation results are obtained for both the bifurcation of attraction and repulsion areas and transitions of attractors and repellers.     

Double Hopf bifurcation and quasi-periodic attractors in delay-coupled limit cycle oscillators

We investigate the double Hopf bifurcation at zero equilibrium point. Firstly, we give the critical values of Hopf and double Hopf bifurcations. Secondly, we implement the normal form method and the center manifold theory for delay-coupled limit cycle oscillators, and derive the universal unfolding and a complete bifurcation diagram of the system. Thirdly, many interesting phenomena, such as attractive periodic motion and three-dimensional invariant torus, are observed using numerical simulation. Finally, the normal forms of several strong resonant cases are listed.     

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.     

非线性颤振系统中既是超临界又是亚临界的Hopf分岔点研究

研究了二元机翼非线性颤振系统的Hopf分岔．应用中心流形定理将系统降维,并利用复数正规形方法得到了以气流速度为分岔参数的分岔方程．研究发现,分岔方程中一个系数不含分岔参数的一次幂,故使得分岔具有超临界和亚临界双重性质．用等效线性化法和增量谐波平衡法验证了所得结果．     

Bifurcation of limit cycles for a quartic near-Hamiltonian system by perturbing a nilpotent center

Hopf bifurcation which produces oscillations is a very important phenomena in the theory and application of dynamical systems. Almost all works available about Hopf bifurcations are related to a non-degenerate focus or center. For the case of a degenerate focus or center, the study of the bifurcations becomes challenge. In this paper, we consider the bifurcation of limit cycles for a quartic near-Hamiltonian system by perturbing a nilpotent center. We take coefficients as parameters, then we can get six limit cycles.     

Bifurcations and fast-slow behaviors in a hyperchaotic dynamical system

This paper reports a four-dimension (4D) fast-slow hyperchaotic system with the structure of two time scales by adding a slow state variable w into a three-dimension (3D) chaotic dynamical system, studies the stability and Hopf bifurcation of origin point. Furthermore, based on the fast-slow dynamical bifurcation analysis and the phase planes analysis, different bursting phenomena, symmetric fold/fold bursting, symmetric sub-Hopf/sub-Hopf bursting and chaotic bursting, as well as chaotic and periodic spiking, are observed in the fast-slow hyperchaotic system. Numerical simulations are presented to show these results.     

Nonlinear dynamics of unicycles in leader–follower formation

In this paper, a dynamical systems analysis is presented for characterizing the motion of a group of unicycles in leader–follower formation. The equilibrium formations are characterized along with their local stability analysis. It is demonstrated that with the variation in control gain, the collective dynamics might undergo Andronov–Hopf and Fold–Hopf bifurcations. The vigor of quasi-periodicity in the regime of Andronov–Hopf bifurcation and heteroclinic bursts between quasi-periodic and chaotic behavior in the regime of Fold–Hopf bifurcation increases with the number of unicycles. Numerical simulations also suggest the occurrence of global bifurcations involving the destruction of heteroclinic orbit.     

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.     

Stochastic Hopf bifurcation and chaos of stochastic Bonhoeffer–van der Pol system via Chebyshev polynomial approximation

In this paper, we analyzed stochastic chaos and Hopf bifurcation of stochastic Bonhoeffer–van der Pol (SBVP for short) system with bounded random parameter of an arch-like probability density function. The modifier ‘stochastic’ here implies dependent on some random parameter. In order to study the dynamical behavior of the SBVP system, Chebyshev polynomial approximation is applied to transform the SBVP system into its equivalent deterministic system, whose response can be readily obtained by conventional numerical methods. Thus, we can further explore the nonlinear phenomena in SBVP system. Stochastic chaos and Hopf bifurcation analyzed here are by and large similar to those in the deterministic mean-parameter Bonhoeffer–van der Pol system (DM–BVP for short) but there are also some featuring differences between them shown by numerical results. For example, in the SBVP system the parameter interval matching chaotic responses diffuses into a wider one, which further grows wider with increasing of intensity of the random variable. The shapes of limit cycles in the SBVP system are some different from that in the DM–BVP system, and the sizes of limit cycles become smaller with the increasing of intensity of the random variable. And some biological explanations are given.     

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 bifurcation control in the XCP for the Internet congestion control system

In this paper, we investigated the Hopf bifurcation of the eXplicit Control Protocol (XCP) for the Internet congestion control system. These bifurcation behaviors may cause heavy oscillation of average queue length and induce network instability. A time-delayed feedback control method was proposed for controlling Hopf bifurcation in the XCP system. Numerical simulation results are presented to show that the time-delayed feedback controller is efficient in controlling Hopf bifurcation.     

A delay differential equation model of HIV infection of CD4+ T-cells with cure rate

A delay differential equation as a mathematical model that described HIV infection of CD4⁺ T-cells is analyzed. When the constant death rate of infected but not yet virus-producing cells is equal to zero, the stability of the non-negative equilibria and the existence of Hopf bifurcation are investigated. A stability switch in the system due to variation of delay parameter has been observed, so is the phenomena of Hopf bifurcation and stable limit cycle. The estimation of the length of delay to preserve stability has been calculated. Further, when the constant death rate of infected but not yet virus-producing cells is not equal to zero, by using the geometric stability switch criterion in the delay differential system with delay dependent parameters, we present that stable equilibria become unstable as the time delay increases. Numerical simulations are carried out to explain the mathematical conclusions.     

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.     

Hopf bifurcation analysis of integro-differential equation with unbounded delay

The complexity of a nonlinear dynamical system is controllable via a selection of system parameters. One representative behavior of such a complex system can be illustrated by Hopf bifurcation. This paper presents a Hopf bifurcation analysis of a kind of integro-differential equations with unbounded delay. Based on the Hopf bifurcation principle, a set of relationships among system parameters are obtained when a periodic orbit exists in the system. A numerical analysis is applied to solve the integro-differential delay equation. This paper proves the existence of Hopf bifurcation in the corresponding difference equations under the same system parameters as that in the integro-differential delay equations.     

Bifurcation and chaos in an epidemic model with nonlinear incidence rates

This paper investigates a discrete-time epidemic model by qualitative analysis and numerical simulation. It is verified that there are phenomena of the transcritical bifurcation, flip bifurcation, Hopf bifurcation types and chaos. Also the largest Lyapunov exponents are numerically computed to confirm further the complexity of these dynamic behaviors. The obtained results show that discrete epidemic model can have rich dynamical behavior.     

Resonance in Periodically Perturbed Hopf Bifurcation

We examine in detail the effect of a periodic perturbation upon a system undergoing a Hopf bifurcation. Particular attention is paid to the case of resonance between the natural Hopf bifurcation frequency and the perturbation frequency. It is shown that for a certain parameter range this Hopf bifurcation is modified by the perturbation to exhibit frequency (phase) pulling and locking, and the nature of this frequency locking is described. In addition, the effect of adding diffusion to the system is discussed, and it is shown that diffusive waves are possible.