Multiple Hopf bifurcations of three coupled van der Pol oscillators with delay

A system of three coupled van der Pol oscillators with delay is considered. Hopf bifurcations at the zero equilibrium as the delay increases are exhibited. The existence and stability of multiple periodic solutions are established using a symmetric Hopf bifurcation result of Wu (Trans. Amer. Math. Soc. 350 (1998) 4799-4838).     

Period-doubling bifurcation in an extended van der Pol system with bounded random parameter

An extended van der Pol system with bounded random parameter subjected to harmonic excitation is investigated by Chebyshev polynomial approximation. Firstly the stochastic extended van der Pol system is reduced into its equivalent deterministic one, solvable by suitable numerical methods. Then we explored nonlinear dynamical behavior about period-doubling bifurcation in stochastic system. Numerical simulations show that similar to the conventional period-doubling phenomenon in deterministic extended van der Pol system, stochastic period-doubling bifurcation may also occur in the stochastic extended van der Pol system. Besides, different from the deterministic case, in addition to the conventional bifurcation parameters, i.e. the amplitude and frequency of harmonic excitation, in the stochastic case the intensity of random parameter should also be taken as a new bifurcation parameter.     

具有时滞耦合的n个van der Pol振子弱共振双Hopf分岔

研究了具有时滞耦合的n个van derPol振子系统中发生的弱共振双Hopf分岔.应用改进的多尺度方法,得到了2:5共振的复振幅方程.通过将复振幅设为极坐标形式,将复振幅方程转化为一个二维的实振幅系统.通过研究实振幅方程的平衡点及其稳定性,对系统在2:5共振点附近的动力学行为进行了开折和分类.得到了一些有趣的动力学现象,如振幅死区、周期解和双稳态解等,相应的数值模拟验证了理论结果的正确性.     

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.     

Boundedness for network of stochastic coupled van der Pol oscillators with time-varying delayed coupling

In this paper, network of stochastic van der Pol oscillators with time-varying delayed coupling is considered. By using graph theory and Lyapunov functional method, the asymptotic boundedness in pth moment of the network is investigated. Moreover, by constructing an appropriate Lyapunov function, sufficient principle in the form of coefficients of network which ensures the asymptotic boundedness is established. Finally, a numerical example is given to show the effectiveness of the proposed criteria.     

时滞微分方程广义Hopf 分岔

本文运用Lyapunov-Schmidt 约化方法研究了一般时滞微分方程的分岔情况, 具体分析了当参数达到一个临界值时, 系统的无穷小生成元具有一对k 重非半单纯虚特征值的情形, 得到了判定分岔周期解存在性和分岔方向的判据, 而且该判据明显依赖于系统参数, 并通过对van der Pol 方程的详细分析进一步验证了我们的结果.     

Stability and bifurcation in a delayed predator-prey system with Beddington-DeAngelis functional response

We consider a delayed predator-prey system with Beddington-DeAngelis functional response. The stability of the interior equilibrium will be studied by analyzing the associated characteristic transcendental equation. By choosing the delay τ as a bifurcation parameter, we show that Hopf bifurcation can occur as the delay τ crosses some critical values. The direction and stability of the Hopf bifurcation are investigated by following the procedure of deriving normal form given by Faria and Magalhães. An example is given and numerical simulations are performed to illustrate the obtained results.     

Horizontal fast excitation in delayed van der Pol oscillator

This paper investigates the interaction effect of horizontal fast harmonic parametric excitation and time delay on self-excited vibration in van der Pol oscillator. We apply the method of direct partition of motion to derive the main autonomous equation governing the slow dynamic of the oscillator. The method of averaging is then performed on the slow dynamic to obtain a slow flow which is analyzed for equilibria and periodic motion. This analysis provides analytical approximations of regions in parameter space where periodic self-excited vibrations can be eliminated. A numerical study is performed on the original oscillator and compared to analytical approximations. It was shown that in the delayed case, horizontal fast harmonic excitation can eliminate undesirable self-excited vibrations for moderate values of the excitation frequency. In contrast, the case without delay requires large excitation frequency to eliminate such motions. This work has application to regenerative behavior in high-speed milling.     

Hopf-transcritical bifurcation in retarded functional differential equations

Firstly, we analyze a codimension-two unfolding for the Hopf-transcritical bifurcation, and give complete bifurcation diagrams and phase portraits. In particular, we express explicitly the heteroclinic bifurcation curve, and obtain conditions under which the secondary bifurcation periodic solutions and the heteroclinic orbit are stable. Secondly, we show how to reduce general retarded functional differential equation, with perturbation parameters near the critical point of the Hopf-transcritical bifurcation, to a 3-dimensional ordinary differential equation which is restricted on the center manifold up to the third order with unfolding parameters, and further reduce it to a 2-dimensional amplitude system, where these unfolding parameters can be expressed by those original perturbation parameters. Finally, we apply the general results to the van der Pol’s equation with delayed feedback, and obtain the existence of stable or unstable equilibria, periodic solutions and quasi-periodic solutions.     

On two-parameter bifurcation analysis of switched system composed of Duffing and van der Pol oscillators

The behaviors of system which alternate between Duffing oscillator and van der Pol oscillator are investigated to explore the influence of the switches on dynamical evolutions of system. Switches related to the state and time are introduced, upon which a typical switched model is established. Poincaré map of the whole switched system is defined by suitable local sections and local maps, and the formal expression of its Jacobian matrix is obtained. The location of the fixed point and associated Floquet multipliers are calculated, based on which two-parameter bifurcation sets of the switched system are obtained, dividing the parameter space into several regions corresponding to different types of attractors. It is found that cascading of period-doubling bifurcations may lead the system to chaos, while fold bifurcations determine the transition between period-3 solution and chaotic movement.     

Models and numerical schemes for generalized van der Pol equations

This paper presents three generalizations of the van der Pol equation (VDPE) using newly proposed three new generalized K-, A- and B-operators. These operators allow kernel to be arbitrary. As a result, these operators provide a greater generalization of the VDPE than the fractional integral and differential operators do. Like the original VDPE, the generalized van der Pol equations (GVDPEs) are also nonlinear equations, and in most cases, they can not be solved analytically. Numerical algorithms are presented and used to solve the GVDPEs. Results for several examples are presented to demonstrate the effectiveness of the numerical algorithms, and to examine the behavior of the GVDPEs and the limit cycles associated with them. Although the numerical algorithms have been used to solve the GVDPEs only, they can also be used to solve many other generalized oscillators and generalized differential equations. This will be considered in the future.     

Hopf-pitchfork bifurcation in van der Pol's oscillator with nonlinear delayed feedback

First, we identify the critical values for Hopf-pitchfork bifurcation. Second, we derive the normal forms up to third order and their unfolding with original parameters in the system near the bifurcation point, by the normal form method and center manifold theory. Then we give a complete bifurcation diagram for original parameters of the system and obtain complete classifications of dynamics for the system. Furthermore, we find some interesting phenomena, such as the coexistence of two asymptotically stable states, two stable periodic orbits, and two attractive quasi-periodic motions, which are verified both theoretically and numerically.     

Periodic bifurcation of Duffing-van der Pol oscillators having fractional derivatives and time delay

In this paper, a Duffing-van der Pol oscillator having fractional derivatives and time delays is investigated by the residue harmonic method. The angular frequencies and limit cycles of periodic motions are expanded into a power series of an order-tracking parameter and the unbalanced residues resulting from the truncated Fourier series are considered iteratively to improve the accuracy. The periodic bifurcations are examined using the fractional order, feedback gain and time delay as continuation parameters. It is shown that jumps and hysteresis phenomena can be delayed or removed. Transition from discontinuous bifurcation to continuous bifurcation is observed. The approximations are verified by numerical integration. We find that the proposed method can easily be programmed and can predict accurate periodic approximations while the system parameters being unfolded.     

A homotopy analysis method for limit cycle of the van der Pol oscillator with delayed amplitude limiting

In this work, a powerful analytical method, called Liao’s homotopy analysis method is used to study the limit cycle of a two-dimensional nonlinear dynamical system, namely the van der Pol oscillator with delayed amplitude limiting. It is shown that the solutions are valid for a wide range of variation of the system parameters. Comparison of the obtained solutions with those achieved by numerical solutions and by other perturbation techniques shows that the utilized method is effective and convenient to solve this type of problems with the desired order of approximation.     

Analysis of the Hopf bifurcation for the family of angiogenesis models

In this paper we study a family of models with delays describing the process of angiogenesis, that is a physiological process involving the growth of new blood vessels from pre-existing ones. This family includes the well-known models of tumour angiogenesis proposed by Hahnfeldt et al. and d?Onofrio-Gandolfi and is based on the Gompertz type of the tumour growth. As a consequence we start our analysis from the influence of delay onto the Gompertz model dynamics. The family of models considered in this paper depends on two time delays and a parameter α∈[0,1] which reflects how strongly the vessels dynamics depends on the ratio between tumour and vessels volume. We focus on the analysis of the model in three cases: one of the delays is equal to 0 or both delays are equal, depending on the parameter α. We study the stability switches, the Hopf bifurcation and the stability of arising periodic orbits for different α∈[0,1], especially for α=1 and α=0 which reflects the Hahnfeldt et al. and the d?Onofrio-Gandolfi models. For comparison we use also the value α=1/2.     

Bifurcation analysis in van der Pol's oscillator with delayed feedback

In this paper, a classical van der Pol's equation with generally delayed feedback is considered. It is shown that there are Bogdanov–Takens bifurcation, triple zero and Hopf-zero singularities by analyzing the distribution of the roots of the associated characteristic equation. In the situation that the zero is as a simple eigenvalue, the normal forms of the reduced equations are obtained by the center manifold theory and normal form method for functional differential equation, and hence the stability of the fixed point is determined, and transcritical and pitchfork bifurcations are found.     

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.     

Analysis of bifurcation in a system of n coupled oscillators with delays

A n-coupled BVP oscillators system with delays is considered. By choosing the delays as the bifurcating parameters, some results of the Hopf bifurcations occurring at the zero equilibrium as the delays increase are exhibited. Using the symmetric functional differential equation theories of Wu [Jianhong Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc. 350 (12) (1998) 4799–4838], the multiple Hopf bifurcations are obtained, and their spatio-temporal patterns: mirror-reflecting waves, standing waves, and discrete waves are demonstrated. Finally, computer simulations are performed to illustrate the analytical results found.     

Global Coupling of an Interface Problem in an Activator-inhibitor System

An activator-inhibitor reaction system with global coupling was introduced in [1]. The authors showed that global coupling suppresses the breathing motion and enhances the propagation of the localized solution. The collision between two traveling waves for a sufficiently strong global coupling is discussed in [2]. If the width of layers is infinitesimally thin, the equation of motion for a pair of the interfaces is derived. We shall study the dynamics of interfaces in the free boundary problem with global coupling and with a strong global coupling.