Resonance,stability and chaotic vibration of a quarter-car vehicle model with time-delay feedback

This paper examines dynamical behavior of a nonlinear oscillator with a symmetric potential that models a quarter-car forced by the road profile. The primary, superharmonic and subharmonic resonances of a harmonically excited nonlinear quarter-car model with linear time delayed active control are investigated. The method of multiple scales is utilized to obtain first order approximation of response. We focus on the influence of delay in the system. This naturally gives rise to a delay deferential equation (DDE) model of the system. The effect of time delay and feedback gains of the steady state responses of primary, superharmonic and subharmonic resonances are investigated. By means of Melnikov technique, necessary condition for onset of chaos resulting from homoclinic bifurcation is derived analytically. We describe a method to identify the critical forcing function and time delay above which the system becomes unstable. It is found that proper selection of time-delay shows optimum dynamical behavior. The accuracy of the method is obtained from the fractal basin boundaries.     

Fractional derivative and time delay damper characteristics in Duffing–van der Pol oscillators

In this paper, we investigate the damping characteristics of two Duffing–van der Pol oscillators having damping terms described by fractional derivative and time delay respectively. The residue harmonic balance method is presented to find periodic solutions. No small parameter is assumed. Highly accurate limited cycle frequency and amplitude are captured. The results agree well with the numerical solutions for a wide range of parameters. Based on the obtained solutions, the damping effects of these two oscillators are investigated. When the system parameters are identical, the steady state responses and their stability are qualitatively different. The initial approximations are obtained by solving a few harmonic balance equations. They are improved iteratively by solving linear equations of increasing dimension. The second-order solutions accurately exhibit the dynamical phenomena when taking the fractional derivative and time delay as bifurcation parameters respectively. When damping is described by time delay, the stable steady state response is more complex because time delay takes past history into account implicitly. Numerical examples taking time delay and fractional derivative are respectively given for feature extraction and convergence study.     

参数激励Duffing-VanderPol振子的动力学响应及反馈控制

研究了Duffing-Van der Pol振子的主参数共振响应及其时滞反馈控制问题．依平均法和对时滞反馈控制项Taylor展开的截断得到的平均方程表明,除参数激励的幅值和频率外,零解的稳定性只与原方程中线性项的系数和线性反馈有关,但周期解的稳定性还与原方程中非线性项的系数和非线性反馈有关．通过调整反馈增益和时滞,可以使不稳定的零解变得稳定．非零周期解可能通过鞍结分岔和Hopf分岔失去稳定性,但选择合适的反馈增益和时滞,可以避免鞍结分岔和Hopf分岔的发生．数值仿真的结果验证了理论分析的正确性．     

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.     

Analytical approximations for the periodic motion of the Duffing system with delayed feedback

In this study, the homotopy analysis method is developed to give periodic solutions of delayed differential equations that describe time-delayed position feedback on the Duffing system. With this technique, some approximate analytical solutions of high accuracy for some possible solutions are captured, which agree well with the numerical solutions in the whole time domain. Two examples of dynamic systems are considered, focusing on the periodic motions near a Hopf bifurcation of an equilibrium point. It is found that the current technique leads to higher accurate prediction on the local dynamics of time-delayed systems near a Hopf bifurcation than the energy analysis method or the traditional method of multiple scales.     

Numerical time-delayed feedback control in a neural network with delay

In this article, by a nonstandard finite-difference method we obtain the general time delayed feedback control numerical discrete scheme for a delayed neural network model. Firstly, the local stability of the equilibria point is discussed according to the Neimark–Sacker bifurcation theory. Then, from the point of view of control, for any step-size, a general time delayed feedback control numerical algorithm is introduced to delay the onset of the Neimark–Sacker bifurcation at a desired point by choosing appropriate control parameters. This controller can deal with the general system that the natural equilibrium cannot be given by analytic expression. Finally, numerical examples are provided to illustrate the theoretical results. The results show that the time delayed feedback numerical scheme is better than a polynomial function time delayed feedback method.     

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.     

Stability and Hopf bifurcation analysis of novel hyperchaotic system with delayed feedback control

In this article, a novel four dimensional autonomous nonlinear systezm called hyperchaotic Rikitake system is proposed. Basic properties of the new system are investigated and the complex dynamical behaviors, such as time series, bifurcation diagram, and Lyapunov exponents are analyzed by dynamic analysis approaches. To control the new hyperchaotic system, the delayed feedback control is introduced. Regarding the time delay as a bifurcation parameter, stability and bifurcations with respect to time delay are investigated. Conditions assuring the existence of Hopf bifurcation and the distribution of roots to the associated characteristic equation are investigated by utilizing the polynomial theorem. Besides, the Hopf bifurcation is proved to occur when the bifurcation parameter (time delay) crosses through derived critical value. Finally, numerical simulations are provided to prove the consistence with the derived theoretical results. © 2015 Wiley Periodicals, Inc. Complexity 21: 180–193, 2016     

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.     

Finding all steady state solutions of chemical kinetic models

Ordinary differential equations are used frequently by theoreticians to model kinetic process in chemistry and biology. These systems can have stable and unstable steady states and oscillations. This paper presents an algorithm to find all steady state solutions to a restricted class of ODE models, for which the right-hand sides are linear combinations of rational functions of variables and parameters. The algorithm converts the steady state equations into a system of polynomial equations and uses a globally convergent homotopy method to find all the roots of the system of polynomials. All steady state solutions of the original ODEs are guaranteed to be present as roots of the polynomial equations. The conversion may generate some spurious roots that do not correspond to steady state solutions. The stability properties of the steady states are not revealed. This paper explains the algorithms used and gives results for a cell cycle modeling problem.     

Bifurcation analysis of delay-induced periodic oscillations

In this paper we consider a generic differential equation with a cubic nonlinearity and delay. This system, in the absence of delay, is known to undergo an oscillatory instability. The addition of the delay is shown to result in the creation of a number of periodic solutions with constant amplitude and a constant frequency; the number of solutions increases with the size of the delay. Indeed, for many physical applications in which oscillatory instabilities are induced by a delayed response or feedback mechanism, the system under consideration forms the underlying backbone for a mathematical model. Our study showcases the effectiveness of performing a numerical bifurcation analysis, alongside the use of analytical and geometrical arguments, in investigating systems with delay. We identify curves of codimension-one bifurcations of periodic solutions. We show how these curves interact via codimension-two bifurcation points: double singularities which organise the bifurcations and dynamics in their local vicinity.     

Response of Duffing system with delayed feedback control under combined harmonic and real noise excitations

This paper presents a procedure for predicting the response of Duffing system with delayed feedback bang–bang control under combined harmonic and real noise excitations by using the stochastic averaging method. First, the time-delayed feedback bang–bang control force is expressed approximately in terms of the system state variables without time delay. Then the averaged Itô stochastic differential equations for the system are derived by using the stochastic averaging method. Finally, the response of the system is obtained by solving the Fokker–Plank–Kolmogorov (FPK) equation associated with the averaged Itô equations. It is shown that the time delay in feedback control can deteriorate the control effectiveness and cause bifurcation of stochastic jump of Duffing system. The validity of the proposed method is confirmed by digital simulation.     

Hopf bifurcation and control for the delayed predator-prey model with nonlinear prey harvesting

In our study, we focused on investigating a delayed differential-algebraic system. The system incorporates a square root functional response and non-linear prey harvesting. Employing the normal form of differential algebraic systems and the central manifold theory, we conducted a detailed analysis of the system"s stability and bifurcation phenomena, with time delay identified as a critical bifurcation parameter. When the time delay reached a critical value, the system"s equilibrium points underwent the Hopf bifurcation, resulting in system instability. To achieve stability, we introduced a feedback controller, successfully transitioning the system from an unstable to a stable state. Through subsequent numerical simulations, we validated the accuracy and correctness of our research conclusions.     

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.     

Symmetric homotopy construction

Homotopy continuation methods can be applied to compute all finite solutions to a given polynomial system. Computations will be performed more efficiently if the symmetric structure of the system can be exploited. This paper presents the construction of a symmetric homotopy. Using this homotopy, only the paths according to the generating solutions have to be traced during continuation.     

Bifurcation analysis in a limit cycle oscillator with delayed feedback

A kind of limit cycle oscillator with delayed feedback is considered. Firstly, the linear stability is investigated. According to the analysis results, the bifurcation diagram is drawn in the parameter plane. It is found that there are stability switches for time delay, and Hopf bifurcations when time delay crosses through some critical values. Then the direction and stability of the Hopf bifurcation are determined, using the normal form method and the center manifold theorem. Finally, some numerical simulations are carried out to illustrate the results found.     

Bifurcations and spatiotemporal patterns in a ratio‐dependent diffusive Holling‐Tanner system with time delay

In this paper, we concentrate on the spatiotemporal patterns of a delayed reaction‐diffusion Holling‐Tanner model with Neumann boundary conditions. In particular, the time delay that is incorporated in the negative feedback of the predator density is considered as one of the principal factors to affect the dynamic behavior. Firstly, a global Turing bifurcation theorem for τ = 0 and a local Turing bifurcation theorem for τ > 0 are given. Then, further considering the degenerated situation, we derive the existence of Bogdanov‐Takens bifurcation and Turing‐Hopf bifurcation. The normal form method is used to study the explicit dynamics near the Turing‐Hopf singularity. It is shown that a pair of stable nonconstant steady states (stripe patterns) and a pair of stable spatially inhomogeneous periodic solutions (spot patterns) could be bifurcated from a positive equilibrium. Moreover, the Turing‐Turing‐Hopf–type spatiotemporal patterns, that is, a subharmonic phenomenon with two spatial wave numbers and one temporal frequency, are also found and explained theoretically. Our results imply that the interaction of Turing and Hopf instabilities can be considered as the simplest mechanism for the appearance of complex spatiotemporal dynamics.     

Hopf bifurcations in a reaction-diffusion population model with delay effect

A reaction-diffusion population model with a general time-delayed growth rate per capita is considered. The growth rate per capita can be logistic or weak Allee effect type. From a careful analysis of the characteristic equation, the stability of the positive steady state solution and the existence of forward Hopf bifurcation from the positive steady state solution are obtained via the implicit function theorem, where the time delay is used as the bifurcation parameter. The general results are applied to a “food-limited” population model with diffusion and delay effects as well as a weak Allee effect population model.     

Bifurcation analysis of HIV infection model with antibody and cytotoxic T‐lymphocyte immune responses and Beddington–DeAngelis functional response

In this paper, a mathematical model for HIV‐1 infection with antibody, cytotoxic T‐lymphocyte immune responses and Beddington–DeAngelis functional response is investigated. The stability of the infection‐free and infected steady states is investigated. The basic reproduction number R₀ is identified for the proposed system. If R₀ < 1, then there is an infection‐free steady state, which is locally asymptotically stable. Further, the infected steady state is locally asymptotically stable for R₀ > 1 in the absence of immune response delay. We use Nyquist criterion to estimate the length of the delay for which stability continues to hold. Also the existence of the Hopf bifurcation is investigated by using immune response delay as a bifurcation parameter. Numerical simulations are presented to justify the analytical results. Copyright © 2014 John Wiley & Sons, Ltd.     

Hopf bifurcation in a delayed reaction–diffusion–advection population model

In this paper, we investigate a reaction–diffusion–advection model with time delay effect. The stability/instability of the spatially nonhomogeneous positive steady state and the associated Hopf bifurcation are investigated when the given parameter of the model is near the principle eigenvalue of an elliptic operator. Our results imply that time delay can make the spatially nonhomogeneous positive steady state unstable for a reaction–diffusion–advection model, and the model can exhibit oscillatory pattern through Hopf bifurcation. The effect of advection on Hopf bifurcation values is also considered, and our results suggest that Hopf bifurcation is more likely to occur when the advection rate increases.