State-dependent delay in regenerative turning processes

Stability of a two degrees of freedom model of the turning process is considered. An accurate modeling of the surface regeneration shows that the regenerative delay, determined by the combination of the workpiece rotation and the tool vibrations, is in fact state-dependent. For that reason, the mathematical model considered in this paper is a delay-differential equation with state-dependent time delay. In order to study linearized stability of stationary cutting processes, an associated linear system, corresponding to the state-dependent delay equation, is derived. Stability analysis of this linear system is performed analytically. A comparison between the state-dependent delay model and the previously used constant or time-periodic delay models shows that the incorporation of the state-dependent delay into the model slightly affects the linear stability properties of the system in certain parameter domains.     

Subcritical Hopf Bifurcation in the Delay Equation Model for Machine Tool Vibrations

We show the existence of a subcritical Hopf bifurcation in thedelay-differential equation model of the so-called regenerative machine toolvibration. The calculation is based on the reduction of the infinite-dimensional problem to a two-dimensional center manifold. Due to the specialalgebraic structure of the delayed terms in the nonlinear part of the equation,the computation results in simple analytical formulas. Numerical simulationsgave excellent agreement with the results.     

Diffusion-driven instability and Hopf bifurcation in Brusselator system

The Hopfbifurcation for the Brusselator ordinary-differential-equation （ODE） model and the corresponding partial-differential-equation （PDE） model are investigated by using the Hopf bifurcation theorem. The stability of the Hopf bifurcation periodic solution is discussed by applying the normal form theory and the center manifold theorem. When parameters satisfy some conditions, the spatial homogenous equilibrium solution and the spatial homogenous periodic solution become unstable. Our results show that if parameters are properly chosen, Hopf bifurcation does not occur for the ODE system, but occurs for the PDE system.     

Regenerative Tool Chatter Near a Codimension 2 Hopf Point Using Multiple Scales

We study a well-known regenerative machine tool vibration model (a delay differential equation) near a codimension 2 Hopf bifurcation point. The method of multiple scales is used directly, bypassing a center manifold reduction. We use a nonstandard choice of expansion parameter that helps understand practically relevant aspects of the dynamics for not-too-small amplitudes. Analytical expressions are then obtained for the double Hopf points. Both sub- and supercritical bifurcations are predicted to occur near the reference point; and analytical conditions on the parameter variations for each type of bifurcation to occur are obtained as well. Analytical approximations are supported by numerics.     

Effect of delay combinations on stability and Hopf bifurcation of an oscillator with acceleration-derivative feedback

This paper presents new observations of delayed AD (acceleration-derivative) controller in active vibration control and in bifurcation control of a Duffing oscillator. Based on the stability analysis of the linear delayed oscillator, it is found that combination of the two delays in acceleration feedback and velocity feedback has a significant influence on the stable region in the parameter plane of the gains. By calculating the real part of the rightmost characteristic roots of the controlled oscillator with fixed delays, it is shown that a delayed acceleration feedback with positive gain can work much better than the corresponding delayed negative acceleration feedback, which is used in classic control theory. For given feedback gains, by calculating the critical delay values, it is shown that a delayed positive acceleration feedback can result in a much larger stable delay interval than the corresponding delayed negative acceleration feedback does. As an application of these results to a delayed Duffing oscillator with acceleration-derivative feedback, a delayed positive acceleration feedback can be well used to postpone the occurrence of Hopf bifurcation in the delayed nonlinear oscillators.     

Hopf bifurcation of an oscillator with quadratic and cubic nonlinearities and with delayed velocity feedback

This paper studies the local dynamics of an SDOF system with quadratic and cubic stiffness terms, and with linear delayed velocity feedback. The analysis indicates that for a sufficiently large velocity feedback gain, the equilibrium of the system may undergo a number of stability switches with an increase of time delay, and then becomes unstable forever. At each critical value of time delay for which the system changes its stability, a generic Hopf bifurcation occurs and a periodic motion emerges in a one-sided neighbourhood of the critical time delay. The method of Fredholm alternative is applied to determine the bifurcating periodic motions and their stability. It stresses on the effect of the system parameters on the stable regions and the amplitudes of the bifurcating periodic solutions. The project supported by the National Natural Science Foundation of China (19972025)     

四维超混沌系统Hopf分岔分析与反控制

对超混沌系统进行分岔反控制的研究已成为当前一个重要研究方向,常采用线性控制器实现反控制。首先,对一个四维超混沌系统的Hopf分岔特性进行了分析,利用高维分岔理论推导出分岔特性与参数之间的关系式,以此判断系统的分岔类型。然后,设计一个由线性与非线性组合成的混合控制器对系统进行分岔反控制,控制参数取值不同时,系统会呈现出不同的分岔特性。通过分析得出,调控线性控制器参数可以使系统Hopf分岔提前或延迟发生;同时,调控混合控制器的两个控制参数,可以改变系统Hopf分岔特性,实现分岔反控制。     

Local Hopf bifurcation and global existence of periodic solutions in TCP system

This paper investigates the dynamics of a TCP system described by a first- order nonlinear delay differential equation. By analyzing the associated characteristic transcendental equation, it is shown that a Hopf bifurcation sequence occurs at the pos- itive equilibrium as the delay passes through a sequence of critical values. The explicit algorithms for determining the Hopf bifurcation direction and the stability of the bifur- cating periodic solutions are derived with the normal form theory and the center manifold theory. The global existence of periodic solutions is also established with the method of Wu （Wu, J. H. Symmetric functional differential equations and neural networks with memory. Transactions of the American Mathematical Society 350（12）, 4799-4838 （1998））.     

Hopf bifurcation for a ecological mathematical model on microbe populations

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Group delay induced instabilities and Hopf bifurcations, of a controlled double pendulum

Digital filters, frequently used in active control of mechanical systems, enable one to improve the signal-to-noise ratio and the control performance, but introduce group delays into the control loops simultaneously. In order to gain an insight into the effects of a digital filter on a controlled mechanical system, this paper presents the stability switches and the corresponding Hopf bifurcations of a double pendulum system with the linear quadratic control having a digital filter via theoretical analysis, numerical simulations and experiments. In this study, the digital filters are used to remove the undesired noise of high frequency, which is embedded in the control signal, and are modeled as the components of pure time delay during the theoretical analysis and numerical simulations. The study shows that a digital filter with moderate specifications can not only improve the vibration reduction effectively, but also save the energy consumption of the servo-motor remarkably. However, over demanding specifications will make the group delay of the filter exceed a critical value and cause either a divergent motion or a self-excited vibration through a Hopf bifurcation, the occurrence of which depends on both the stability and the size of the basin of attraction of the bifurcating periodic motion. The experimental results well coincide with the theoretical and numerical ones, and strongly support the simplification of the digital filters as the components of pure time delay. Finally, some suggestions are made to avoid the group delay induced instability.     

A new approach for the computation of Hopf bifurcation points

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Stochastic Hopf bifurcation in quasi-integrable-Hamiltonian systems

A new procedure is developed to study the stochastic Hopf bifurcation in quasiintegrable-Hamiltonian systems under the Gaussian white noise excitation. Firstly, the singular boundaries of the first-class and their asymptotic stable conditions in probability are given for the averaged Ito differential equations about all the sub-system‘s energy levels with respect to the stochastic averaging method. Secondly, the stochastic Hopf bifurcation for the coupled sub-systems are discussed by defining a suitable bounded torus region in the space of the energy levels and employing the theory of the torus region when the singular boundaries turn into the unstable ones. Lastly, a quasi-integrable-Hamiltonian system with two degrees of freedom is studied in detail to illustrate the above procedure.Moreover, simulations by the Monte-Carlo method are performed for the illustrative example to verify the proposed procedure. It is shown that the attenuation motions and the stochastic Hopf bifurcation of two oscillators and the stochastic Hopf bifurcation of a single oscillator may occur in the system for some system‘s parameters. Therefore, one can see that the numerical results are consistent with the theoretical predictions.     

Hopf bifurcation in general Brusselator system with diffusion

The general Brusselator system is considered under homogeneous Neumann boundary conditions. The existence results of the Hopf bifurcation to the ordinary differential equation (ODE) and partial differential equation (PDE) models are obtained. By the center manifold theory and the normal form method, the bifurcation direction and stability of periodic solutions are established. Moreover, some numerical simulations are shown to support the analytical results. At the same time, the positive steady-state solutions and spatially inhomogeneous periodic solutions are graphically shown to supplement the analytical results.     

On the appearance of a Hopf bifurcation in a non-ideal mechanical problem

In this paper we studied a non-ideal system with two degrees of freedom consisting of a dumped nonlinear oscillator coupled to a rotatory part. We investigated the stability of the equilibrium point of the system and we obtain, in the critical case, sufficient conditions in order to obtain an appropriate Normal Form. From this, we get conditions for the appearance of Hopf Bifurcation when the difference between the driving torque and the resisting torque is small. It was necessary to use the Bezout Theorem, a classical result of Algebraic Geometry, in the obtaining of the foregoing results.     

时滞Rucklidge系统Hopf分岔分析与电路实现

以一类新的单时滞Rucklidge系统为分析对象,通过计算时滞系统的平衡点,分析该系统在各平衡点的稳定性和Hopf分岔的存在性,得到其发生Hopf分岔的条件。Matlab多组数值仿真验证了理论分析的正确性。基于此设计了一种可切换时滞与非时滞的混沌电路,并运用Multisim14.0进行仿真,实验结果表明,该电路可行且有效。     

Cyclic motions near a Hopf bifurcation of a four-dimensional system

We study motions near a Hopf bifurcation of a representative nonconservative four-dimensional autonomous system with quadratic nonlinearities. Special cases of the four-dimensional system represent the envelope equations that govern the amplitudes and phases of the modes of an internally resonant structure subjected to resonant excitations. Using the method of multiple scales, we reduce the Hopf bifurcation problem to two differential equations for the amplitude and phase of the bifurcating cyclic solutions. Constant solutions of these equations provide asymptotic expansions for the frequency and amplitude of the bifurcating limit cycle. The stability of the constant solutions determines the nature of the bifurcation (i.e., subcritical or supercritical). For different choices of the control parameter, the range of validity of the analytical approximation is ascertained using numerical simulations. The perturbation analysis and discussions are also pertinent to other autonomous systems.     

Estimating Critical Hopf Bifurcation Parameters for a Second-Order Delay Differential Equation with Application to Machine Tool Chatter

Nonlinear time delay differential equations are well known to havearisen in models in physiology, biology and population dynamics. Theyhave also arisen in models of metal cutting processes. Machine toolchatter, from a process called regenerative chatter, has been identifiedas self-sustained oscillations for nonlinear delay differentialequations. The actual chatter occurs when the machine tool shifts from astable fixed point to a limit cycle and has been identified as arealized Hopf bifurcation. This paper demonstrates first that a class ofnonlinear delay differential equations used to model regenerativechatter satisfies the Hopf conditions. It then gives a precisecharacterization of the critical eigenvalues on the stability boundaryand continues with a complete development of the Hopf parameter, theperiod of the bifurcating solution and associated Floquet exponents.Several cases are simulated in order to show the Hopf bifurcationoccurring at the stability boundary. A discussion of a method ofintegrating delay differential equations is also given.     

Stability and bifurcation of a human respiratory system model with time delay

The stability and bifurcation of the trivial solution in the two-dimensional differential equation of a model describing human respiratory system with time delay were investigated. Formulas about the stability of bifurcating periodic solution and the directionof Hopf bifurcation were exhibited by applying the normal form theory and the center manifold theorem.Furthermore, numerical simulation was carried out.     

Hopf bifurcation for functional differential equations of mixed type

We prove Hopf bifurcation and center manifold theorems for functional differential equations of mixed type. An application to the dynamic behavior of a competitive economy (business cycle) is provided.     

A modified averaging scheme with application to the secondary Hopf bifurcation of a delayed van der Pol oscillator

In this paper, a modified averaging scheme is presented for a class of time-delayed vibration systems with slow variables. The new scheme is a combination of the averaging techniques proposed by Hale and by Lehman and Weibel, respectively. The averaged equation obtained from the modified scheme is simple enough but it retains the required information for the local nonlinear dynamics around an equilibrium. As an application of the present method, the delay value for which a secondary Hopf bifurcation occurs is successfully located for a delayed van der Pol oscillator.