Oscillation of Higher-Order Delay Differential Equation with Applications to Hyperbolic Equations

In this paper, we consider an odd-order delay differential equation with positive and negative coefficients. New sufficient conditions that guarantee the oscillation of all solutions are presented. Our results extend and improve some known results. Next, these results are used for establishing oscillation criteria for hyperbolic delay differential equations with positive and negative coefficients corresponding to three sets of boundary conditions.     

The sufficient and necessary conditions of unconditional stability and the delay bound of the third-order neutral delay differential equation

In this paper,the sufficient and necessary conditions of the unconditional stability.andthe delay bound of the third-order neutral delay differential equation with real constantcoefficients are given.The conditions are brief and practical algebraic criterions.Furthermore,we get the delay bound     

Bifurcation analysis for T system with delayed feedback and its application to control of chaos

In this paper, a three-dimensional autonomous nonlinear system called the T system which has potential application in secure communications is considered. Regarding the delay as parameter, we investigate the effect of delay on the dynamics of T system with delayed feedback. Firstly, by employing the polynomial theorem to analyze the distribution of the roots to the associated characteristic equation, the conditions of ensuring the existence of Hopf bifurcation are given. Secondly, by using the normal form theory and center manifold argument, we derive the explicit formulas determining the stability, direction and other properties of bifurcating periodic solutions. Finally, we give several numerical simulations, which indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable steady state or a periodic orbit.     

COMPLICATED BURSTING BEHAVIORS AS WELL AS THE MECHANISM OF A THREE DIMENSIONAL NONLINEAR SYSTEM 1)

由多时间尺度耦合效应引起的簇发振荡行为是非线性动力学研究的重要课题之一.本文针对一类参数激励下的三维非线性电机系统(该系统可以描述两种自激同极发电机系统的动力学行为,两种系统在数学上等效),研究了当参数激励频率远小于系统自然频率时的各种复杂簇发振荡行为及其产生机理.通过快慢分析方法, 将参数激励作为慢变参数,得到了非自治系统对应的广义自治系统及快子系统和慢变量,并给出了快子系统的稳定性和分岔条件以及系统关于典型参数的单参数分岔图.借助转换相图与分岔图的叠加, 分析了对称式delayed subHopf/fold cycle簇发振荡的产生机理及其动力学转迁, 即delayed subHopf/fold cycle簇发振荡、焦点/焦点型对称式叉形分岔滞后簇发振荡和焦点/焦点型叉形分岔滞后簇发振荡.研究结果表明, 系统会出现两种不同的分岔滞后形式, 一种是亚临界Hopf分岔滞后,另一种是叉形分岔滞后,而且控制参数显著影响平衡点的稳定性和分岔滞后区间的宽度.同时初始点的选取则会影响系统动力学行为的对称性.本文的研究进一步加深了对由分岔滞后引起的簇发振荡的认识和理解.     

一类三维非线性系统的复杂簇发振荡行为及其机理

由多时间尺度耦合效应引起的簇发振荡行为是非线性动力学研究的重要课题之一.本文针对一类参数激励下的三维非线性电机系统(该系统可以描述两种自激同极发电机系统的动力学行为,两种系统在数学上等效),研究了当参数激励频率远小于系统自然频率时的各种复杂簇发振荡行为及其产生机理.通过快慢分析方法, 将参数激励作为慢变参数,得到了非自治系统对应的广义自治系统及快子系统和慢变量,并给出了快子系统的稳定性和分岔条件以及系统关于典型参数的单参数分岔图.借助转换相图与分岔图的叠加, 分析了对称式delayed subHopf/fold cycle簇发振荡的产生机理及其动力学转迁, 即delayed subHopf/fold cycle簇发振荡、焦点/焦点型对称式叉形分岔滞后簇发振荡和焦点/焦点型叉形分岔滞后簇发振荡.研究结果表明, 系统会出现两种不同的分岔滞后形式, 一种是亚临界Hopf分岔滞后,另一种是叉形分岔滞后,而且控制参数显著影响平衡点的稳定性和分岔滞后区间的宽度.同时初始点的选取则会影响系统动力学行为的对称性.本文的研究进一步加深了对由分岔滞后引起的簇发振荡的认识和理解.      

Controllability of delay degenerate control systems with independent subsystems

IntroductionAsthestudyforthetheoryofcontrolsystemsdeeplygoesonandtheneedforthestudyandtheapplicationofmanypracticalsystemssuchaspowersystems,ecosystems,economicmanagementsystemsandsoon ,peoplerequestthattheprecisionforthedescribing ,analysisanddesignabout…     

Dynamics of microbubble oscillators with delay coupling

Two vibrating bubbles submerged in a fluid influence each others’ dynamics via sound waves in the fluid. Due to finite sound speed, there is a delay between one bubble’s oscillation and the other’s. This scenario is treated in the context of coupled nonlinear oscillators with a delay coupling term. It has previously been shown that with sufficient time delay, a supercritical Hopf bifurcation may occur for motions in which the two bubbles are in phase. In this work, we further examine the bifurcation structure of the coupled microbubble equations, including analyzing the sequence of Hopf bifurcations that occur as the time delay increases, as well as the stability of this motion for initial conditions which lie off the in-phase manifold. We show that in fact the synchronized, oscillating state resulting from a supercritical Hopf is attracting for such general initial conditions.     

Sufficient conditions for the oscillation of solutions of first-order neutral delay impulsive differential equations with constant coefficients

This paper is dealing with the oscillatory properties of first-order neutral delay impulsive differential equations and the corresponding inequalities with constant coefficients. The established sufficient conditions ensure the oscillation of every solution of equations of this type.     

A NECESSARY AND SUFFICIENT CONDITION FOR THEOSCILLATION OF SOLUTIONS OF LIENARD TYPE SYSTEM WITH MULTIPLE SINGULAR POINTS

I.IntroductionJ.R.Graer']andG.VillariI2]gavenecessar\JandsufficientconditionsfortheoscillationofsolutionofLienardsystemf~])~F(x),y~~glXJ(l.l)HanMaoanI3]gaveoscillatirytheoremsforLienardt}'pesystemt'.ithasingularpointi~h(.t/)~F(x),#~~g(x)(1.2)Whereas,aloto…     

Relaxation oscillation and attractive basins of a two-neuron Hopfield network with slow and fast variables

In this paper, the nonlinear dynamics of a two-neuron Hopfield network with slow and fast variables is investigated. By means of the geometric singular perturbation theory, the condition that ensures the existence of the relaxation oscillation is obtained, the period of the relaxation oscillation is determined analytically, and the shape of attractive basin of every stable equilibrium is figured out. By using the method of stability switches, the delay effect on the characteristics of the relaxation oscillation and the attractive basins is studied. Case studies are given to demonstrate the validity of theoretical analysis.     

Non-resonant response, bifurcation and oscillation suppression of a non-autonomous system with delayed position feedback control

The maglev system with delayed position feedback control is excitated by the deflection of flexible guideway and resonant response may take place. This paper concerns the non-resonant response of the system by employing centre manifold reduction and method of multiple time scales. The dynamical model is presented and expanded to the third-order Taylor series. Taking time delay as its bifurcation parameter, the condition with which the Hopf bifurcation may occur is investigated. Centre manifold reduction is applied to get the Poincaré normal form of the nonlinear system so that we can study the relationship between periodic solution and system parameter. At first, the non-resonant periodic solution of the normal form is calculated based on the method of multiple time scales. Then the bifurcation condition of the free oscillation in the solution is analyzed, and we get the conditions with which the free oscillation has maximum and minimum values. The relationship between external excitation and the periodic solution is also discussed in this paper. Finally, numerical simulation results show how system and excitation parameters affect the system response. It is shown that the existence of the free oscillation and the amplitude of the forced oscillation can be determined by time delay and control parameters. So felicitously selecting them can suppress the oscillation effectively.     

Bursting-like motion induced by time-varying delay in an internet congestion control model

Time delay is an important parameter in the problem of internet congestion control. According to some researches, time delay is not always constant and can be viewed as a periodic function of time for some cases. In this work, an internet congestion control model is considered to study the time-varying delay induced bursting-like motion, which consists of a rapid oscillation burst and quiescent steady state. Then, for the system with periodic delay of small amplitude and low frequency, the method of multiple scales is employed to obtain the amplitude of the oscillation. Based on the expression of the asymptotic solution, it can be found that the relative length of the steady state increases with amplitude of the variation of time delay and decreases with frequency of the variation of time delay. Finally, an effective method to control the bursting-like motion is proposed by introducing a periodic gain parameter with appropriate amplitude. Theoretical results are in agreement with that from numerical method.     

Oscillation behavior of solutions for even order neutral functional differential equations

Even order neutral functional differential equations are considered. Sufficient conditions for the oscillation behavior of solutions for this differential equation are presented. The new results are presented and some examples are also given.     

On the Absolute Exponential Stability of Solutions of Systems of Linear Parabolic Differential Equations with One Delay

We investigate necessary conditions for the absolute exponential stability of a system of linear parabolic differential equations with one delay.     

Absolute Exponential Stability of Solutions of Linear Parabolic Differential Equations with Delay

We find necessary and sufficient conditions for the absolute exponential stability of solutions of linear parabolic differential equations with delay in a pair of norms.     

Nonlinear aeroelastic behavior of an oscillating airfoil during stall-induced vibration

In this paper, nonlinear aeroelastic behavior of a two-dimensional symmetric rotor blade in the dynamic stall regime is investigated. Two different oscillation models have been considered here: pitching oscillation and flap–edgewise oscillation. Stall aeroelastic instability in such systems can potentially lead to structural damage. Hence it is an important design concern, especially for wind turbines and helicopter rotors, where such modes of oscillation are likely to take place. Most previous analyses of such dynamical systems are not exhaustive. System parameters like structural nonlinearity or initial conditions have not been studied which could play a significant role on the overall dynamics. In the present paper, a parametric study on the aeroelastic instability and the nonlinear dynamical behavior of the system has been performed. Emphasis is given on the effect of structural nonlinearity and initial conditions. The aerodynamic loads in the dynamic stall regime have been computed using the Onera model. The qualitative influence of the system parameters is different in the two systems studied. The effect of structural nonlinearity on the bifurcation pattern of the system response is significant in the case of pitching oscillation. The initial condition plays an important role on the aeroelastic stability as well as on the bifurcation pattern in both the systems. In the forced response study, interesting dynamical behavior, like period-3 response, has been observed in the pitching oscillation case. On the other hand, for the flap–edgewise oscillation case, super-harmonic and quasi-harmonic response have been found.     

Dynamic behavior in a delayed stage-structured population model with stochastic fluctuation and harvesting

In this paper, the dynamic behavior of a stage-structured population model involving gestation delay is investigated within stochastically fluctuating environment and harvesting. Firstly, the stability and Hopf bifurcation condition are described on the delayed population model within deterministic environment. Secondly, the stochastic population model systems are discussed by incorporating white noise terms to the deterministic system model. Finally, numerical simulations show that the gestation delay with larger magnitude has ability to drive the system from stable to unstable within the same fluctuating environment and the frequency and amplitude of oscillation for the population density is enhanced as environmental driving forces increase. These indicate that the magnitude of gestation delay plays a crucial role to determine the stability or instability and the magnitude of environmental driving forces plays a crucial role to determine the magnitude of oscillation of the population model system within fluctuating environment.     

Finite Dimensional State Representation of Linear and Nonlinear Delay Systems

We consider the question of when delay systems, which are intrinsically infinite dimensional, can be represented by finite dimensional systems. Specifically, we give conditions for when all the information about the solutions of the delay system can be obtained from the solutions of a finite system of ordinary differential equations. For linear autonomous systems and linear systems with time-dependent input we give necessary and sufficient conditions and in the nonlinear case we give sufficient conditions. Most of our results for linear renewal and delay differential equations are known in different guises. The novelty lies in the approach which is tailored for applications to models of physiologically structured populations. Our results on linear systems with input and nonlinear systems are new.     

Stability and oscillations in a slow-fast flexible joint system with transformation delay

Flexible joints are usually used to transfer velocities in robot systems and may lead to delays in motion transformation due to joint flexibility. In this paper, a linkrotor structure connected by a flexible joint or shaft is firstly modeled to be a slow-fast delayed system when moment of inertia of the lightweight link is far less than that of the heavy rotor. To analyze the stability and oscillations of the slowfast system, the geometric singular perturbation method is extended, with both slow and fast manifolds expressed analytically. The stability of the slow manifold is investigated and critical boundaries are obtained to divide the stable and the unstable regions. To study effects of the transformation delay on the stability and oscillations of the link, two quantitatively different driving forces derived from the negative feedback of the link are considered. The results show that one of these two typical driving forces may drive the link to exhibit a stable state and the other kind of driving force may induce a relaxation oscillation for a very small delay. However, the link loses stability and undergoes regular periodic and bursting oscillation when the transformation delay is large. Basically, a very small delay does not affect the stability of the slow manifold but a large delay affects substantially.     

Prediction of the dynamic oscillation threshold in a clarinet model with a linearly increasing blowing pressure: influence of noise

This paper presents an analysis of the effects of noise and precision on a simplified model of the clarinet driven by a variable control parameter. When the control parameter is varied, the clarinet model undergoes a dynamic bifurcation. A consequence of this is the phenomenon of bifurcation delay: the bifurcation point is shifted from the static oscillation threshold to a higher value, called dynamic oscillation threshold. In a previous work Bergeot et al. in Nonlinear Dyn. doi:10.1007/s11071-013-0806-y, (2013), the dynamic oscillation threshold is obtained analytically. In the present article, the sensitivity of the dynamic threshold on precision is analyzed as a stochastic variable introduced in the model. A new theoretical expression is given for the dynamic thresholds in presence of the stochastic variable, providing a fair prediction of the thresholds found in finite-precision simulations. These dynamic thresholds are found to depend on the increase rate and are independent on the initial value of the parameter, both in simulations and in theory.