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1.
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. 相似文献
2.
Multiple Scales without Center Manifold Reductions for Delay Differential Equations near Hopf Bifurcations 总被引:3,自引:0,他引:3
We study small perturbations of three linear Delay DifferentialEquations (DDEs) close to Hopf bifurcation points. In analytical treatments of such equations, many authors recommend a center manifold reductionas a first step. We demonstrate that the method of multiple scales, onsimply discarding the infinitely many exponentially decaying components of the complementary solutionsobtained at each stage of the approximation,can bypass the explicit center manifold calculation.Analytical approximations obtained for the DDEs studied closely matchnumerical solutions. 相似文献
3.
时滞对于参数激励系统周期运动的影响 总被引:2,自引:0,他引:2
本文以结构物承受周期激励地震波和具有时滞的弹性地基为背景,研究时滞对于结构物振动响应的影响规律。问胚的数学模型是一个非线性参数激励时滞系统,采用中心流形和平均法得到Hopf分岔方程,研究结果表明时滞量对结构物的地震响应有重要影响,它可使响应振幅增大,意味着结构物遭破坏的危险性增大,同时,结果也表明可以通过阻尼控制器克服时滞引起的危险性。最后,对于不同的时滞量,比较理论分析与数值模拟结果基本吻合,说明了本文结果的可靠性。 相似文献
4.
A kind of 2-dimensional neural network model with delay is considered. By analyzing the distribution of the roots of the characteristic
equation associated with the model, a bifurcation diagram was drawn in an appropriate parameter plane. It is found that a
line is a pitchfork bifurcation curve. Further more, the stability of each fixed point and existence of Hopf bifurcation were
obtained. Finally, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions were determined
by using the normal form method and centre manifold theory.
Foundation item: the National Natural Science, Foundation of China (19831030)
Biography: WEI Jun-jie, Professor, Doctor, E-mail: weijj@hit.edu.cn 相似文献
5.
BIFURCATION IN A TWO-DIMENSIONAL NEURAL NETWORK MODEL WITH DELAY 总被引:1,自引:0,他引:1
IntroductionForunderstandingthedynamicsofneuralnetworks ,thepropertiesofstabilityandbifurcationinasimplifiednon_self_connectionneuralnetwork u1(t) =-μ1u1(t) aF(u2 (t-τ2 ) ) , u2 (t) =-μ2 u2 (t) bG(u1(t-τ1) ) ( 1 )hasbeenstudied .Forexample ,inRef.[1 ]ChenandWustudiedtheexistenceoftheslowlyoscillatingperiodicsolutionbyusingthemethodofdiscreteLiapunovfunction .InRef.[2 ]thesumoftimedelaysτ=τ1 τ2 beingregardedasabifurcationparameter,theexistenceoflocalHopfbifurcationandthepropertiesof… 相似文献
6.
Sue Ann Campbell Victor G. LeBlanc 《Journal of Dynamics and Differential Equations》1998,10(2):327-346
A second-order delay differential equation (DDE) which models certain mechanical and neuromechanical regulatory systems is analyzed. We show that there are points in parameter space for which 1:2 resonant Hopf–Hopf interaction occurs at a steady state of the system. Using a singularity theoretic classification scheme [as presented by LeBlanc (1995) and LeBlanc and Langford (1996)], we then give the bifurcation diagrams for periodic solutions in two cases: variation of the delay and variation of the feedback gain near the resonance point. In both cases, period-doubling bifurcations of periodic solutions occur, and it is argued that two tori can bifurcate from these periodic solutions near the period doubling point. These results are then compared to numerical simulations of the DDE. 相似文献
7.
A planar model with multiple delays is studied. The singularities of the model and the corresponding bifurcations are investigated
by using the standard dynamical results, center manifold theory and normal form method of retarded functional differential
equations. It is shown that Bogdanov–Takens (BT) singularity for any time delays, and a serious of pitchfork and Hopf bifurcation
can co-existent. The versal unfoldings of the normal forms at the BT singularity and the singularity of a pure imaginary and
a zero eigenvalue are given, respectively. Numerical simulations have been provided to illustrate the theoretical predictions. 相似文献
8.
Pavol Brunovský Alexander Erdélyi Hans-Otto Walther 《Journal of Dynamics and Differential Equations》2004,16(2):393-432
A delay differential equation is presented which models how the behavior of traders influences the short time price movements of an asset. Sensitivity to price changes is measured by a parameter a. There is a single equilibrium solution, which is non-hyperbolic for all a>0. We prove that for a< 1 the equilibrium is asymptotically stable, and that for a>1 a 2-dimensional global center-unstable manifold connects the equilibrium to a periodic orbit. Its birth at a=1 is not of Hopf type and seems part of a Takens–Bogdanov scenario. 相似文献
9.
Aldo Rustichini 《Journal of Dynamics and Differential Equations》1989,1(2):145-177
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. 相似文献
10.
The vector field induced on the finite-dimensional inertial manifold of a delay equation with small delay is proved to agree, up to the order of the expansion, with the vector field induced on a slow manifold of the differential equation obtained from the delay equation by expanding to some finite order in powers of the delay. In addition, the smoothness of inertial vector fields, the smoothness of slow vector fields, and the existence of combinatorial-style identities obtained by equating the series expansions of the slow and inertial vector fields are discussed.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday.AMS Subject Classification: 34K19. 相似文献
11.
Hopf Bifurcation and Stability of Periodic Solutions for van der Pol Equation with Distributed Delay 总被引:2,自引:0,他引:2
The van der Pol equation with a distributed time delay is analyzed. Itslinear stability is investigated by employing the Routh–Hurwitzcriteria. Moreover, the local asymptotic stability conditions are alsoderived. By using the mean time delay as a bifurcation parameter, themodel is found to undergo a sequence of Hopf bifurcations. The directionand the stability criteria of the bifurcating periodic solutions areobtained by the normal form theory and the center manifold theorem. Somenumerical simulation examples for justifying the theoretical analysisare also given. 相似文献
12.
Melanie Rupflin 《Journal of Dynamics and Differential Equations》2009,21(1):45-71
For a certain class of delay equations with piecewise constant nonlinearities we prove the existence of a rapidly oscillating
stable periodic solution and a rapidly oscillating unstable periodic solution. Introducing an appropriate Poincaré map, the
dynamics of the system may essentially be reduced to a two dimensional map, the periodic solutions being represented by a
stable and a hyperbolic fixed point. We show that the two dimensional map admits a one dimensional invariant manifold containing
the two fixed points. It follows that the delay equations under consideration admit a one parameter family of rapidly oscillating
heteroclinic solutions connecting the rapidly oscillating unstable periodic solution with the rapidly oscillating stable periodic
solution.
相似文献
13.
Daniel Stoffer 《Journal of Dynamics and Differential Equations》2008,20(1):201-238
We prove analytically that there exist delay equations admitting rapidly oscillating stable periodic solutions. Previous results
were obtained with the aid of computers, only for particular feedback functions. Our proofs work for stiff equations with
several classes of feedback functions. Moreover, we prove that for negative feedback there exists a class of feedback functions
such that the larger the stiffness parameter is, the more stable rapidly oscillating periodic solutions there are. There are
stable periodic solutions with arbitrarily many zeros per unit time interval if the stiffness parameter is chosen sufficiently
large. 相似文献
14.
This paper is concerned with the bifurcation of a complex Swift-Hohenberg equation. The attractor bifurcation of the complex Swift-Hohenberg equation on a one- dimensional domain (0, L) is investigated. It is shown that the n-dimensional complex Swift-Hohenberg equation bifurcates from the trivial solution to an attractor under the Dirichlet boundary condition on a general domain and under a periodic boundary condition when the bifurcation parameter crosses some critical values. The stability property of the bifurcation attractor is analyzed. 相似文献
15.
In this paper, a physiological model of invasive blood-glucose (BG) measurement is employed to consider the diabetic treatment by the external auxiliary system, i.e., artificial pancreas (AP). For such system, there are two time delays, i.e., technological and liver's physiological delay, where the former comes from external auxiliary system with the active pancreas inputting. The technological delay and the infection degree of patients are considered as two controlled parameters to regulate the BG level of patients. This two parameters can also lead to the non-resonant double Hopf bifurcations. The classification and unfolding for the non-resonant double Hopf bifurcation are performed in terms of non-linear dynamics. The results show that such controlled parameters are very important. They can determine the efficiency for the diabetic treatment. It implies whether the diabetic patients recover or are still tormented by the simple or complex glucose fluctuation. The results have also been promising applications on analyzing, predicting and optimizing the medical outcome, evaluating the medical risk and feasibility. The physiological meaning in this paper is that one is able to achieve the better medical outcomes for the different patients by controlling the technological delay qualitatively. 相似文献
16.
Hopf bifurcation of an oscillator with quadratic and cubic nonlinearities and with delayed velocity feedback 总被引:2,自引:0,他引:2
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) 相似文献
17.
Jaume Llibre Alexandrina–Alina Tarţa 《Journal of Dynamics and Differential Equations》2007,19(1):199-213
In this paper, we give sufficient conditions for the existence of periodic orbits of some systems of delay differential equations with a unique delay having 3, 4 or n equations. Moreover, we provide examples of delay systems satisfying the different sets of sufficient conditions. 相似文献
18.
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. 相似文献
19.
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. 相似文献
20.
We investigated the structure of the so-called first Hopf bifurcation surface associated to a differential equation with two time delays. A geometrical approach leading naturally to a number theoretic approach provides rigourous results which are corroborated by previous numerical and experimental (optical compound resonator) results. 相似文献