时变小扰动Hamilton系统的Hopf分岔

运用Melnikov方法研究了时变小扰动Ｈamilton系统周期轨道发生Ｈopf分岔的条件,并将这些条件应用到一类三维时变小扰动非自治系统,数值结果验证了本文理论的正确性,进一步数值积分表明,所研究的系统还存在复杂而有规律的环面分岔行为。     

Analytical solutions for asymmetric periodic motions to chaos in a hardening Duffing oscillator

In this paper, the analytical dynamics of asymmetric periodic motions in the periodically forced, hardening Duffing oscillator is investigated via the generalized harmonic balance method. For the hardening Duffing oscillator, the symmetric periodic motions were extensively investigated with the aim of a good understanding of solutions with jumping phenomena. However, the asymmetric periodic motions for the hardening Duffing oscillators have not been obtained yet, and such asymmetric periodic motions are very important to find routes of periodic motions to chaos in the hardening Duffing oscillator analytically. Thus, the bifurcation trees from asymmetric period-1 motions to chaos are presented. The corresponding unstable periodic motions in the hardening Duffing oscillator are presented, and numerical illustrations of stable and unstable periodic motions are carried out as well. This investigation provides a comprehensive understanding of chaos mechanism in the hardening Duffing oscillator.     

The “resultant bifurcation diagram” method and its application to bifurcation behaviors of a symmetric railway bogie system

The concept of symmetric bifurcation for a symmetric wheel-rail system is defined. After that, the time response of the system can be achieved by the numerical integration method, and an unfixed and dynamic Poincaré section and its symmetric section for the symmetric wheel-rail system are established. Then the ??resultant bifurcation diagram?? method is constructed. The method is used to study the symmetric/asymmetric bifurcation behaviors and chaotic motions of a two-axle railway bogie running on an ideal straight and perfect track, and a variety of characteristics and dynamic processes can be obtained in the results. It is indicated that, for the possible sub-critical Hopf bifurcation in the railway bogie system, the stable stationary solutions and the stable periodic solutions coexist. When the speed is in the speed range of Hopf bifurcation point and saddle-node bifurcation point, the coexistence of multiple solutions can cause the oscillating amplitude change for different kinds of disturbance. Furthermore, it is found that there are symmetric motions for lower speeds, and then the system passes to the asymmetric ones for wide ranges of the speed, and returns again to the symmetric motions with narrow speed ranges. The rule of symmetry breaking in the system is through a blue sky catastrophe in the beginning.     

冲击消振器的概周期碰振运动分析

建立了冲击消振器对称周期运动的Poincar啨映射方程 ,讨论了对称周期运动的稳定性与局部分岔。通过数值仿真研究了冲击消振器在非共振、弱共振和强共振条件下的概周期碰振运动及其向混沌的转迁过程。     

CO-DIMENSION 2 BIFURCATIONS AND CHAOS IN CANTILEVERED PIPE CONVEYING TIME VARYING FLUID WITH THREE-TO-ONE INTERNAL RESONANCES

The nonlinear behavior of a cantilevered fluid conveying pipe subjected to principal parametric and internal resonances is investigated in this paper. The flow velocity is divided into constant and sinusoidai parts. The velocity value of the constant part is so adjusted such that the system exhibits 3:1 internal resonances for the first two modes. The method of multiple scales is employed to obtain the response of the system and a set of four first-order nonlinear ordinary-differential equations for governing the amplitude of the response. The eigenvalues of the Jacobian matrix are used to assess the stability of the equilibrium solutions with varying parameters. The codimension 2 derived from the double-zero eigenvaiues is analyzed in detail. The results show that the response amplitude may undergo saddle-node, pitchfork, Hopf, homoclinic loop and period-doubling bifurcations depending on the frequency and amplitude of the sinusoidal flow. When the frequency of the sinusoidal flow equals exactly half of the first-mode frequency, the system has a route to chaos by period-doubling bifurcation and then returns to a periodic motion as the amplitude of the sinusoidal flow increases.     

Bifurcation trees of period-3 motions to chaos in a time-delayed Duffing oscillator

Nonlinear Dynamics - In this paper, bifurcation trees of period-3 motions to chaos in a periodically forced, time-delayed, hardening Duffing oscillator are investigated by a semi-analytical method....     

Chaos controlling self-sustained electromechanical seismograph system based on the Melnikov theory

This paper deals with problems for which it is necessary to represent the oscillations of the electromechanical seismograph about their position of equilibrium as regards the synchronous condition. The loss of stability of the system occurs through a saddle-node bifurcation, where there is a collision of the stable orbit with an unstable one. Then, global bifurcations and chaotic dynamics of an electromechanical seismograph are the aims of this study. The electrical part of the model is described by an extended force Rayleigh oscillator with Φ ⁶-potential, while the mechanical part is described by a damped and driven linear oscillator. By using the direct perturbation technique, we analytically obtain the general solution of the first-order equation. Through the boundedness condition of the general solution we get the famous Melnikov function predicting the onset of chaos in the case where the Φ ⁶-potential is three wells, which are complemented by numerical simulations by which we illustrate the bifurcation curves and the fractality of the basins of attraction. The results show that the threshold amplitude of harmonic excitation for the onset of instability will move upwards as the amplitude intensity of the ground motion increases. These results suggest that much attention should be paid to controlling the increase of the amplitude of the ground motion, especially when the harmonic excited electromechanical seismograph system as a main device is applied to some practical systems.     

Bifurcations in the dynamics of an orthogonal double pendulum

The weakly nonlinear resonant response of an orthogonal double pendulum to planar harmonic motions of the point of suspension is investigated. The two pendulums in the double pendulum are confined to two orthogonal planes. For nearly equal length of the two pendulums, the system exhibits 1:1 internal resonance. The method of averaging is used to derive a set of four first order autonomous differential equations in the amplitude and phase variables. Constant solutions of the amplitude and phase equations are studied as a function of physical parameters of interest using the local bifurcation theory. It is shown that, for excitation restricted in either plane, there may be as many as six pitchfork bifurcation points at which the nonplanar solutions bifurcate from the planar solutions. These nonplanar motions can become unstable by a saddle-node or a Hopf bifurcation, giving rise to a new branch of constant solutions or limit cycle solutions, respectively. The dynamics of the amplitude equations in parameter regions of the Hopf bifurcations is then explored using direct numerical integration. The results indicate a complicated amplitude dynamics including multiple limit cycle solutions, period-doubling route to chaos, and sudden disappearance of chaotic attractors.     

Bifurcations and chaotic dynamics in suspended cables under simultaneous parametric and external excitations

The bifurcations and chaotic dynamics of parametrically and externally excited suspended cables are investigated in this paper. The equations of motion governing such systems contain quadratic and cubic nonlinearities, which may result in two-to-one and one-to-one internal resonances. The Galerkin procedure is introduced to simplify the governing equations of motion to ordinary differential equations with two-degree-of-freedom. The case of one-to-one internal resonance between the modes of suspended cables, primary resonant excitation, and principal parametric excitation of suspended cables is considered. Using the method of multiple scales, a parametrically and externally excited system is transformed to the averaged equations. A pseudo arclength scheme is used to trace the branches of the equilibrium solutions and an investigation of the eigenvalues of the Jacobian matrix is used to assess their stability. The equilibrium solutions experience pitchfork, saddle-node, and Hopf bifurcations. A detailed bifurcation analysis of the dynamic (periodic and chaotic) solutions of the averaged equations is presented. Five branches of dynamic solutions are found. Three of these branches that emerge from two Hopf bifurcations and the other two are isolated. The two Hopf bifurcation points, one is supercritical Hopf bifurcation point and another is primary Hopf bifurcation point. The limit cycles undergo symmetry-breaking, cyclic-fold, and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging, boundary crises. Simultaneous occurrence of the limit cycle and chaotic attractors, homoclinic orbits, homoclinic explosions and hyperchaos are also observed.     

Amplitude modulated dynamics of a resonantly excited autoparametric two degree-of-freedom system

Forced, weakly nonlinear oscillations of a two degree-of-freedom autoparametric vibration absorber system are studied for resonant excitations. The method of averaging is used to obtain first-order approximations to the response of the system. A complete bifurcation analysis of the averaged equations is undertaken in the subharmonic case of internal and external resonance. The locked pendulum mode of response is found to bifurcate to coupled-mode motion for some excitation frequencies and forcing amplitudes. The coupled-mode response can undergo Hopf bifurcation to limit cycle motions, when the two linear modes are mistuned away from the exact internal resonance condition. The software packages AUTO and KAOS are used and a numerically assisted study of the Hopf bifurcation sets, and dynamic steady solutions of the amplitude or averaged equations is presented. It is shown that both super-and sub-critical Hopf bifurcations arise and the limit cycles quickly undergo period-doubling bifurcations to chaos. These imply chaotic amplitude modulated motions for the system.     

Symmetry, cusp bifurcation and chaos of an impact oscillator between two rigid sides

Both the symmetric period n-2 motion and asymmetric one of a one-degree- of-freedom impact oscillator are considered.The theory of bifurcations of the fixed point is applied to such model,and it is proved that the symmetric periodic motion has only pitchfork bifurcation by the analysis of the symmetry of the Poincarémap.The numerical simulation shows that one symmetric periodic orbit could bifurcate into two antisymmet- ric ones via pitchfork bifurcation.While the control parameter changes continuously, the two antisymmetric periodic orbits will give birth to two synchronous antisymmetric period-doubling sequences,and bring about two antisymmetric chaotic attractors subse- quently.If the symmetric system is transformed into asymmetric one,bifurcations of the asymmetric period n-2 motion can be described by a two-parameter unfolding of cusp, and the pitchfork changes into one unbifurcated branch and one fold branch.     

Chaos and instability in a power system: Subharmonic-resonant case

The response of a single-machine quasi-infinite busbar system to the simultaneous occurrence of principal parametric resonance and subharmonic resonance of order one-half is investigated. By numerical simulations we show the existence of oscillatory solutions (limit cycles), period-doubling bifurcations, chaos, and unbounded motions (loss of synchronism). The method of multiple scales is used to derive a second-order analytical solution that predicts (a) the onset of period-doubling bifurcations, which is a precursor to chaos and unbounded motions (loss of synchronism), and (b) saddle-node bifurcations, which may be precursors to loss of synchronism.     

Modified post-bifurcation dynamics and routes to chaos from double-Hopf bifurcations in a hyperchaotic system

In order to understand the onset of hyperchaotic behavior recently observed in many systems, we study bifurcations in the modified Chen system leading from simple dynamics into chaotic regimes. In particular, we demonstrate that the existence of only one fixed point of the system in all regions of parameter space implies that this simple point attractor may only be destabilized via a Hopf or double Hopf bifurcation as system parameters are varied. Saddle-node, transcritical and pitchfork bifurcations are precluded. The normal form immediately following double Hopf bifurcations is constructed analytically by the method of multiple scales. Analysis of this generalized double Hopf normal form along standard lines reveals possible regimes of periodic solutions, two-period tori, and three-period tori in parameter space. However, considering these more carefully, we find that only certain combinations or sequences of these dynamical regimes are possible, while others derived and considered in earlier work are in fact mathematically impossible. We also discuss the post-bifurcation dynamics in the context of two intermittent routes to chaos (routes following either (i) subcritical or (ii) supercritical Hopf or double Hopf bifurcations). In particular, the route following supercritical bifurcations is somewhat subtle. Such behavior following repeated Hopf bifurcations is well-known and widely observed, including in the classical Ruelle?CTakens and quasiperiodic routes to chaos. However, to the best of our knowledge, it has not been considered in the context of the double-Hopf normal form, although it has been numerically observed and tracked in the post-double Hopf regime. Numerical simulations are employed to corroborate these various predictions from the normal form. They reveal the existence of stable periodic and toroidal attractors in the post-supercritical-Hopf cases, and either attractors at infinity or bounded chaotic dynamics following subcritical Hopf bifurcations. Future work will map out the remainder of the routes into the chaotic regimes, including further bifurcations of the post-supercritical-Hopf two- and three-tori via either torus doubling or breakdown.     

Coupled, forced response of an axially moving strip with internal resonance

In this paper, the forced response of a non-linear axially moving strip with coupled transverse and longitudinal motions is studied. In particular, the response of the system is examined in the neighborhood of a 3 : 1 internal resonance between the first two transverse modes. The equations of motion are derived using the Hamilton's Principle and discretized by the Galerkin's method. First, with the longitudinal motion neglected, the forced transverse response is investigated by applying the method of multiple scales to assess the effects of speed and the internal resonance. In general, the speed is shown to affect each mode differently. The internal resonance results in the constant solutions having transition to instability of both a saddle-node type and a Hopf bifurcation. In the region where the Hopf bifurcation occurs, steady-state periodic motion does not exist. Instead the stable motion is amplitude- and phase-modulated. When the coupled system with longitudinal motion is examined with internal resonance, results reveal that the modulated motions disappear. Thus, the presence of the longitudinal motion has a stabilizing effect on the transverse modes in the Hopf bifurcation region. The second longitudinal mode is shown to drift due primarily to a direct excitation of the first transverse mode. Effects of the longitudinal motion on the transverse response are shown to be significant for speeds both away from and close to the critical speed.     

DC-DC开关功率变换器的非线性动力学行为研究

DC-DC开关功率变换器是一种典型的分段光滑动力学系统, 在一定的工作和参数条件下, 系统会出现各种分岔如倍周期分岔、Hopf分岔、边界碰撞分岔和混沌运动. 系统评述了DC-DC开关功率变换器的非线性动力学行为的研究进展;介绍了离散非线性映射、分段线性模型、平均值模型等3种建模方法;分析了这种电路系统中的分岔特点及通向混沌的途径与机制;结合我们的研究工作, 讨论了对这种电路系统进行混沌控制的必要性及相关策略;最后, 从应用的角度提出了未来的若干研究方向.      

Periodic-impact motions and bifurcations in dynamics of a plastic impact oscillator with a frictional slider

A two-degree-of-freedom plastic impact oscillator with a frictional slider is considered. Dynamics of the plastic impact oscillator are analyzed by a three-dimensional map, which describes free flight and sticking solutions of two masses of the system, between impacts, supplemented by transition conditions at the instants of impacts. Piecewise property and singularity are found to exist in the impact Poincaré map. The piecewise property of the map is caused by the transitions of free flight and sticking motions of two masses immediately after the impact, and the singularity of the map is generated via the grazing contact of two masses immediately before the impact. These properties of the map have been shown to exhibit particular types of sliding and grazing bifurcations of periodic-impact motions under parameter variation. The influence of piecewise property, grazing singularity and parameter variation on dynamics of the vibro-impact system is analyzed. The global bifurcation diagrams of before-impact velocity as a function of the excitation frequency are plotted to predict much of the qualitative behavior of the system. The global bifurcations of period-N single-impact motions of the plastic impact oscillator are found to exhibit extensive and systematic characteristics. Dynamics of the impact oscillator, in the elastic impact case, is also analyzed. This type of impact is modelled by using the conditions of conservation of momentum and an instantaneous coefficient of restitution rule. The differences in periodic-impact motions and bifurcations are found by making a comparison between dynamic behaviors of the plastic and elastic impact oscillators with a frictional slider. The best progression of the plastic impact oscillator is found to occur in period-1 single-impact sticking motion with large impact velocity. The largest progression of the elastic impact oscillator occurs in period-1 multi-impact motion. The simulative results show that the plastic impact feature for the impact-progressive oscillator is of a considerable importance in minimizing adverse effects such as high noise levels, wear and tear caused by impacts.     

Bifurcation and path-following continuation analysis of periodic orbits of an extended Fermi oscillator model

In this research, we offer eigenvalue analysis and path following continuation to describe the impact, stick, and non-stick between the particle and boundaries to understand the nonlinear dynamics of an extended Fermi oscillator. The principles of discontinuous dynamical systems will be utilized to explain the moving process in such an extended Fermi oscillator. The motion complexity and stick mechanism of such an oscillator are demonstrated using periodic and chaotic motions. The major parameters are the frequency, amplitude in periodic excitation force, and the gap between the top and bottom boundary. We employ path-following analysis to illustrate the bifurcations that lead to solution destabilization. We present the evolution of the period solutions of the extended Fermi oscillator as the parameter varies. From the viewpoint of eigenvalue analysis, the essence of period-doubling, saddle-node, and Torus bifurcation is revealed. Numerical continuation methods are used to do a complete one- and two-parameter bifurcation analysis of the extended Fermi oscillator. The presence of codimension-one bifurcations of limit cycles, such as saddle-node, period-doubling, and Torus bifurcations, is shown in this work. Bifurcations cause all solutions to lose stability, according to our findings. The acquired results provide a better understanding of the extended Fermi oscillator mechanism and demonstrate that we may control the system dynamics by modifying the parameters.

     

On the Hamiltonian Hopf Bifurcations in the 3D Hénon–Heiles Family

An axially symmetric perturbed isotropic harmonic oscillator undergoes several bifurcations as the parameter adjusting the relative strength of the two terms in the cubic potential is varied. We show that three of these bifurcations are Hamiltonian Hopf bifurcations. To this end we analyse an appropriately chosen normal form. It turns out that the linear behaviour is not that of a typical Hamiltonian Hopf bifurcation as the eigen-values completely vanish at the bifurcation. However, the nonlinear structure is that of a Hamiltonian Hopf bifurcation. The result is obtained by formulating geometric criteria involving the normalized Hamiltonian and the reduced phase space.     

非自治时滞反馈控制系统的周期解分岔和混沌

研究时滞反馈控制对具有周期外激励非线性系统复杂性的影响机理,研究对应的线性平衡态失稳的临界边界,将时滞非线性控制方程化为泛函微分方程,给出由Hopf分岔产生的周期解的解析形式．通过分析周期解的稳定性得到周期解的失稳区域,使用数值分析观察到时滞在该区域可以导致系统出现倍周期运动、锁相运动、概周期运动和混沌运动以及两条通向混沌的道路：倍周期分岔和环面破裂．其结果表明,时滞在控制系统中可以作为控制和产生系统的复杂运动的控制“开关”．     

Codimension-two bifurcation analysis in two-dimensional Hindmarsh–Rose model

In this paper, we analyze the codimension-2 bifurcations of equilibria of a two-dimensional Hindmarsh–Rose model. By using the bifurcation methods and techniques, we give a rigorous mathematical analysis of Bautin bifurcation. The main result is that no more than two limit cycles can be bifurcated from the equilibrium via Hopf bifurcation; sufficient conditions for the existence of one or two limit cycles are obtained. This paper also shows that the model undergoes a Bogdanov–Takens bifurcation which includes a saddle-node bifurcation, an Andronov–Hopf bifurcation, and a homoclinic bifurcation. In some case, the globally asymptotical stability is discussed.