一类碰撞振动系统在内伊马克沙克-音叉分岔点附近的局部两参数动力学

考虑一类具有对称性的三自由度碰撞振动系统.系统的庞加莱映射在一定条件下存在对称不动点,对应于系统的对称周期运动.根据对称性导出庞加莱映射P是另外一个隐式虚拟映射Q的二次迭代.推导了庞加莱映射对称不动点的解析表达式.根据映射不动点的稳定性及分岔理论,映射P的对称不动点发生内伊马克沙克-音叉(Neimark--Saker-pitchfork)分岔对应于映射Q发生内伊马克沙克-倍化(Neimark--Sakerflip)分岔.利用隐式虚拟映射Q,通过对范式作两参数开折分析,研究了映射P的对称不动点在内伊马克沙克-音叉分岔点附近的局部动力学行为.碰撞振动系统在这个余维二分岔点附近的局部动力学行为可能表现为投影后的庞加莱截面上的单一对称不动点、一对共轭不动点、单一对称拟周期吸引子以及一对共轭拟周期吸引子.数值模拟得到了内伊马克沙克-音叉分岔点附近的各种可能情况.内伊马克沙克-分岔和音叉分岔互相作用可能产生新的结果:对称不动点虽然首先分岔为两个共轭不动点,但是这两个共轭不动点是不稳定的,最终收敛到同一个对称拟周期吸引子.     

Analytical period-3 motions to chaos in a hardening Duffing oscillator

In this paper, bifurcation trees of period-3 motions to chaos in the periodically forced, hardening Duffing oscillator are investigated analytically. Analytical solutions for period-3 and period-6 motions are used for the bifurcation trees of period-3 motions to chaos. Such bifurcation trees are based on the Hopf bifurcations of asymmetric period-3 motions. In addition, an independent symmetric period-3 motion without imbedding in chaos is discovered, and such a symmetric period-3 motion possesses saddle-node bifurcations only. The switching of symmetric to asymmetric period-3 motions is completed through saddle-node bifurcations, and the onset of asymmetric period-6 motions occurs at the Hopf bifurcations of asymmetric period-3 motions. Continuously, the onset of period-12 motions is at the Hopf bifurcation of asymmetric period-6 motions. With such bifurcation trees, the chaotic motions relative to asymmetric period-3 motions can be determined analytically. This investigation provides a systematic way to study analytical dynamics of chaos relative to period-m motions in nonlinear dynamical systems.     

RESEARCH IN STABILITY OF PERIODIC MOTION,BIFURCATION AND CHAOS IN A VIBRATORY SYSTEM WITH A CLEARANCE

A two-degrees-of-freedom vibratory system with a clearance or gap is under consideration based on the Poincard map. Stability and local bifurcation of the period-one doubleimpact symmetrical motion of the system are analyzed by using the equation of map. The routes from periodic impact motions to chaos, via pitchfork bifurcation, period-doubling bifurcation and grazing bifurcation, are studied by numerical simulation. Under suitable system parameter conditions, Neimark-Sacker bifurcations associated with periodic impact motion can occur in the two-degrees-of-freedom vibro-impact system.     

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.     

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.      

Two-parameter bifurcation analysis of limit cycles of a simplified railway wheelset model

The effect of the nonlinear terms on bifurcation behaviors of limit cycles of a simplified railway wheelset model is investigated. At first, the stable equilibrium state loses its stability via a Hopf bifurcation. The bifurcation curve is divided into a supercritical branch and a subcritical one by a generalized Hopf point, which plays a key role in determining the occurrence of flange contact and derailment of high-speed railway vehicles, and the occurrence of this critical situation is an important decision-making criteria for design parameters. Secondly, bifurcations of limit cycles are discussed by comparing the bifurcation behavior of cycles for two different nonlinear parameters. Unlike local Hopf bifurcation analysis based on a single bifurcation parameter in most papers, global bifurcation analysis of limit cycles based on two bifurcation parameters is investigated, simultaneously. It is shown that changing nonlinear parameter terms can affect bifurcation types of cycles and division of parameter domains. In particular, near the branch points of cycles, two symmetrical limit cycles are created by a pitchfork bifurcation and then two symmetrical cycles both undergo a period-doubling bifurcation to form two stable period-two cycles. Around the resonant points, period orbits can make several turns, whose number of turns corresponds to the ratio of resonance. Thirdly, near the Neimark–Sacker bifurcation of cycles, a stable torus is created by a supercritical Neimark–Sacker bifurcation, which shows that the orbit of the model exhibits modulated oscillations with two frequencies near the limit cycle. These results demonstrate that nonlinear parameter terms can produce very complex global bifurcation phenomena and make obvious effects on possible hunting motions even though a simple railway wheelset model is concerned.

     

Local bifurcation analysis and ultimate bound of a novel 4D hyper-chaotic system

This paper presents a new four-dimensional smooth quadratic autonomous hyper-chaotic system which can generate novel two double-wing periodic, quasi-periodic and hyper-chaotic attractors. The Lyapunov exponent spectrum, bifurcation diagram and phase portrait are provided. It is shown that this system has a wide hyper-chaotic parameter. The pitchfork bifurcation and Hopf bifurcation are discussed using the center manifold theory. The ellipsoidal ultimate bound of the typical hyper-chaotic attractor is observed. Numerical simulations are given to demonstrate the evolution of the two bifurcations and show the ultimate boundary region.     

Global Bifurcation Analysis of a Controlled Underactuated Mechanical System

In this paper, bifurcation theory is employed to classify different dynamical behaviors arising in an underactuated mechanical system subject to bounded controls. The methodology is applied to an inertia wheel pendulum consisting of a simple pendulum with a rotating disk at the end. Restricting the magnitude of the control action places an important obstacle to the design of a continuous controller capable of swinging-up and stabilize the pendulum at the inverted position: the arm only can reach that position by means of oscillations of increasing amplitude. The controller is derived from a simple nonlinear state-feedback law, followed by a saturating device that limits the maximum amplitude of the control action applied to the system. This bound gives birth to a rich dynamical behavior, including pitchfork and Hopf bifurcations of equilibria, saddle-node bifurcations of periodic orbits, homoclinic and heteroclinic bifurcations. The global dynamics is analyzed in terms of certain control gains and a two-parameter bifurcation diagram is derived. It is shown that the dynamics on this bifurcation diagram is organized in a pair of codimension-two rotationally symmetric bifurcation points. Finally, it is found out that when the control gains lie on a certain region in the parameter space simultaneous stabilization of the upright position together with a large basin of attraction is obtained. Simulation results show that almost global stabilization of the system can be achieved.     

BIFURCATIONS OF AIRFOIL IN INCOMPRESSIBLE FLOW

Bifurcations of an airfoil with nonlinear pitching stiffness in incompressible flow are investigated. The pitching spring is regarded as a spring with cubic stiffness. The motion equations of the airfoil are written as the four dimensional one order differential equations. Taking air speed and the linear part of pitching stiffness as the parameters, the analytic solutions of the critical boundaries of pitchfork bifurcations and Hopf bifurcations are obtained in 2 dimensional parameter plane. The stabilities of the equilibrium points and the limit cycles in different regions of 2 dimensional parameter plane are analyzed. By means of harmonic balance method, the approximate critical boundaries of 2-multiple semi-stable limit cycle bifurcations are obtained, and the bifurcation points of supercritical or subcritical Hopf bifurcation are found. Some numerical simulation results are given.     

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。     

一类二维非自治离散系统中的复杂簇发振荡结构1）

簇发振荡是多时间尺度系统复杂动力学行为的典型代表,簇发振荡的动力学机制与分类问题是簇发研究的重要问题之一,但当前学者们所揭示的簇发振荡的结构大多较为简单.研究以非自治离散Duffing系统为例,探讨具有复杂分岔结构的新型簇发振荡模式,并将其分为两大类,一类经由Fold分岔所诱发的对称式簇发,另一类经由延迟倍周期分岔所诱发的非对称式簇发.快子系统的分岔表现为典型的含有两个Fold分岔点的S形不动点曲线,其上、下稳定支可经由倍周期（即Flip）分岔通向混沌.当非自治项（即慢变量）穿越Fold分岔点时,系统的轨线可以向上、下稳定支的各种吸引子（例如,周期轨道和混沌）进行转迁,因此得到了经由Fold分岔所诱发的各种对称式簇发;而当非自治项无法穿越Fold分岔点,但可以穿越Flip分岔点时,系统产生了延迟Flip分岔现象.基于此,得到了经由延迟Flip分岔所诱发的各种非对称簇发.特别地,文中所报道的簇发振荡模式展现出复杂的反向Flip分岔结构.研究结果表明,这与非自治项缓慢地反向穿越快子系统的Flip分岔点有关.研究结果丰富了离散系统簇发的动力学机理和分类.     

Dynamics of nonlocal and localized spatiotemporal solitons for a partially nonlocal nonlinear Schrödinger equation

Modern methods of nonlinear dynamics including time histories, phase portraits, power spectra, and Poincaré sections are used to characterize the stability and bifurcation regions of a cantilevered pipe conveying fluid with symmetric constraints at the point of contact. In this study, efforts are made to demonstrate the importance of characterizing the system at the arbitrarily positioned symmetric constraints rather than at the tip of the cantilevered pipe. Using the full nonlinear equations of motion and the Galerkin discretization, a nonlinear analysis is performed. After validating the model with previous results using the bifurcation diagrams and achieving full agreement, the bifurcation diagram at the point of contact is further investigated to select key flow velocities of interest. In addition to demonstrating the progression of the selected regions using primarily phase portraits, a detailed comparison is made between the tip and the point of contact at the key flow velocities. In doing so, period doubling, pitchfork bifurcations, grazing bifurcations, sticking, and chaos that occur at the point of contact are found to not always occur at the tip for the same flow speed. Thus, it is shown that in the case of cantilevered pipes with constraints, more accurate characterization of the system is obtained in a specified range of flow velocities by characterizing the system at the point of contact rather than at the tip.     

Quasiperiodic and exponential transient phase waves and their bifurcations in a ring of unidirectionally coupled parametric oscillators

Phase waves rotating in a ring of unidirectionally coupled parametric oscillators are studied. The system has a pair of spatially uniform stable periodic solutions with a phase difference and an unstable quasiperiodic traveling phase wave solution. They are generated from the origin through a period doubling bifurcation and the Neimark?CSacker bifurcation, respectively. In transient states, phase waves rotating in a ring are generated, the duration of which increases exponentially with the number of oscillators (exponential transients). A power law distribution of the duration of randomly generated phase waves and the noise-sustained propagation of phase waves are also shown. These properties of transient phase waves are well described with a kinematical equation for the propagation of wave fronts. Further, the traveling phase wave is stabilized through a pitchfork bifurcation and changes into a standing wave through pinning. These bifurcations and exponential transient rotating waves are also shown in an autonomous system with averaging and a coupled map model, and they agree with each other.     

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 analysis of the motion of an asymmetric double pendulum subjected to a follower force: Codimension three problem

Local and global bifurcations in the motion of a double pendulum subjected to a follower force have been studied when the follower force and the springs at the joints have structural asymmetries. The bifurcations of the system are examined in the neighborhood of double zero eigenvalues. Applying the center manifold and the normal form theorem to a four-dimensional governing equation, we finally obtain a two-dimensional equation with three unfolding parameters. The local bifurcation boundaries can be obtained for the criteria for the pitchfork and the Hopf bifurcation. The Melnikov theorem is used to find the global bifurcation boundaries for appearance of a homoclinic orbit and coalescence of two limit cycles. Numerical simulation was performed using the original four-dimensional equation to confirm the analytical prediction.     

Vibro-impact dynamics near a strong resonance point

Two typical vibratory systems with impact are considered, one of which is a two-degree-of-freedom vibratory system impacting an unconstrained rigid body, the other impacting a rigid amplitude stop. Such models play an important role in the studies of dynamics of mechanical systems with repeated impacts. Two-parameter bifurcations of fixed points in the vibro-impact systems, associated with 1:4 strong resonance, are analyzed by using the center manifold and normal form method for maps. The single-impact periodic motion and Poincaré map of the vibro-impact systems are derived analytically. Stability and local bifurcations of a single-impact periodic motion are analyzed by using the Poincaré map. A center manifold theorem technique is applied to reduce the Poincaré map to a two-dimensional one, and the normal form map for 1:4 resonance is obtained. Local behavior of two vibro-impact systems, near the bifurcation points for 1:4 resonance, are studied. Near the bifurcation point for 1:4 strong resonance there exist a Neimark–Sacker bifurcation of period one single-impact motion and a tangent (fold) bifurcation of period 4 four-impact motion, etc. The results from simulation show some interesting features of dynamics of the vibro-impact systems: namely, the “heteroclinic” circle formed by coinciding stable and unstable separatrices of saddles, T _in, T _on and T _out type tangent (fold) bifurcations, quasi-periodic impact orbits associated with period four four-impact and period eight eight-impact motions, etc. Different routes of period 4 four-impact motion to chaos are obtained by numerical simulation, in which the vibro-impact systems exhibit very complicated quasi-periodic impact motions. The project supported by National Natural Science Foundation of China (50475109, 10572055), Natural Science Foundation of Gansu Province Government of China (3ZS061-A25-043(key item)). The English text was polished by Keren Wang.     

INSTABILITY AND CHAOS IN A PIPE CONVEYING FLUID WITH ADDED MASS AT FREE END

This paper shows the mechanism of instability and chaos in a cantilevered pipe conveying steady fluid. The pipe under consideration has added mass or a nozzle at the free end. The Galerkin method is used to transform the original system into a set of ordinary differential equations and the standard methods of analysis of the discrete system are introduced to deal with the instability. With either the nozzle parameter or the flow velocity increasing, a route to chaos can be observed very clearly: the pipe undergoing buckling (pitchfork bifurcation), flutter (Hopf bifurcation), doubling periodic motion (pitchfork bifurcation) and chaotic motion occurring finally. The project supported by the National Key Projects of China under grant No. PD9521907 and Science Foundation of Tongji University under grant No. 1300104010.     

3-torus,quasi-periodic bursting,symmetric subHopf/fold-cycle bursting,subHopf/fold-cycle bursting and their relation

In this paper, the dynamical behaviors of a perturbed hyperchaotic system is studied. The fast subsystem is examined using local stability and bifurcations, including simple bifurcation, Hopf bifurcation, and fold bifurcation of limit cycle. The results of these analysis are applied to the perturbed hyperchaotic system, where two types of periodic bursting, i.e., symmetric subHopf/fold-cycle bursting and subHopf/fold-cycle bursting, can be observed. In particular, the symmetric subHopf/fold-cycle bursting is new and has not been reported in previous work. With variation of the parameter, subHopf/fold-cycle bursting with symmetric structure may bifurcate into two coexisted subHopf/fold-cycle bursting symmetric to each other. Moreover, 3-torus and quasi-periodic bursting (2-torus) are presented. The relation among 3-torus, quasi-periodic bursting, and symmetric subHopf/fold-cycle bursting is discussed, which suggests that 3-torus may develop to quasi-periodic bursting, while quasi-periodic bursting may further evolve to symmetric subHopf/fold-cycle bursting.