A codimension-two resonant bifurcation from a heteroclinic cycle with complex eigenvalues

Robust heteroclinic cycles between equilibria lose stability either through local bifurcations of their equilibria or through global bifurcations. This paper considers a global loss of stability termed a 'resonant' bifurcation. This bifurcation is usually associated with the birth or death of a nearby periodic orbit, and generically occurs in either a supercritical or subcritical manner. For a specific robust heteroclinic cycle between equilibria with complex eigenvalues we examine the codimension-two point that separates the supercritical and subcritical. We investigate the bifurcation structure and show the existence of further bifurcations of periodic orbits.     

双跨松动转子-轴承系统周期运动稳定性

建立了带有支承松动故障的具有三轴承支承双跨弹性转子-轴承系统非线性动力学模型,利用求解非线性非自治系统周期解的延拓打靶法和Floquet理论,研究了系统周期运动的稳定性及失稳规律。双跨松动转子-轴承系统响应存在着周期运动、拟周期运动和混沌运动等复杂的运动现象,系统以鞍结分岔形式失稳。在不同的转速下,系统会出现鞍结分岔和Hopf分岔等不同的分岔形式;在高转速区,松动端轴颈的运动轨迹呈现出特有的形状。研究结果为有效识别转子-轴承系统的基础松动故障提供了一定的参考。     

非共振双Hopf分叉系统的规范形及其应用

利用接近恒同的非线性变换，计算出了非共振双Ｈｏｐｆ分叉系统规范形和系数。利用广义坐标变换，将非共振单自由度非线性强迫振动系统变换为双Ｈｏｐｆ分叉系统，用规范形理论给出了一种计算该类系统定常解及分叉特性的方法。     

一类离心调速器的Hopf分岔及其混沌控制

研究了受外部扰动的离心调速器系统的复杂动力学行为。通过系统运动的拉格朗日方程和牛顿第二定律,建立了离心调速器系统的动力学方程,应用Lyapunov直接方法分析了该系统平衡点的稳定性。利用相图分析了系统超混沌吸引子的特性,通过Poincaré截面和Lyapunov指数研究了系统的超混沌行为,通过仿真系统的分岔图和相图分析了系统通向混沌的道路,并且验证了该系统的分岔图与Lyapunov指数谱是完全吻合的。通过对系统施加非线性反馈控制器,并选取合适的反馈系数,可以获得各种不同的所需的稳定周期轨道。对受外部扰动的离心调速器系统施于此控制,计算机数值模拟结果表明,这种控制方法简便有效,控制范围广。     

Non-linear dynamics of space-independent xenon oscillations

The space-independent xenon oscillation problem relevant to power nuclear reactors is studied. Xenon oscillations - both, without temperature feedback and in the presence of temperature feedback - have been analyzed semi-analytically and numerically. The effects of various parameters on the nature of bifurcation are studied. Bifurcation analysis shows that Hopf bifurcation occurs, and it is found that both sub- critical and super-critical Hopf bifurcation can occur in different regions of parameter space. Numerical experiments show that 'outsidey the unstable periodic solutions that exist for the sub-critical Hopf bifurcation case, there exist large amplitude, stable periodic solutions to which the initial conditions, outside the basin of attraction of the stable fixed point, evolve. Though the existence of sub-critical Hopf bifurcation indicates that large amplitude perturbations even in the stable region may lead to initially diverging oscillations, it is reassuring that the oscillation amplitude remains bounded     

碰撞振动系统四阶共振下的Hopf分岔和次谐分岔

建立了一类三自由度含间隙碰撞振动系统的力学模型,求解了系统六维n?1周期运动的周期解及其Poincaré映射。通过理论分析和数值模拟相结合的方法,分析了该系统在强共振点附近,系统两参数控制的局部动力学行为。即在两参数平面上共振点的附近变化两控制参数,进行数值模拟并划分两参数平面的拓扑区域;分析了以“四方形”和“四叶形”异宿轨道为特征的存在于强共振点附近的Hopf分岔不变圈和次谐分岔4?4周期运动,并进一步分析了四阶次谐分岔向混沌的演化过程。     

Bifurcations in dynamical systems with interior symmetry

We propose a definition of interior symmetry in the context of general dynamical systems. This concept appeared originally in the theory of coupled cell networks, as a generalization of the idea of symmetry of a network. The notion of interior symmetry introduced here can be seen as a special form of forced symmetry breaking of an equivariant system of differential equations. Indeed, we show that a dynamical system with interior symmetry can be written as the sum of an equivariant system and a ‘perturbation term’ which completely breaks the symmetry. Nonetheless, the resulting dynamical system still retains an important feature common to systems with symmetry, namely, the existence of flow-invariant subspaces. We define interior symmetry breaking bifurcations in analogy with the definition of symmetry breaking bifurcation from equivariant bifurcation theory and study the codimension one steady-state and Hopf bifurcations. Our main result is the full analogues of the well-known Equivariant Branching Lemma and the Equivariant Hopf Theorem from the bifurcation theory of equivariant dynamical systems in the context of interior symmetry breaking bifurcations.     

多自由度含间隙振动系统周期运动的二重Hopf分岔

基于Poincaré映射方法和数值仿真分析了多自由度含间隙振动系统对称型周期碰撞运动的稳定性与二重Hopf分岔。应用映射的中心流形和范式方法,研究了高维映射在其Jacobian矩阵两对复共轭特征值同时穿越复平面单位圆周情况下的余维二分岔,分析了映射在二重Hopf分岔点附近的双参数开折,揭示了含间隙振动系统在二重Hopf分岔点附近的动力学行为。含间隙振动系统在二重Hopf分岔点附近存在对称型周期碰撞运动、对称型周期碰撞运动的Hopf分岔、环面分岔及“轮胎”型概周期吸引子。环面分岔导致了半吸引不变环和复杂的“轮胎”型概周期吸引子。     

铁道车辆系统周期解的延续算法

用有限差分格式离散常微分方程组的周期解,形成一个含参数的非线性代数方程组,用DERPAR算法对该含参数代数方程组进行延续求解,不但可计算稳定的周期解,而且不稳定的周期解也可求出,采用了vandePol方程和Lorenz方程验证了该方法的可行性。用上述方法计算了一个17自由度铁路客车模型的周期解,得到了一个大范围的车辆系统周期解的解图,包括稳定的和不稳定的周期解。确定了客车系统Hopf分叉点及系统的非线性临界速度,分析了车辆系统周期和轮对横移幅值与车速的关系。     

松动裂纹转子轴承系统周期运动分岔及稳定性分析

根据松动裂纹耦合故障转子轴承系统的非线性动力学方程,利用求解非线性非自治系统周期解的延拓打靶方法,研究了系统周期运动的分岔特性及其稳定性。研究发现,在较大和较小的偏心量作用下,系统的周期运动都由倍周期分岔而失稳,在适当的偏心量下,系统的周期运动以Hopf分岔形式失稳且稳定性较强。转轴裂纹和基础松动故障都使系统周期运动稳定性降低、系统Hopf分岔存在的偏心量范围变大。结论为转子轴承系统的安全稳定运行和振动的抑制及控制提供了理论参考。     

多自由度含间隙振动系统周期运动的Hopf-pitchfork余维二分岔

建立了多自由度含间隙振动系统对称型周期碰撞运动及Poincaré映射的解析表达式,讨论了该映射不动点的稳定性与局部分岔。应用映射的中心流形和范式方法,研究了映射在Hopf-pitchfork余维二分岔点附近的参数开折,揭示了含间隙振动系统在余维二分岔点附近的动力学行为。在该类余维二分岔点附近,不仅存在对称型周期碰撞运动、Hopf分岔和叉式分岔,还存在非对称型周期碰撞运动及其Hopf分岔。通过数值仿真研究了余维二分岔点附近含间隙振动系统对称型周期碰撞运动经叉式分岔和Hopf分岔向混沌的转迁过程。     

A tertiary hopf bifurcation with applications to problems with symmetries

We develop explicit criteria for the occurrence of a tertiary Hopf bifurcation, and stability of the bifurcating orbits, in a special class of two-parameter systems of ordinary differential equations. We use these results to discuss tertiary Hopf and torus bifurcations in some bifurcation problems with symmetries such as steady-state-Hopf and Hopf-Hopf interaction problems. To analyse (and even detect) these bifurcations we use invariant coordinates and rescaling techniques     

Isolas, cusps and global bifurcations in an electronic oscillator

The aim of the present work is to describe the bifurcation behaviour of a class of asymmetric periodic orbits, in an electronic oscillator. The first time we detected them they were organized in a closed branch: that is, their bifurcation diagram showed an eight-shaped isola, with a nice structure of secondary branches emerging from period-doubling bifurcations. In a two-parameter bifurcation set, the isola structure persists. We find the regions of its existence, and describe its destruction in an isola centre with a cusp of periodic orbits. Finally, the introduction of a third parameter allows us to find the relation of our orbits to symmetric periodic orbits (via a symmetry-breaking bifurcation) and to homoclinic connections of the non-trivial equilibria. The isolas are successively created by collision of two adjacent limbs of the wiggly bifurcation curve. The Shil?nikov homoclinic and heteroclinic connections, related to the symmetric and asymmetric periodic orbits, emerge from T-points and end at Shil?nikov-Hopf singularities     

Sup-Resonant Response of a Nonautonomous Maglev System With Delayed Acceleration Feedback Control

A maglev system with delayed acceleration feedback control is disturbed by the deflection of flexible guideway, and resonant response may take place. We have investigated sup-resonant response of the maglev system by employing center manifold reduction and the method of multiple scales. We present the dynamic model and expand it to a third-order Taylor series. Taking time delay as its bifurcation parameter, we discuss the condition for the occurring of Hopf bifurcation. We apply center manifold reduction to get the PoincarÉ normal form of the nonlinear system and employ the perturbation technique to study sup-resonant response of the system. This yields the sup-resonant periodic solution of the normal form. We analyze the stability condition of the free oscillation in the solution and discuss the relatonship between guideway excitation and periodic solution. Finally, numerical results show how time delay, control, and excitation parameters affect the system response. With the proper system parameter, the free oscillation may vanish and only the periodic solution plays a part. Time delay can control amplitude of the forced oscillation. The appearance of the chaos phenomenon can also be governed by regulating time delay. And judiciously selecting a control parameter makes it possible to suppress the response.      

Subcritical and supercritical bifurcations of the first- and second-order Benney equations

The problem of the stability threshold of thin-film dynamics as described by the Benney equation of both first and second orders is revisited. The main result is that the primary Hopf bifurcation of the Benney equation of first order is supercritical for smaller values of Reynolds number and subcritical for its larger values. This result is numerically validated and further investigated analytically to reveal coexisting stable and unstable traveling waves. However, the primary bifurcation of the second-order Benney equation is supercritical for any Reynolds numbers. Sideband instability of traveling-wave regimes whose amplitude and frequency arise from the corresponding complex Ginzburg-Landau equation (CGLE) is found for the Benney equation of both first and second orders.     

Modulated rotating waves in O(2) mode interactions

The interaction of steady-state and Hopf bifurcations in the presence of O(2) symmetry generically gives a secondary Hopf bifurcation to a family of 2-tori, from the primary rotating wave branch. We present explicit formulas for the coefficients which determine the direction of bifurcation and the stability of the 2-tori. These formulas show that the tori are determined by third-degree terms in the normal-form equations, evaluated at the origin. The flow on the torus near criticality has a small second frequency, and is close to linear flow, without resonances. Existence of an additional SO(2) symmetry, as in the Taylor-Couette problem, forces the flow to be exactly linear; however, the tori are unstable at bifurcation in the Taylor-Couette case. More generally, these tori may reveal themselves physically as slowly modulated rotating waves, for example in reaction-diffusion problems.     

Modulated rotating waves in O(2) mode interactions

The interaction of steady-state and Hopf bifurcations in the presence of O(2) symmetry generically gives a secondary Hopf bifurcation to a family of 2-tori, from the primary rotating wave branch. We present explicit formulas for the coefficients which determine the direction of bifurcation and the stability of the 2-tori. These formulas show that the tori are determined by third-degree terms in the normal-form equations, evaluated at the origin. The flow on the torus near criticality has a small second frequency, and is close to linear flow, without resonances. Existence of an additional SO(2) symmetry, as in the Taylor-Couette problem, forces the flow to be exactly linear; however, the tori are unstable at bifurcation in the Taylor-Couette case. More generally, these tori may reveal themselves physically as slowly modulated rotating waves, for example in reaction-diffusion problems.     

松动碰摩转子轴承系统周期运动稳定性研究

根据松动碰摩耦合故障转子轴承系统的非线性动力学方程，利用求解非线性非自治系统周期解的延拓打靶方法，对系统周期运动的稳定性及其失稳规律进行了研究，得到了系统在不平衡量-转速、碰摩间隙-转速等参数域内的分岔集。分析表明：在较大和较小的不平衡量下，系统的周期运动分别以Hopf分岔形式和倍周期分岔形式失稳；耦合故障转子轴承系统表现出与碰摩转子轴承系统相似的分岔失稳规律；随着系统动静件之间的碰摩间隙减小，系统的Hopf分岔集区间变大而且失稳转速降低。该结论可以为转子系统的故障诊断、安全稳定运行及振动控制提供理论依据。