WORKSHOP: Resonance oscillations and stability of nonsmooth systems

有时，对机械或电气系统施加很小的力，就会引起稳定振荡幅度的显著增长。一个熟悉的例子是推秋千：（平稳地）推秋千（几乎）与它的固有频率一致，将使秋千的振幅增加到某个最大振幅振荡，通常称为共振振荡。对应的数学模型是频率为w_0的谐振子受频率为w的周期性力扰动的数学模型，其中|w_0-w|很小。该现象的实质是谐振子的一个周期解（来自Lyapunov族）在时间周期扰动下变得渐近稳定，且该解的振幅随着|w_0-w|的减小而增大。经典文献中共振振荡的严格证明是基于一种平均方法，得到一个分岔函数M_w，其零点A_w对应于解，如果（M_w）′（A_w）的特征值具有负实部，则具有渐近稳定性。这里A_w表示振荡的振幅，当w接近w_0时，振幅增加。我们注意到经典理论在本质上使用了无摄动系统和摄动的平滑性。非光滑系统的共振振荡理论是在1940年塔霍马窄桥坍塌后的第二次世界大战期间开始发展起来的。直到1989年，Glover等人才证明了坍塌是由于吊索从下方限制了桥梁的运动。这种单侧悬架引入了一个不可微（非光滑）函数到相关的数学模型中，使得经典的平均理论不适用。这是将分岔理论中的经典结果扩展到非光滑系统（通过放宽光滑假设）的努力的起点。已知非光滑系统中的共振振荡能够表现出非光滑系统特有的行为，但在光滑近似中消失。证明这一点的机械系统是一个身体连接到一个刚性固定梁上的移动带。如果物体与皮带之间的摩擦是干的，并且摩擦系数与外力振幅之间的某些关系是正确的，那么物体可能周期性地粘在皮带上，并表现出所谓的粘滑振荡。采用符号函数模型的光滑近似，干摩擦破坏了这种现象。由于相应的导数（M_w）'在A_w不存在，因此不能在这里应用类平均方法（即使是形式上的）。这一观察结果导致了大量的研究工作，特别是解释粘滑运动的稳定性。如上所述，在强迫摆动的例子中，当某个非渐近稳定的周期解变得渐近稳定时，共振振荡就发生了。因此，稳定性和稳定性的变化——在光滑系统中——是分岔理论的一个重要方面，也是一个活跃的研究领域，特别是当光滑系统的技术和概念不能应用时。近年来，对非光滑系统中共振振荡的数学研究得到了越来越多的关注，由于最近的进展，一些与非光滑机械和物理系统的共振有关的技术相关的开放问题已经接近解决。讲习班将关注这些与工业有关的问题。也许是由于这门学科的强劲增长，不同的学校在研究非光滑系统和稳定性的情况下取得了相似的结果，而彼此却没有意识到。本次研讨会在创造认识和刺激讨论和合作不同的研究小组之间的工作在振荡和非光滑系统的稳定性的实质性尝试。

Sometimes the application of a small force to a mechanical or electrical system causes a considerable growth in the amplitude of stable oscillations. A familiar example is a pushed playground swing: (smoothly) pushing a swing (almost) in tune with its natural frequency will make the swing's amplitude increase to some maximal amplitude oscillation, commonly known as a resonance oscillation. The corresponding mathematical model is that of a harmonic oscillator with frequency w_0 perturbed by a periodic force of frequency w, where |w_0-w| is small. The essence of the phenomenon is that one of the periodic solutions (from a Lyapunov family) of the harmonic oscillator becomes asymptotically stable under the time-periodic perturbation and the amplitude of this solution increases as |w_0-w| decreases. The rigorous justification of resonance oscillations in the classical literature is based on an averaging method yielding a bifurcation function M_w whose zeroes A_w correspond to solutions and asymptotic stability follows if the eigenvalues of (M_w)'(A_w) have negative real part. Here A_w represents the amplitude of the oscillation which increases as w approaches w_0. We note that the classical theory uses smoothness of the unperturbed system and the perturbations in an essential way. The theory of resonance oscillations in nonsmooth systems started to be developed around the second world war following the Tahoma Narrow Bridge collapse in 1940. Only in 1989, Glover et al. proved that the collapse was due to suspension cables which restrict the motion of the bridge from below. This one-sided suspension introduces a nondifferentiable (nonsmooth) function to the relevant mathematical model, rendering the classical averaging theory non-applicable. This has been the starting point of an efort to extend classical results in bifurcation theory to nonsmooth systems (by relaxation of smoothness assumptions). Resonance oscillations in nonsmooth systems are known to be able to exhibit behaviour that is unique to nonsmooth systems but disappears in smooth approximations. A mechanical system that demonstrates this is a body attached to a rigid fixed beam on a moving belt. If the friction between the body and the belt is dry and some relations between the friction coefficient and the amplitude of the external forcing hold true, then the body may stick to the belt periodically and exhibit so-called stick-slip oscillations. Taking a smooth approximation of the sign-function modelling the dry friction destroys the phenomenon. Averaging-like methods can not be applied here (even formally) since the corresponding derivative (M_w)' does not exist at A_w. This observation has lead to a substantial research effort, in particular to explain the stability of stick-slip motions.As discussed above in the example of the forced swing, the resonance oscillations occurs when some non-asymptotically stable periodic solution becomes asymptotically stable. Stability and change of stability is thus - as in smooth systems - an essential aspect of the bifurcation theory, and an active field of research, in particular when techniques and concepts from smooth systems cannot be applied.Research into the mathematics of resonance oscillations in nonsmooth systems has seen a steady growth in attention lately, and due to recent progress a number of technically relevant open problems concerning resonances of nonsmooth mechanical and physical systems have become in reach of being resolved. The workshop will pay attention to such problems with industrial relevance.Perhaps due to the strong growth in the subject, the situation has arisen where different schools working on nonsmooth systems and stability achieved similar results without being aware of each other. This workshop makes a substantial attempt in creating awareness and stimulate discussion and collaboration between different research groups working on oscillations and stability of nonsmooth systems.