Pattern Formation in Periodicaly Forced Oscillatory Systems
Pattern Formation in Periodicaly Forced Oscillatory Systems
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发表时间：
2004
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通讯作者：
A. Yochelis
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作者：
A. Yochelis
振荡系统是我们生活中的常见特征，人们可以在简单的时钟或更复杂的技术设备中找到它们的实现，如基于卫星的导航系统和植入心脏起搏器以防止心律失常。生态和生物世界也揭示了不同的振荡动力学行为，如季节性开花，捕食者-猎物关系，或细胞分裂。在大尺度物体的背景下，很容易注意到地球（像许多其他行星一样）正在围绕太阳振荡，大约周期为一年（365天）。还指出，星系围绕黑洞进行旋转周期运动，假设存在于它们的中心。在小尺度系统中，基本粒子、电磁、晶体等的量子力学描述涉及振荡行为。振荡动力学的科学意义吸引了理论研究和特定的物理、化学、生物、生态甚至经济系统的研究。当振荡系统受到时间强迫时，它们可能会表现出夹带或频率锁定现象。当一个系统的振荡频率被调整到强迫频率的一个不可约分数时，该系统被频率锁定。这种共振条件允许在强迫振幅和频率所跨越的平面中存在舌状域。在共振舌外，系统呈现准周期振荡。频率锁定现象已被广泛研究的单振子类型的系统，然而，空间扩展系统的共振现象的基本描述是缺失的。本论文主要研究空间延伸介质中的频率锁定现象，并探讨图案形成对共振行为的影响。我们研究图案的形成机制和参数范围内的谐振和非谐振模式的发展。在我们的结果中，我们表明，在扩展系统：（a）驻波是唯一的模式（除了均匀振荡），满足频率锁定条件：空间中的每个点可以振荡或共振。(b)空间结构和不稳定性可以减小或扩展频率锁定的边界，使得单个振荡器的谐振范围不总是与扩展系统中的谐振范围一致。这项研究的动机是最近的实验时间驱动Belousov-Zhabotinsky（BZ）反应扩散系统。观察到，当系统以其自然频率的大约两倍周期性地受力时（下文称为2：1共振），驻波图案可以从螺旋波发展，并且驻波仅占据2：1共振舌的一部分。实验还表明，电磁驻波模式可能以两种不同的方式发展。指状不稳定性和未锁定振荡的条纹形核可能会形成Labelths。
Oscillatory systems are common features of our lives, one may find their realizations in simple clocks or in more complicated technological equipments like satellite-based navigation systems and pacemakers implanted to prevent cardiac arrhythmia. The ecological and biological worlds also reveal diverse oscillatory dynamical behaviors like seasonal flower blooms, predator-prey relations, or cell divisions. In the context of large-scale objects, it is easy to notice that the earth (like many other planets) is making oscillations around a sun with an approximate period of one year (365 days). It was also indicated that the galaxies perform rotational periodic motion around black holes, assumed to exist in their center. In small-scale systems, the Quantum mechanical descriptions of the elementary particles, Electromagnetism, crystals etc. involve oscillatory behaviors. The scientific significance of oscillatory dynamics have attracted both theoretical studies and studies of specific physical, chemical, biological, ecological and even economical systems. When oscillatory systems are subjected to temporal forcing they may exhibit entrainment or frequency locking phenomena. A system is frequency locked when its oscillation frequency is adjusted to an irreducible fraction of the forcing frequency. This resonance condition admits a tongue-like domain in the plane spanned by the forcing amplitude and frequency. Outside the resonance tongue the system exhibits quasiperiodic oscillations. Frequency locking phenomena have been extensively studied for single oscillator type systems, however, the fundamental description of resonance phenomena for spatially extended systems is missing. This thesis is concerned with frequency locking phenomena in spatially extended media and addresses the effects of pattern formation on resonance behavior. We study pattern formation mechanisms and parameters ranges where resonant and non-resonant patterns are developed. Among our results we show that in extended systems: (a) Standing waves are the only patterns (besides uniform oscillations) that satisfy the frequency locking condition: each point in space can oscillate either in or out of resonance. (b) Spatial structures and instabilities may reduce or extend the boundaries of frequency locking so that the resonance ranges for a single oscillator do not always coincide with resonance ranges in extended systems. This research has been motivated by recent experiments on temporally driven Belousov-Zhabotinsky (BZ) reaction-diffusion systems. It was observed that standing– wave patterns may develop from spiral waves when the system is periodically forced at approximately twice its natural frequency (hereafter 2:1 resonance), and that the standing waves occupy only a part of the 2:1 resonance tongue. The experiments also indicate that labyrinthine standing–wave patterns may develop in two distinct ways. Labyrinths may develop by a fingering instability, and by nucleation of stripes from unlocked oscillations.