Department of Solar Energy and Environmental Physics, Blaustein Institutes for Desert Research (BIDR), Ben-Gurion University of the Negev, Sede Boqer Campus, Midreshet Ben-Gurion 8499000, Israel.
Chaos. 2020 Feb;30(2):023120. doi: 10.1063/1.5130618.
Frequency locking in forced oscillatory systems typically organizes in "V"-shaped domains in the plane spanned by the forcing frequency and amplitude, the so-called Arnol'd tongues. Here, we show that if the medium is spatially extended and monotonically heterogeneous, e.g., through spatially dependent natural frequency, the resonance tongues can also display "U" and "W" shapes; we refer to the latter as an "inverse camel" shape. We study the generic forced complex Ginzburg-Landau equation for damped oscillations under parametric forcing and, using linear stability analysis and numerical simulations, uncover the mechanisms that lead to these distinct resonance shapes. Additionally, we study the effects of discretization by exploring frequency locking of oscillator chains. Since we study a normal-form equation, the results are model-independent near the onset of oscillations and, therefore, applicable to inherently heterogeneous systems in general, such as the cochlea. The results are also applicable to controlling technological performances in various contexts, such as arrays of mechanical resonators, catalytic surface reactions, and nonlinear optics.
在受迫振荡系统中，频率锁定通常在由强迫频率和幅度构成的平面上组织成“V”形域，即所谓的阿诺尔德舌。在这里，我们表明如果介质是空间扩展的且单调非均匀的，例如，通过空间相关的固有频率，共振舌也可以显示“U”和“W”形状；我们将后者称为“反向骆驼”形状。我们研究了参数激励下阻尼振荡的通用强迫复金兹堡-朗道方程，并通过线性稳定性分析和数值模拟，揭示了导致这些不同共振形状的机制。此外，我们通过探索振荡器链的频率锁定来研究离散化的影响。由于我们研究的是规范形式的方程，因此在振荡开始时，结果在模型上是独立的，因此通常适用于固有异质系统，例如耳蜗。这些结果还适用于控制各种情况下的技术性能，例如机械谐振器阵列、催化表面反应和非线性光学。
