A local sensitivity analysis in Landau damping for the kinetic Kuramoto equation with random inputs

IF 0.9 4区 数学 Q3 MATHEMATICS, APPLIED
Zhiyan Ding, Seung‐Yeal Ha, Shi Jin
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引用次数: 0

Abstract

We present a local sensitivity analysis in Landau damping for the kinetic Kuramoto equation with random inputs. The kinetic Kuramoto equation governs the temporal-phase dynamics of the one-oscillator distribution function for an infinite ensemble of Kuramoto oscillators. When random inputs are absent in the coupling strength and initial data, it is well known that the incoherent state is nonlinearly stable in a subscritical regime where the coupling strength is below the critical coupling strength which is determined by the geometric shape of the distribution function for natural frequency. More precisely, Kuramoto order parameter measuring the fluctuations around the incoherent state tends to zero asymptotically and its decay mode depends on the regularity(smoothness) of natural frequency distribution function and initial datum. This phenomenon is called as Landau damping in the Kuramoto model in analogy with Landau damping arising from plasma physics. Our analytical results show that Landau damping is structurally robust with respect to random inputs at least in subscritical regime. As in the deterministic setting, the decay mode for the derivatives of the order parameter in random space can be either algebraic or exponential depending on the regularities of the initial datum and natural frequency distribution, respectively, and the smoothness for the order parameter in random space is determined by the smoothness of the coupling strength
随机输入动力Kuramoto方程Landau阻尼的局部灵敏度分析
我们对具有随机输入的动力学Kuramoto方程进行了Landau阻尼的局部灵敏度分析。动力学Kuramoto方程支配Kuramoto振荡器的无限系综的单振子分布函数的时间相位动力学。当耦合强度和初始数据中不存在随机输入时,众所周知,非相干状态在亚临界状态下是非线性稳定的,其中耦合强度低于由固有频率分布函数的几何形状确定的临界耦合强度。更准确地说,测量非相干态周围波动的Kuramoto阶参数趋于渐近零,其衰减模式取决于固有频率分布函数和初始数据的规律性(平滑性)。这种现象在Kuramoto模型中被称为朗道阻尼,类似于等离子体物理产生的朗道阻尼。我们的分析结果表明,至少在亚临界状态下,朗道阻尼对随机输入具有结构鲁棒性。与确定性设置一样,随机空间中阶参数导数的衰减模式可以是代数的,也可以是指数的,这分别取决于初始基准和固有频率分布的规律,而随机空间中的阶参数的平滑度由耦合强度的平滑度决定
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来源期刊
Quarterly of Applied Mathematics
Quarterly of Applied Mathematics 数学-应用数学
CiteScore
1.90
自引率
12.50%
发文量
31
审稿时长
>12 weeks
期刊介绍: The Quarterly of Applied Mathematics contains original papers in applied mathematics which have a close connection with applications. An author index appears in the last issue of each volume. This journal, published quarterly by Brown University with articles electronically published individually before appearing in an issue, is distributed by the American Mathematical Society (AMS). In order to take advantage of some features offered for this journal, users will occasionally be linked to pages on the AMS website.
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