Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, Tennessee 37996, USA.
Chaos. 2021 Feb;31(2):023131. doi: 10.1063/5.0036508.
Many reduced order modeling techniques for oscillatory dynamical systems are only applicable when the underlying system admits a stable periodic orbit in the absence of input. By contrast, very few reduction frameworks can be applied when the oscillations themselves are induced by coupling or other exogenous inputs. In this work, the behavior of such input-induced oscillations is considered. By leveraging the isostable coordinate framework, a high-accuracy reduced set of equations can be identified and used to predict coupling-induced bifurcations that precipitate stable oscillations. Subsequent analysis is performed to predict the steady state phase-locking relationships. Input-induced oscillations are considered for two classes of coupled dynamical systems. For the first, stable fixed points of systems with parameters near Hopf bifurcations are considered so that the salient dynamical features can be captured using an asymptotic expansion of the isostable coordinate dynamics. For the second, an adaptive phase-amplitude reduction framework is used to analyze input-induced oscillations that emerge in excitable systems. Examples with relevance to circadian and neural physiology are provided that highlight the utility of the proposed techniques.
许多针对振荡动力系统的降阶建模技术仅在没有输入时，基础系统允许存在稳定的周期性轨道的情况下适用。相比之下，当振荡本身是由耦合或其他外生输入引起时，很少有减少框架可以应用。在这项工作中，考虑了这种输入引起的振荡的行为。通过利用等稳定坐标框架，可以确定高精度的约化方程组，并用于预测引发稳定振荡的耦合诱导分岔。随后进行了分析以预测稳态锁相关系。对于两类耦合动力系统，考虑了输入诱导的振荡。对于第一种情况，考虑了参数接近 Hopf 分岔的系统的稳定平衡点，以便使用等稳定坐标动力学的渐近展开来捕捉显著的动力学特征。对于第二种情况，使用自适应相幅约减框架来分析兴奋性系统中出现的输入诱导振荡。提供了与生理节律和神经生理学相关的示例，突出了所提出技术的实用性。
