Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata-700108, India.
Department of Mathematics, Bar-Ilan University, Ramat Gan 52900, Israel.
Sci Rep. 2017 Apr 5;7:45909. doi: 10.1038/srep45909.
In this report, we investigate the stabilization of saddle fixed points in coupled oscillators where individual oscillators exhibit the saddle fixed points. The coupled oscillators may have two structurally different types of suppressed states, namely amplitude death and oscillation death. The stabilization of saddle equilibrium point refers to the amplitude death state where oscillations are ceased and all the oscillators converge to the single stable steady state via inverse pitchfork bifurcation. Due to multistability features of oscillation death states, linear stability theory fails to analyze the stability of such states analytically, so we quantify all the states by basin stability measurement which is an universal nonlocal nonlinear concept and it interplays with the volume of basins of attractions. We also observe multi-clustered oscillation death states in a random network and measure them using basin stability framework. To explore such phenomena we choose a network of coupled Duffing-Holmes and Lorenz oscillators which are interacting through mean-field coupling. We investigate how basin stability for different steady states depends on mean-field density and coupling strength. We also analytically derive stability conditions for different steady states and confirm by rigorous bifurcation analysis.
在本报告中，我们研究了在个体振荡器表现出鞍点固定点的耦合振荡器中稳定鞍点固定点的问题。耦合振荡器可能具有两种结构上不同类型的抑制状态，即振幅死亡和振荡死亡。鞍点平衡点的稳定是指振幅死亡状态，其中振荡停止，所有振荡器通过逆叉形分岔收敛到单个稳定的稳态。由于振荡死亡状态的多稳定性特征，线性稳定性理论无法对这些状态进行分析，因此我们通过基区稳定性测量来量化所有状态，基区稳定性测量是一种通用的非局部非线性概念，它与吸引域的体积相互作用。我们还在随机网络中观察到多簇的振荡死亡状态，并使用基区稳定性框架对其进行测量。为了探索这些现象，我们选择了一个由耦合的杜芬-霍尔姆斯和洛伦兹振荡器组成的网络，它们通过平均场耦合相互作用。我们研究了不同稳态的基区稳定性如何随平均场密度和耦合强度而变化。我们还通过严格的分岔分析解析推导出了不同稳态的稳定性条件，并进行了验证。
