Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80309, USA.
Chaos. 2013 Sep;23(3):033142. doi: 10.1063/1.4822017.
While considerable progress has been made in the analysis of large systems containing a single type of coupled dynamical component (e.g., coupled oscillators or coupled switches), systems containing diverse components (e.g., both oscillators and switches) have received much less attention. We analyze large, hybrid systems of interconnected Kuramoto oscillators and Hopfield switches with positive feedback. In this system, oscillator synchronization promotes switches to turn on. In turn, when switches turn on, they enhance the synchrony of the oscillators to which they are coupled. Depending on the choice of parameters, we find theoretically coexisting stable solutions with either (i) incoherent oscillators and all switches permanently off, (ii) synchronized oscillators and all switches permanently on, or (iii) synchronized oscillators and switches that periodically alternate between the on and off states. Numerical experiments confirm these predictions. We discuss how transitions between these steady state solutions can be onset deterministically through dynamic bifurcations or spontaneously due to finite-size fluctuations.
虽然在分析包含单一类型耦合动力学组件（例如，耦合振荡器或耦合开关）的大型系统方面已经取得了相当大的进展，但包含多种组件（例如，振荡器和开关）的系统却受到的关注要少得多。我们分析了具有正反馈的互连 Kuramoto 振荡器和 Hopfield 开关的大型混合系统。在这个系统中，振荡器的同步促进了开关的开启。反过来，当开关打开时，它们增强了与之耦合的振荡器的同步性。根据参数的选择，我们从理论上找到了共存的稳定解，要么是（i）不同步的振荡器和所有的开关永久关闭，要么是（ii）同步的振荡器和所有的开关永久打开，要么是（iii）同步的振荡器和开关周期性地在打开和关闭状态之间交替。数值实验证实了这些预测。我们讨论了这些稳定状态解之间的转换如何通过动态分岔或由于有限大小的波动而自发地决定。
