开关与振子混合复杂系统的动力学
Dynamics in hybrid complex systems of switches and oscillators
摘要：
虽然在分析包含单一类型耦合动态分量的大型系统（例如，耦合振荡器或耦合开关），包含不同组件的系统（例如，振荡器和开关）受到的关注要少得多。我们分析了大的，混合动力系统的相互连接的仓本振荡器和Hopfield开关的正反馈。在该系统中，振荡器同步促进开关接通。反过来，当开关接通时，它们增强与它们耦合的振荡器的同步。根据参数的选择，我们发现理论上共存的稳定的解决方案，（i）非相干振荡器和所有开关永久关闭，（ii）同步振荡器和所有开关永久打开，或（iii）同步振荡器和开关，周期性地交替之间的开和关状态。数值实验证实了这些预测。我们讨论了如何在这些稳定状态的解决方案之间的过渡可以发病确定性通过动态分叉或自发地由于有限大小的波动。
英文摘要：
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.
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