Title: Dynamics of a time-delayed relay system Show Chinese title

Title: 时滞继电器系统的动力学

Abstract: We study the dynamics of a piecewise-linear second-order delay differential equation that is representative of feedback systems with relays (switches) that actuate after a fixed delay. The system under study exhibits strong multirhythmicity, the coexistence of many stable periodic solutions for the same values of the parameters. We present a detailed study of these periodic solutions and their bifurcations. Starting from an integro-differential model, we show how to reduce the system to a set of finite-dimensional maps. We then demonstrate that the parameter regions of existence of periodic solutions can be understood in terms of discontinuity induced bifurcations and their stability is determined by smooth bifurcations. Using this technique we are able to show that slowly oscillating solutions are always stable if they exist. We also demonstrate the coexistence of stable periodic solutions with quasiperiodic solutions. Show Chinese abstract

Abstract: 我们研究了一个分段线性二阶延迟微分方程的动力学，该方程代表了具有继电器（开关）的反馈系统，这些继电器在固定延迟后动作。 所研究的系统表现出强烈的多节奏性，在相同参数值下存在许多稳定的周期解。 我们对这些周期解及其分岔进行了详细研究。 从一个积分微分模型出发，我们展示了如何将系统简化为一组有限维映射。 然后我们证明，周期解的存在参数区域可以通过不连续诱导的分岔来理解，其稳定性由平滑分岔决定。 使用这种方法，我们能够证明如果缓慢振荡解存在，它们总是稳定的。 我们还展示了稳定周期解与准周期解的共存。