Department of Mathematics, Huzhou University, Huzhou, Zhejiang 313000, People's Republic of China.
Chaos. 2022 Dec;32(12):123137. doi: 10.1063/5.0125266.
Various periodic motions are investigated for a mass-spring-damper oscillator described by second-order discontinuous delayed systems with a parabolic boundary. The G-function related to a delayed variable is first developed to judge the crossing, sliding, and grazing motions at the boundary points. Then, basic and local mappings, together with their motion equations, are classified by current and delayed states of a local flow. It is shown that periodic motions can be predicted analytically in terms of the return mapping and the initial value function provided posteriorly. Finally, numerical simulations are given to illustrate the existence of slowly oscillating periodic solutions without or with sliding portions as well as a crossing periodic solution not slowly oscillating. These results not only enrich and extend switchability theory of discontinuous delayed systems, but also reveal some complicated periodic behaviors which cannot be found in continuous delayed systems and discontinuous systems without time delay.
研究了具有抛物线边界的二阶不连续时滞系统描述的质量-弹簧-阻尼振荡器的各种周期运动。首先开发了与时滞变量相关的 G 函数，以判断边界点处的穿越、滑动和滑掠运动。然后，根据局部流的当前和时滞状态对基本和局部映射及其运动方程进行分类。结果表明，可以根据返回映射和随后提供的初始值函数来分析预测周期运动。最后，给出了数值模拟结果，说明了存在无滑动部分或具有滑动部分的缓慢振荡周期解以及不缓慢振荡的穿越周期解。这些结果不仅丰富和扩展了不连续时滞系统的开关性理论，而且揭示了一些在连续时滞系统和无时滞不连续系统中无法找到的复杂周期行为。
