论文标题
延迟的动力学，瞬时谐振振荡
Delayed Dynamics with Transient Resonating Oscillations
论文作者
论文摘要
最近，我们研究了一个延迟微分方程，该方程的系数是时间的线性函数。方程表明，随着延迟的增加到零延迟之间的延迟增加，振荡瞬态动力学出现并消失。我们在这里提出和研究另一个显示相似瞬时振荡的方程式。在延迟的反馈术语中，它具有额外的指数高斯因素。结果表明，通过使用Lambert $ w $函数，该方程在分析上是可以分析的。还对该方程进行了数值研究，以确认从分析解决方案推断出的某些属性。我们还发现，瞬态振荡的幅度会发生变化，并且随着我们增加延迟值的最大值。从这个意义上讲，提出的方程是最简单的动态方程之一，它带出没有任何外部振荡输入的共振行为。
Recently, we have studied a delay differential equation which has a coefficient that is a linear function of time. The equation has shown the oscillatory transient dynamics appear and disappear as the delay is increased between zero to asymptotically large delay. We here propose and study another equation that shows similar transient oscillations. It has an extra exponential gaussian factor on the delayed feedback term. It is shown that this equation is analytically tractable with the use of the Lambert $W$ function. This equation is also studied numerically to confirm some of the properties inferred from the analytical solution. We also have found that the amplitude of transient oscillation changes and goes through a maximum as we increase the value of the delay. In this sense, the proposed equation is one of the simplest dynamical equations that brings out a resonant behavior without any external oscillating inputs.
