具有瞬态共振振荡的延迟动力学
Delayed Dynamics with Transient Resonating Oscillations
AI总结：
本文提出一个含高斯指数延迟反馈项的延迟微分方程，利用朗伯W函数解析求解并结合数值验证，发现瞬态振荡振幅随延迟增大出现最大值，从而在无外部振荡输入下展现共振行为。
摘要：
最近，我们研究了一个延迟微分方程，该方程具有一个随时间线性变化的系数。该方程表明，当延迟从零增加到渐近大延迟时，振荡瞬态动力学出现又消失。我们在此提出并研究了另一个表现出类似瞬态振荡的方程。它在延迟反馈项上有一个额外的高斯指数因子。结果表明，利用朗伯W函数，该方程在解析上是可处理的。我们还对该方程进行了数值研究，以确认从解析解推断出的一些性质。我们还发现，随着延迟值的增加，瞬态振荡的振幅发生变化并经历一个最大值。在这个意义上，所提出的方程是在没有任何外部振荡输入的情况下产生共振行为的最简单动力学方程之一。
英文摘要：
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.
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