In this paper, the nonsmooth compound transmission control protocol (TCP) with the gentle random early detection (GRED) system with the state-dependent round-trip delay is investigated in detail. Uniqueness of the positive equilibrium is proved firstly. Then, the closed approximate periodic solutions in this state-dependent delayed nonsmooth compound TCP with the GRED model are obtained by employing the harmonic balance and alternating frequency/time domain (HB-AFT) method. Then, we compare the results generated by numerical simulations with those of the closed approximate expressions obtained by HB-AFT. It indicates that HB-AFT is simple, correct, and powerful for state-dependent delayed nonsmooth dynamical systems. Finally, we find the complicated dynamic: chaos. It is a grazing chaos with a hybrid property, i.e., where usually oscillates at a very low frequency and oscillates at a very high frequency. And, the route to chaos is a very rare route, i.e., the instantaneous and local transition of stable equilibrium to chaos. So, to the end of stability and good performance, we should adjust the parameters carefully to avoid the periodic and chaotic oscillations.

中文翻译：

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带有HBRED的GRED拥塞控制系统的状态依赖性往返延迟非光滑复合TCP从平衡到混合混沌的突然过渡和周期振荡

本文研究了带有状态依赖往返延迟的温和随机早期检测（GRED）系统的非光滑复合传输控制协议（TCP）。首先证明了正平衡的唯一性。然后，通过采用谐波平衡和交替频/时域（HB-AFT）方法，获得具有GRED模型的该状态依赖的延迟非光滑复合TCP中的闭合近似周期解。然后，我们将数值模拟生成的结果与HB-AFT获得的封闭近似表达式的结果进行比较。它表明HB-AFT对于状态相关的延迟非光滑动力系统而言是简单，正确且功能强大的。最后，我们发现了复杂的动态：混乱。这是具有混合属性的放牧混乱，即通常以非常低的频率振荡，并以非常高的频率振荡。并且，通向混沌的途径是非常罕见的途径，即，稳定平衡向混沌的瞬时和局部转变。因此，为了达到稳定和良好的性能，我们应该仔细调整参数以避免周期性和混沌的振荡。