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AIMD and RED are two dominant algorithms for controlling Internet congestion. So this paper explores the periodic solutions and complex dynamical phenomena in the state-dependent round-trip delayed AIMD/RED network congestion model with heterogeneous flows, and its improved model. We first use the semi-analytical and semi-numerical method, known as the harmonic balance method with alternating frequency/time (HB-AFT) domain technique, to derive the analytical approximations of periodic oscillations of the system. The obtained results are compared with the numerical results by WinPP, and they show good consistency. At the same time, this suggests that the method used in this paper is correct and valid. Then for the sake of making the system more realistic, we improve the model by using the hyperbolic tangent function. We obtain the approximate solutions, and find some rich dynamical behaviors of this delayed heterogeneous system, including Period-1 to torus, Period-1 to Period-2 to Period-3 motions and two kinds of mechanisms of chaos, i.e. the windows of Period-2 and Period-3 orbits to chaos, where to the best knowledge of the authors, the former route has never been reported. The periodic oscillations may induce synchronization and further congestion, where chaotic oscillation usually means that the system is unstable and may even collapse. Hence, we need to avoid these abundant dynamics discovered in this paper because they are undesirable and harmful. The derived results can help researchers better understand the performance of the AIMD/RED system, and they can be a guide for choosing parameters in a suitable range in order to maintain the network stability and optimize system performance.

Periodic, Quasi-Periodic and Chaotic Oscillations in Two Heterogeneous AIMD/RED Network Congestion Models with State-Dependent Round-Trip Delays