Approximate Expression Of Periodic Solution And Complex Dynamics Of State-dependent Time-delayed Non-smooth CTCP-GRED Congestion Control System

Compound TCP(CTCP)is a transmission control protocol which combines time delay and packet loss and is suitable for high-speed and long-distance networks.Gentle RED(GRED)can reduce the amplitude of queue length,which has a positive impact on the network.The stability and performance of the network are closely related to the dynamics of congestion control system,and complex phenomena such as periodic solution and chaos will degrade or even collapse the network performance,so it is essential to study the dynamics of network congestion control system.With the support of NSFC(No.11372282 and No.11972327),we mainly investigate the approximate expressions of periodic solutions and complex dynamics of non-smooth CTCP-GRED congestion control system with state-dependent roundtrip delay in two different scenarios in this dissertation.For the first scenario,the average queue size is considered in the queue management algorithm.Firstly,we prove the uniqueness of the positive equilibrium of the system.Then we use the software DDE-BIFTOOL to analyze its Hopf bifurcation and employ software Win PP for numerical verification.Finally we obtain the approximate expressions of periodic solutions by HB-AFT method.And we compare the numerical simulation results with those obtained by HB-AFT method,which proves that HB-AFT method is simple and effective for state-dependent time-delayed non-smooth systems.For the second scenario,the average queue size is ignored in the queue management algorithm.Firstly,we prove the uniqueness of the positive equilibrium of the system.Then we use the software DDE-BIFTOOL to analyze its Hopf bifurcation and adopt software Win PP for numerical verification.Then we obtain the approximate expressions of periodic solutions by HB-AFT method and compare the results of numerical simulation with those obtained by HB-AFT method.Finally,we find the complex dynamics of the system,chaos,and its route to chaos is the sudden transition from equilibrium point to chaos.Since the periodic and chaotic oscillations of congestion control system will hinder the network performance,the research on periodic solutions and chaotic phenomena may have a good guidance for researchers to adjust parameters properly to avoid these phenomena.The features and innovations of this dissertation are as follows: Firstly,we use the software DDE-BIFTOOL to study Hopf bifurcation and stability of nonsmooth CTCP-GRED system,which shows its potential in bifurcation research of non-smooth delay differential equations.Secondly,we prove the uniqueness of the positive equilibrium of the CTCP-GRED systems.Thirdly,we accurately obtain the approximate expressions of periodic solutions of state-dependent time-delayed nonsmooth CTCP-GRED systems by HB-AFT method.Last but not least,we find that there may be a rare sudden transition from equilibrium to chaos in CTCP-GRED system without considering the average queue length.