Nonlinear Stability Research Of Internet Network Congestion Control Systems

With the development of Internet's scale and quality of service (QoS), its congestion control will be more important than before. Impacted by communication delay and time-variant network topology, Internet congestion control system shows obvious nonlinear characteristics such as bifurcation and chaos, which means that the router's queue length and the user'window size will oscillate radically, and network resource cannot be used effectively. Around such a problem, after studying on the chaos and synchronization of nonlinear and time-delayed systems, we use signal processing and modifying models methods to study how to put off the bifurcation outset or how to enlarge the stable range. The main contents and contributions of this dissertation are as follows.(1) Synchronizations between same or different chaotic systems are studied. First, taking Chen's system and Chua's system as instances, adaptive synchronizations of 3-dimension chaotic systems with uncertain or unknown parameters are investigated, where adaptive controllers and parameter update laws are designed with Lyapunov's stability theorem. Next, general formula for full state hybrid projective synchronization (FSHPS) is derived. Based on the proposed formula, a concrete synchronization controller about 4-dimension chaotic systems, i.e. Chen's hyper-chaotic system and Lü's hyper-chaotic system is given. Finally, synchronization results are proved by numerical simulations. Studies show that two same or different chaotic systems can be synchronized asymptotically by the proposed controller.(2) Synchronization and consensus of time-delayed multi-agent network models are investigated. First, several prior networks'synchronization characteristics with or without time delay are studied. Then, in the aspect of enlarging models'stable domain, an improved model is presented, whose main characteristics lie in two aspects: One is that a proportional coefficient k is introduced to raise the model's flexibility, the other is that time delay is introduced into each communication receiver. Finally, by geometrical method and frequency-domain method, the synchronization condition is derived, how to choose time delay is also provided and verification is come out by numerical simulations. It is found that synchronization domain of the novel model is larger than those of some prior models and it can be regulated.(3) Nonlinear stability of time-delayed dual model of Internet congestion control system is studied. Firstly, some notions and typical examples of the time-delayed dual model are reviewed. It's found that these models'stable domains are narrow and prone to enter into bifurcated or chaotic states. Then an improved model is proposed correspondingly, and the Hopf bifurcation phenomenon is studied by setting time delay as the bifurcation parameter. Both perturbation method and center manifest theorem and normal form theory are used to complete the nonlinear stability analysis, including the bifurcation direction, the stability and the change of the periodic solution. In addition, we try to use homotopy analysis method (HAM) to calculate bifurcation solution of the improved model, and the calculation's complexity is compared with some traditional methods. In the final, the theoretical conclusions above are verified by numerical experiments.(4) Nonlinear stability of a fluid flow model in Internet congestion control is studied. First, the simplification and evolvement of fluid flow models are analyzed. Then a modified fluid flow model is presented in order to enlarge its stable range. Finally the linear and nonlinear stability analyses are finished, respectively. Its nonlinear stability analysis is based on perturbation method, and the conclusion can be described by its Floquet exponent's sign. Fluid flow model's research is summarized as two theorems which are verified by numerical simulations.(5) Bifurcation control and nonlinear stability of TCP/RED stroboscopic model in Internet congestion control are studied. First basic characteristics of TCP/RED stroboscopic model are reviewed. By numerical simulation, it is shown that TCP/RED stroboscopic model is unstable when the parameter w passes through some critical bifurcation value, and the critical bifurcation value is too small to satisfy the general request of Internet congestion control. Then, in order to enlarge the model's stable domain, a moving average filter is added to the TCP/RED stroboscopic mode. Studies show that the moving average filter can be used to put off the bifurcation's outset, and the model's stable domain can be regulated by choosing different filter sizes. In addition, the computation complexity after being added to the filter is analyzed and the conclusion in point is verified by numerical simulations.