Stability And On Control Of Hopf Bifu Rcation Analysis In Three Classes Of Complex Dynamic Networks

The progress of genome sequencing and gene recognition has been quit sig-nificant in the last few years. While the amount of available gene expressiondata has been increasing rapidly, the required mathematical techniques to ana-lyze such data is still in development. Particularly, deriving a gene regulatorynetwork from gene expression data has proven to be di?cult. The contents of thispaper include: the stability and on control of Hopf bifurcation analysis in threeclasses of complex dynamic networks, which infer to stability and Hopf bifurca-tion in two classes of genetic regulatory networks and on control of bifurcationin a class of TCP/AQM networks.The main contents in this paper can be summarized as follows:1. In the first chapter, we introduce the developmental process and sig-nificance of the genetic regulatory networks; And we summarize some famousmathematical model of genetic regulatory networks and some research resultsfor the networks; Then we brie?y introduces some rudimentary knowledge aboutthe hopf bifurcation control; The organization of this paper is given in the end.2. In the second chapter, we deal with a model of single genetic negativefeedback autoregulatory system, Choosing the transcriptional rate k as the bi-furcation parameter, we derive some su?cient conditions on local stability andHopf bifurcation. Further, we derive the explicit formulas which determine thebifurcation direction and the stability of bifurcated periodic solution. we give aexamples to illustrate the e?ciency of our results. In the existing literature ismostly with time delay as bifurcation parameters, While ignoring the transcrip-tional rate changes to the network impact, this is the innovation of this paper place.3. In the third chapter, we investigate the Goodwin model. Choosing thesum of the delays as bifurcation parameter and using characteristic equationand Cardano's formula, we analysis the distribution of characteristic root, thenwe derive some su?cient conditions on local stability and Hopf bifurcation. Anumerical simulation is given in the end.4. In the fourth chapter, we consider a class of TCP/AQM networks. Basedon the proposed model in [10], We use wash-out filter method to control Hopfbifurcation. Finally a numerical simulation verify the e?ectiveness of the control.