| Chaos has been studied for more than forty years. The study emphasis have developed to controlling and use of chaos from the initial finding and explaining chaos. Since 1990 the famous OGY method is proposed, many controlling methodologies have been developed and applied widely. In this paper, R?ssler system from delayed feedback controlling method by Pyragas developed is mainly studied.When parameters satisfy some assumptions, R(o|¨)ssler system is a chaotic dynamical system. In order to discuss the dynamical character of R(o|¨)ssler system and use delayed feedback control chaos, in the first, we give the formulation of the model. Secondly, we analyze the stability of the equilibrium and Hopf bifurcations by using linearizing stability method. We study the stability of the model by linearizing local the nonlinear system at the equilibrium. The specific statements are as follows:Firstly, we takeτ1 as a parameter whenτ2 is equal to zero, by analyzing the roots of the characteristic equation, and obtain the stability of zero. Secondly, by using the same method, whenτ1 is in its stable interval, we takeτ2 as a parameter, obtaining the stability of the zero solution and the existence of the Hopf bifurcation. In the end, we derive the explicit formula for determining the direction of the Hopf bifurcation and the stability of these periodic solutions bifurcating from the steady states, by using the normal form and center manifold theorem. Some numerical simulations are carried out for supporting the analytic results found.
|
PDF Full Text Request