Hopf Bifurcation Investigation For A New Chaos Entanglement System

In this paper we propose,discuss dynamic analysis that new chaos entanglement system is chaotic and all equilibria are unstable saddle points when chaos entanglement is achieved.Demonstrate a computationally numerical meaning of chaos which can be practical very largely to situations that are commonly encountered,comprising attractors and non-periodically forced systems.And the route from periodic-doubling to chaos is established by partial enlarged bifurcation illustrations.We furthermore present the system has two positive Lyapunov exponents which recommends chaos and the existence of Hopf bifurcation are investigated.Details as follow:1.Bifurcation theory delivers a necessary condition which is obviously the failure of the Implicit Function Theorem.Bifurcation results is due to mathematical existence of bifurcation scenarios experimental in various methods and tests,most experts that reduction principle that it preserves the existence of a potential if the parameter values are chosen well.The principle of Exchange of stability holds for non-degenerate as well as for degenerate Hopf Bifurcation for parabolic problems.The way to produce the preferred result is to suppress chaos to change the system parameters.Dynamical systems have been model both the linear and nonlinear phenomenon with technological discipline.Laser technology,automated circuits,population dynamics and turbulence has been witnessed in modern centuries as deterministic chaos.2.The purposes and present development of the system investigated about chaos entanglement and Hopf bifurcation theory time delay system as well as expounds the concept and definition of stability,bifurcation,flip bifurcation theory,Hopf bifurcation theory,center manifold theorem,Hurwitz criterion and Lyapunov coefficient method.Briefly summarized and illustrates the conditions of flip bifurcation and Hopf bifurcation occurs,and the stability of the equilibrium.Finally,this article made a brief introduction for the main work.3.The section also introduces the reader to a class of nonlinear systems and changes variables that transform the nonlinear system into an equivalent linear system.Transversely conditions have been carefully studied to enable numerical analysis.Eigenvalues are sufficiently helpful to determine equilibrium of nonlinear systems.Most of these chaotic indications display randomness and can have a more complicated steady –state behavior that is not equilibrium or almost-period alternation.The situation where Jacobian matrix has eigenvalues on the imaginary axis then qualitative behavior of the nonlinear state equation is close to the equilibrium point and could be quite distinct from that of the linearized state equation.Numerical construction of phase portraits is to find all equilibrium points and determine the type trajectories.4.The Hopf bifurcation and stability of system are analyzed in detail.There are abundant and complex dynamical behaviors which the new autonomous system produces,despite its apparent simplicity,is investigated and expatiated in this paper,but the attractors and their forming mechanism need study and explore further,and their topological structure should be completely and thoroughly investigated.Apparently there are more interesting problems about this chaotic system in terms of complexity,control and synchronization,which deserve further investigation.