Stability And Bifurcation Analysis In A Kind Of Second Order Neutral Differential Equations

Delay phenomenon exists in the real word extensively. The neutral delay system iswidely used in numerous fields, such as control theory, ecology, mechanics,management science, physics, etc. Neutral differential equation describes the problemnot only considering the current state and effects of the history on the current situation,but depending on the partial of state in the past with respect to time, which can bebasically consistent with the actual conditions. So the study of neutral system has animportant value and significance, no matter in theory or in application.Along with the development of high-tech, stability of the system plays anincreasingly important role in engineering practice. However, the existence oftime-delay often causes instability to the system. Then the study of the solutions of thedelay differential equations with the dynamic character has become a hot problem inmodern applied mathematics, while bifurcation is an important part of the dynamicssystem theory.In this paper, a kind of second order neutral differential equation is considered.Firstly, a kind of equation which has the general form is discussed. Based on the methodof the exponential polynomial function zero distribution, the associated characteristicequation is analyzed with respect to two parameter and p, respectively. Secondly,applying results in the previous chapter to the neutral equation which has practicalbackground. Thus, the sufficient conditions of equation’s asymptotic stability and theexistence of Hopf bifurcation are obtained. Moreover, interesting phenomenon such asstability switches can also occur for this equation. In addition, the Hopf bifurcationcalculation formula is received by means of Center Manifold Theorem and NormalForm. That is, generating the condition to decide the direction of bifurcation andstability of periodic solution. Last but not least, some numerical simulations usingMatlab are given to illustrate the analytic results by selecting specific functions andparameter values.