A Direct Algorithm for the Stability and Hopf Bifurcation of A Logistic Differential Equation Versus Time Delay

Abstract

This paper investigates the local stability, the existence and the stability of Hopf bifurcation of typical logistic differential equation versus time delay. The local stability is considered within the framework of the $\tau$ decomposition method, which involves the calculation of the PIR and the determination of the cross direction around it. And the complete and exact delay stable interval is given analytically. Subsequently, the nonlinear dynamics of bifurcating solutions are reviewed carefully by center manifold theorem and normal form theory. By the way, a new simple bilinear form is presented for reducing the computation of projection eigenvector according to the adjoint operator theory. In addition, the eigendecomposition strategy is more clearly characterized by the fact that it is applied to compute extensions of normal forms. And all the bifurcating parameters (Coefficients of normal form) are provided in the explicit expressions of the systems’ parameter, directly. At last, a direct algorithm is proposed to summarize the procedure for the computation of Hopf bifurcation. Typical example is introduced to show the correctness and effectiveness.