Hopf bifurcation and chaos analysis of Chen's system with distributed delays

In this paper, a general model of nonlinear systems with distributed delays is studied. Chen's system can be derived from this model with the weak kernel. After the local stability is analyzed by using the Routh-Hurwitz criterion, Hopf bifurcation is studied, where the direction and the stability of the bifurcating periodic solutions are determined by using the normal form theory and the center manifold theorem. Some numerical simulations for justifying the theoretical analysis are also presented. Chaotic behavior of Chen's system with the strong kernel is also found through numerical simulation, in which some waveform diagrams, phase portraits, and bifurcation plots are presented and analyzed.