Stability and Hopf bifurcation analysis in a delay Swarms model

In this paper, the problem of Hopf bifurcation for a swarm model of self-propelled agents in the presence of noise and communication time delay is studied. When the value of delayed communication time passes through a critical value, Hopf bifurcation will occur, where the center of mass undergoes a Hopf bifurcation from steady state to a limit cycle, the swarm is transformed from a misaligned state into an aligned state. Using a mean field model, we prove the existence of bifurcation and gave the existence conditions of bifurcation. Numerical results have been presented to verify the theoretical results.