Local bifurcations in delayed chaos anticontrol systems

In this paper, we analyze the local bifurcation phenomena in a simple system described by equation x˙(t)=-ax(t)+bsin(x(t-τ)), which is an one-dimensional linear system with nonlinear delayed feedback. Such systems have been proven to exhibit chaotic behavior, and thus can be viewed as the so-called chaos anticontrol systems. In this paper, the nonlinearity is chosen as the trigonometric function sin(·), different from the existing ones. By local analysis we prove that with increasing parameters, the number of equilibria increases and Hopf bifurcation occurs near some equilibria. This complex bifurcation phenomenon can help to understand the complex behavior of such models. To illustrate the theoretical results, bifurcation diagrams are numerically calculated and Hopf bifurcation and chaotic behavior are identified.