Using center manifold theory, we investigate complex dynamics in a segmented disc dynamo with mechanical friction by making a determination of equilibrium states and analyzing local stability. We find that the system undergoes Hopf bifurcation for a key parameter. Normal form theory produces the formulae for determining a superor sub-critical Hopf bifurcation; the stability of the bifurcating limit cycle is also determined. In addition, integrability of the system is studied carefully.

Based on the fact that Chua's system is a classic model system of electronic circuits, we first present modified Chua's system with a smooth nonlinearity, described by a cubic polynomial in this paper. Then, we explore the distribution of the equilibrium points of the modified Chua circuit system. By using the averaging theory, we consider zero-Hopf bifurcation of the modified Chua system. Moreover, the existence of periodic solutions in the modified Chua system is derived from the classical Hopf bifurcation theorem.