The Normal Form of the Pitchfork Bifurcation for Fractional-Order Systems

摘要

This paper investigates the pitchfork bifurcation for the fractional di甧rential system involving one parameter in the variable space. Applying the Lyapunov direct method for the fractional differential system to prove the stability or the instability of the equilibria, the phase portrait of the pitchfork bifurcation is obtained. With the help of the Taylor's expansion and the Implicit Function Theorem in the classical sense, the normal form of the pitchfork bifurcation for the fractional differential system is calculated. By using the topological equivalent map which depend on one parameter, the topological normal form of the pitchfork bifurcation for the fractional differential system is derived.