The Bifurcation of the Topological Structure in the Sunspot's Electric   Topological Current with Locally Gauge-invariant Maxwell-Chern-Simons Term
  The topological structure of the electric topological current of the locally gauge invariant Maxwell-Chern-Simons Model and its bifurcation is studied. The electric topological charge is quantized in term of winding number. The Hopf indices and Brouwer degree labeled the local topological structure of the electric topological current. Using $\Phi $-mapping method and implicity theory, the electric topological current is found generating or annihilating at the limit points and splitting or merging at the bifurcate points. The total electric charge holds invariant during the evolution. 
Sheng Li and Yishi Duan