  We present a new generalized topological current in terms of the order
parameter field $\vec \phi$ to describe the arbitrary dimensional topological
defects. By virtue of the $% \phi$-mapping method, we show that the topological
defects are generated from the zero points of the order parameter field $\vec
\phi$, and the topological charges of these topological defects are topological
quantized in terms of the Hopf indices and Brouwer degrees of $\phi$-mapping
under the condition that the Jacobian $% J(\frac \phi v)\neq 0$. When $J(\frac
\phi v)=0$, it is shown that there exist the crucial case of branch process.
Based on the implicit function theorem and the Taylor expansion, we detail the
bifurcation of generalized topological current and find different directions of
the bifurcation. The arbitrary dimensional topological defects are found
splitting or merging at the degenerate point of field function $\vec \phi $ but
the total charge of the topological defects is still unchanged.
