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| Using the $phi$-mapping method, we have introduce a topological current to describe the topological feature of the vector field. In this description, the equilibria can be viewed as the particles, which carry the winding numbers as their topological charges and move in the phase space as the parameter varies. Based on this description, we qualitatively discuss various steady-state bifurcations of the vector field, and find a general method to determine the numbers and the directions of the bifurcation curves at the limit points and the bifurcation points. |
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| Keywords:mathematical physics, vector field, topological current, steady-state bifurcations |
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