This first of three related books examines local dynamics of planar nonlinear systems with univariate vector fields through polynomialization. Singular sink source and saddle flows and singular parabola flows are discussed first, followed by local singular arrays of 1-dimensional flows in planar function systems with constant and self and crossing-univariate vector fields. The local singular flows are the appearing and switching bifurcations of the 1-dimensional flows. The local arrays of 1-dimensional flows include the local arrays of sink, source and saddle flows, and the local arrays of parabola and inflections flows. The singular self and crossing-flows with singular infinite-equilibriums existing in single-variable systems and the infinite-equilibriums for singular self and crossing-flows are also presented. The singular hybrid arrays of self and crossing 1-dimensional flows in single-univariate planar systems are discussed, and the infinite-equilibriums in the local singular hybrid flow arrays are determined. The switching bifurcations of two local hybrid flow arrays are determined through infinite-equilibriums. Local homoclinic networks without centers in self-univariate systems are discussed, and the singular self-equilibriums are for appearing and switching bifurcation of local simple and singular homoclinic network without centers. Local homoclinic networks with centers in crossing-univariate systems are discussed, and the singular crossing-equilibriums are for appearing and switching bifurcation of local simple and singular homoclinic network with centers.
