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.

Professor Albert C. J. Luo is currently the most distinguished research professor at Southern Illinois University Edwardsville (SIUE), USA, and an internationally renowned expert in the field of nonlinear dynamical systems theory and applications. Over the past 30 years, Professor Luo has made seminal contributions to theoretical physics, nonlinear dynamics, and applied mathematics. These contributions include: the theory of planar polynomial dynamical systems and solutions to Hilbert’s 16th problem; stability and bifurcation theory for nonlinear continuous dynamical systems; stability and bifurcation theory for nonlinear discrete dynamical systems; bifurcation dynamics; discontinuous dynamical system theory; generalized synchronization theory for dynamical systems; analytical and semi-analytical methods for periodic and chaotic motions in nonlinear dynamical systems; stochastic layer and resonance layer theory in nonlinear Hamiltonian systems; and dynamics of nonlinear deformable bodies. Professor Luo has published more than 400 peer-reviewed papers in prestigious journals and conference proceedings. He is the author of over 60 monographs and editor of more than 20 volumes. He serves as the editor-in-chief for the series Nonlinear Physical Science (co-published by Higher Education Press and Springer), the series Nonlinear Systems and Complexity (Springer), and the series Chaos, Nonlinearity, and Complexity (World Scientific). He is currently an associate editor for the International Journal of Bifurcation and Chaos and an editorial board member for the AIP journal Chaos.