SINGLE LINEAR-BIVARIATE CUBIC SYSTEMS

This book is about the nonlinear dynamics of single linear-bivariate cubic dynamical systems. For such cubic dynamical systems, the inflection-flows and third-order parabola flows exist for appearing bifurcations. The inflection-flows are for appearing bifurcations of two parabola flows on the same direction. The third-order parabola flows are for the appearing bifurcation of inflection and parabola flows, and for the appearing bifurcations of up, down and up-parabola flows or down, up and down-parabola flows. Third-order parabola flow are for the appearing bifurcation among the up and down-parabola flows. There are four types of infinite-equilibriums: (i) The inflection-source and sink infinite-equilibriums are for the switching bifurcations of two parabola flows on the two-directions. (ii) The parabola-source and sink infinite-equilibriums are for the switching bifurcations of parabola and inflection flows on the two-directions. (iii) The inflection-saddle infinite-equilibriums are for the switching bifurcation of two inflection flows in two directions. (iv) The parabola-saddle infinite-equilibriums are for the switching bifurcations of parabola and inflection flows on the two-directions are the switching bifurcations for parabola and inflection flows. Such switching bifurcations for 1-dimensional flow are based on the infinite-equilibriums, which will help one understand global dynamics in nonlinear cubic systems.