Constant and Self-Cubic Vector Fields

In this chapter, dynamical systems with constant and self-univariate cubic vector fields are considered. The corresponding stability and bifurcation of the one-dimensional flow are discussed. The first-order and third-order sink and source flows in such a cubic nonlinear system with single-variable functions are presented from the first integral manifolds. The upper-saddle and lower-saddle flows are the appearing and switching bifurcations of the sink and source flows. The third-order source flow bifurcations are the appearing bifurcations from a source flow to source, sink, and source flows. The third-order sink flow bifurcations are the appearing bifurcations from a sink flow to sink, source, and sink flows. The third-order source flows are the switching bifurcations for source, sink, and source flows. The third-order sink flows are also the switching bifurcations for sink, source, and sink flows.