Double $S$-Breaking Cubic Turning Points and Their Computation

Abstract

In the paper we are concerned with double $S$-breaking cubic turning points of two-parameter nonlinear problems in the presence of ${\cal Z}_2$-symmetry. Three extended systems are proposed to determine double $S$-breaking cubic turning points. We show that there exist two kinds of singular point path passing through double $S$-breaking cubic turning point. One is the simple quadratic turning point path, the other is the pitchfork bifurcation point path.