Miscellaneous Aspects of Bifurcation Phenomena

Abstract

Bifurcation behaviors involve various uncertainties and complexities, only a fraction of which has been investigated in the previous chapters of Part III. At the end of this part, we present the miscellaneous aspects of bifurcation phenomena that have not been exhausted by this investigation: A clustered bifurcation point, which is a set of closely located simple bifurcation points, appears extensively for bifurcation of materials. This subject is dealt with in §15.2. Size effect is observed in experiments for rocks (Grosvenor, 1963 [62]), concretes (Sangha and Dhir, 1972 [158]), and sands (Goto, 1986 [59]; Ikeda and Goto, 1993 [79]). In §15.3 the size effect of materials is investigated from the standpoints of fracture, imperfection sensitivity, hilltop bifurcation, and mode switching. A hilltop bifurcation point, which is a parametric double critical point occurring as a coincidence of a limit point and a simple pitchfork bifurcation point, is observed for an atomic lattice (Thompson and Schorrock, 1975 [176]) and for a slender steel specimen (Okazawa et al., 2002 [141]). We deal with this subject in §15.3.3. An interesting bifurcation phenomenon, called an explosive bifurcation, is observed (Ikeda, Providéncia, and Hunt, 1993 [97]). We see this in §15.4.