On the Abnormality of Period Doubling Bifurcations ---In Connection with the Bifurcation Structure in the Belousov-Zhabotinsky Reaction System---
Two kinds of abnormal characters of period doubling bifurcations are studied as compared with Feigenbaum's universality theory. One kind of abnormality is that Feigenbaum constant distributes although the cascade of 2n type bifurcation continues to infinity. The other kind of abnormality is that the cascade of 2n type bifurcation is interrupted at finite n by an emergence of a bifurcation of a different type. The latter abnormality is obtained in our previous Lorenz map determined empirically in the Belousov-Zhabotinsky reaction. Furthermore, a perturbation approach to the abnormalities in period doubling bifurcations is developed. With this perturbation approach, it is proved that a breakdown of the third order balanced equation at the onset regime of 2n-periodic solution implies a breakdown of the Schwarzian condition.