Suppose $F_{C}=(x-C)^{2}$ is a family of functions. I found that at $C=-\frac{1}{4}$ there is a saddle-node bifurcation. So for $C<-\frac{1}{4}$ there are $0$ fixed points, $C=-\frac{1}{4}$ there is one fixed point and $C>-\frac{1}{4}$ there are two fixed points. Now I am trying to figure out if there is a $C-value$ for which a period-doubling bifurcation exists. My question is, if a family of functions can experience a saddle-node bifurcation, can it also experience a period -doubling bifurcation?
2026-03-25 11:03:24.1774436604
Can a family of functions $F_{\lambda}$ experience a saddle-node bifurcation and also experience a period -doubling bifurcation?
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