Bifurcation in a family of maps

Remarks

Example: for $f_\mu(x) = \mu x$ on $\R$ with $\mu \in (0, \infty)$, the value $\mu_0 = 1$ is a bifurcation value. The map $f_1$ is the identity, so every point is fixed, while for $\mu \ne 1$ the only fixed point is $0$. A conjugacy maps fixed points bijectively to fixed points, so $f_\mu$ is not conjugate to $f_1$.

Non-example: in the same family, $\mu_0 = 1/2$ is not a bifurcation value. For $0 < a, b < 1$ the homeomorphism $h(x) = \sgn(x) \abs{x}^{\ln b / \ln a}$ satisfies $h(a x) = b\, h(x)$, so all $f_\mu$ with $\mu \in (0, 1)$ are conjugate to each other.

Many texts use a local version, in which only the behaviour near a fixed point or a periodic orbit is compared. A local bifurcation of a fixed point $x^*$ of a smooth family can only occur when an eigenvalue of $D f_{\mu_0}(x^*)$ lies on the unit circle. For one-dimensional maps this means $f_{\mu_0}'(x^*) = 1$ or $f_{\mu_0}'(x^*) = -1$.