  In this paper we establish an invariant set bifurcation theory for the
nonautonomous dynamical system $(\va_\lam,\0)_{X,\cH}$ generated by the
evolution equation \be\label{e0}u_t+Au=\lam u+p(t,u),\hs p\in
\cH=\cH[f(\.,u)]\ee on a Hilbert space $X$, where $A$ is a sectorial operator,
$\lam$ is the bifurcation parameter, $f(\.,u):\R\ra X$ is translation compact,
$f(t,0)\equiv0$ and $\cH[f]$ is the hull of $f(\.,u)$. Denote by
$\va_\lam:=\va_\lam(t,p)u$ the cocycle semiflow generated by the equation.
Under some other assumptions on $f$, we show that as the parameter $\lam$
crosses an eigenvalue $\lam_0\in\R$ of $A$, the system bifurcates from
  $0$ to a nonautonomous invariant set $B_\lam(\.)$ on one-sided neighborhood
of $\lam_0$. Moreover, $$\lim_{\lam\ra\lam_0}H_{X^\a}\(B_\lam(p),0\)=0,\hs p\in
P,$$ where $H_{X^\a}(\.,\.)$ denotes the Hausdorff semidistance in $X^\a$ (here
$X^\alpha$ ($\a\geq0$) defined below is the fractional power spaces associated
with $A$).
  Our result is based on the pullback attractor bifurcation on the local
central invariant manifolds $\cM^\lam_{loc}(\.)$.
