  This paper is devoted to the study of the singularity phenomenon of timelike
extremal hypersurfaces in Minkowski spacetime $\mathbb{R}^{1+3}$. We find that
there are two explicit lightlike self-similar solutions to a graph
representation of timelike extremal hypersurfaces in Minkowski spacetime
$\mathbb{R}^{1+3}$, the geometry of them are two spheres. The linear mode
unstable of those lightlike self-similar solutions for the radially symmetric
membranes equation is given. After that, we show those self-similar solutions
of the radially symmetric membranes equation are nonlinearly stable inside a
strictly proper subset of the backward lightcone. This means that the dynamical
behavior of those two spheres is as attractors. Meanwhile, we overcome the
double roots case (the theorem of Poincar\'{e} can't be used) in solving the
difference equation by construction of a Newton's polygon when we carry out the
analysis of spectrum for the linear operator.
