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 Poincaré 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.

本文致力于研究Minkowski时空\（{\ mathbb {R}} ^ {1 + 3} \）中时态极值超曲面的奇异现象。我们发现，在Minkowski时空\（{\ mathbb {R}} ^ {1 + 3} \）中，时态极值超曲面的图形表示有两种显式的光似自相似解。，它们的几何形状是两个球体。对于径向对称膜方程，给出了那些类似光的自相似解的线性模态。之后，我们证明了径向对称膜方程的自相似解在后向光锥的严格适当子集内是非线性稳定的。这意味着这两个球体的动力学行为是吸引子。同时，当我们为线性算子进行频谱分析时，通过构造牛顿多边形克服了双根情况（不能使用庞加莱定理）来求解差分方程。