Dual Nature of the Electronic Structure of $\left({\mathrm{La}}_{2-\mathit{x}-\mathit{y}}{\mathrm{Nd}}_{\mathit{y}}{\mathrm{Sr}}_{\mathit{x}}\right){\mathrm{CuO}}_4$ and ${\mathrm{La}}_{1.85}{\mathrm{Sr}}_{0.15}{\mathrm{CuO}}_4$

High resolution angle-resolved photoemission measurements have been carried out on $\left({\mathrm{La}}_{1.4-x}-{\mathrm{Nd}}_{0.6}{\mathrm{Sr}}_x\right){\mathrm{CuO}}_4$, a model system with static one-dimensional (1D) charge ordering (stripe), and $\left({\mathrm{La}}_{1.85}-{\mathrm{Sr}}_{0.15}\right){\mathrm{CuO}}_4$, a high temperature superconductor $(T_c=40\mathrm{}\mathrm{K})$ with possible dynamic stripes. In addition to the straight segments near ( $\pi ,0\mathrm{}$) and ( $0,\pi$) antinodal regions, we have identified the existence of spectral weight along the $[1,1]$ nodal direction in the electronic structure of both systems. This observation of nodal state, together with the straight segments near antinodal regions, reveals the dual nature of the electronic structure of stripes due to the competition of order and disorder.