<p>Chaotic systems attract significant attention due to the rich dynamics induced by their nonlinear terms. While most existing systems are modeled using continuous nonlinear functions, only a few chaotic systems exhibit singularities. These singular systems often fall into categories of jump or removable continuity. Chaotic models involving poles or infinite continuity remain largely unexplored. To address this gap, this paper introduces a novel chaotic system characterized by a Singular Plane (SP), which features a pole. The dynamics analysis proves the chaotic nature of this system. Owing to the presence of the SP, the system exhibits two distinct types of Lyapunov exponent (LE) behaviors depending on the initial conditions. Furthermore, a modified version of the system is proposed to investigate how trajectories approach or cross the SP. In doing so, the blow-up behavior and transient divergence are observed–phenomena that are rarely reported in other chaotic systems. Then, the original system is implemented on an FPGA platform, and its hardware-level performance is thoroughly evaluated. The period–precision rate is introduced to quantify the degree of chaos degradation in hardware implementations, providing practical guidance for selecting appropriate data precision in chaos-based applications. Finally, an image encryption algorithm utilizing the proposed chaotic system is presented.</p>