In this paper, we have presented a novel one-dimensional discrete-time chaotic map. This chaotic map has a uni-modal transfer characteristic. Unlike common practices, the mechanism of generating the positive slope and the negative slope of the uni-modal transfer characteristic are split in this map circuit. In this way, a stiffer transfer characteristic is achieved which results in a more complex chaotic behavior compared to already published one-dimensional maps. The design methodology of this split-slope chaotic map is presented with the help of the stability analysis of fixed points in non-linear dynamics. The design methodology is generally applicable to a wide variety of non-linear circuits. The chaotic complexity of the proposed circuit is analyzed with bifurcation plot, correlation-coefficient, and Lyapunov Exponent. The results are compared with reported works to demonstrate a significant improvement that is achieved from the proposed design. Along with high chaotic complexity, this split-slope chaotic map provides a wide chaotic range covering 100\% of the region of operation with a low-overhead CMOS circuit. The high chaotic complexity and wide chaotic range of the proposed circuit can be applicable for hardware-security applications including, random number generation, chaotic logic circuits, and so on for resource-constrained devices.