<p>Chaotic maps are known for their unpredictability, randomness, and sensitivity to initial conditions, making them valuable for cryptographic applications. However, the presence of fixed points can pose challenges, potentially leading to system failures. This study introduces a novel one-dimensional chaotic map that does not possess any fixed point, making the map very efficient for practical applications. In addition, the map possesses high computational efficiency and exhibits robust chaotic behavior. The effectiveness of the proposed map is demonstrated through bifurcation diagrams, Lyapunov exponents, and entropy, and validated using NIST SP800-22 tests. Furthermore, the application of this chaotic map in image encryption is explored, and the corresponding encryption scheme is evaluated using metrics such as correlation coefficient, mean squared error, peak signal-to-noise ratio, entropy, key sensitivity, histogram analysis, and robustness against attacks. The results show that the proposed map and encryption scheme outperform existing methods, demonstrating its potential for securing digital images.</p>