Filtered Quadratic Map with Triple Positive Lyapunov Exponents: Route to Hyperchaos and Torus Doubling

This paper investigates the dynamical effects of Finite Impulse Response (FIR) filtering with two coefficients on a three-dimensional quadratic mapping. Chaotic and hyperchaotic maps are widely used in nonlinear science and engineering applications such as secure communication and image encryption; however, the impact of signal filtering on higher-dimensional hyperchaotic discrete systems remains largely unexplored. Motivated by the emergence of extreme hyperchaos characterized by all three positive Lyapunov Exponents (LEs) in the unfiltered system and by the need for bandwidth control and signal smoothing in practical applications, we systematically explore how the introduction of FIR filtering alters the bifurcation structure and associated dynamical regimes. Fixed points of the filtered system are computed analytically, and their stability is analyzed in detail. The organization of attractors is further investigated in relevant two-parameter spaces, revealing rich bifurcation scenarios including period-doubling and Neimark–Sacker bifurcations. Notably, despite filtering’s smoothing effect, the system preserves hyperchaotic dynamics with all three positive LEs across a wide range of parameter values. The persistence of ergodic and resonant torus-doubling bifurcations is also demonstrated in the presence of filtering. As a practical application, the chaotic sequences generated by the filtered map are employed for secure grayscale and color image encryption, with statistical analyses confirming high security and accurate decryption.