Robust chaos in the Tinkerbell map and its applications

Purpose This article aims to present a robust chaotic dynamic analysis of the Tinkerbell map and to investigate the persistence of chaos over wide ranges of its bifurcation parameters. Design/methodology/approach An analytical and numerical approach is employed, including Lyapunov exponent analysis, phase space characterization and stability analysis of fixed points, to examine the dynamical behavior of the system. Findings The results demonstrate that the discrete-time nonlinear Tinkerbell map exhibits strongly chaotic behavior over wide parameter ranges without periodic windows. The study confirms the structural stability of the system, showing that its strange attractor remains unchanged despite parameter variations. Furthermore, the analysis identifies the key bifurcation mechanisms governing transitions between stability, bifurcations and chaos, as well as the sensitivity to initial conditions. Social implications The findings of this study indicate that the application of the Tinkerbell map to image encryption may carry significant social implications. In particular, enhancing the security of digital images can contribute to the protection of privacy, the safeguarding of sensitive visual data, and the strengthening of trust in technological systems that depend on the secure processing, storage, and transmission of images. Originality/value This work highlights the Tinkerbell map as a significant benchmark for robust chaos in discrete dynamical systems, with potential applications in physics, engineering and information security.