Degn–Harrison map: Dynamical and network behaviours with applications in image encryption

This paper focuses on the discrete dynamics of the Degn-Harrison model, which is used to explain the respiration rate oscillation behaviour seen in the continuous cultures of Klebsiella aerogenes bacteria. The system’s transition between periodic, quasiperiodic, and chaotic states with respect to system parameters is studied by employing the techniques of non-invertibility, fixed-point analysis, and bifurcation theory. The model demonstrates complex behaviours, including saddle-node bifurcation, period-doubling bifurcations, and Neimark-Sacker bifurcation, providing a comprehensive understanding of the mechanisms underlying these dynamical changes. Furthermore, we also explore the Degn-Harrison model’s application in a ring-star network topology. Various rich spatiotemporal patterns, such as synchronized state, unsynchronized state and chimera states, are detected and discussed. Moreover, we show that these patterns arise due to the network’s different coupling strengths. Finally, we introduced an image encryption technique based on the discrete Degn-Harrison model of chaotic dynamics, the well-known framework of complex dynamical behaviours. It is used to construct a scheme of insensitive encryption relating to initial conditions and transition from periodic to chaotic states whereby it ensures strong security in image data. The results of the paper advance the knowledge regarding discrete dynamical systems in chemical and biological systems, especially in the ones that encounter oscillation and synchronization, and also reveal that Degn-Harrison models have the potential for the safe transmission of images.