Discrete Lorenz Attractors in 3D Sinusoidal Maps

Abstract

Discrete Lorenz attractors can be found in three-dimensional discrete maps. Discrete Lorenz attractors have similar topology to that of the continuous Lorenz attractor exhibited by the well studied 3D Lorenz system. However, the routes to the formation of discrete Lorenz attractor in 3D maps is different from that of the routes of formation of continuous Lorenz flow attractor. This paper explores the exotic dynamics of a three-dimensional sinusoidal map, highlighting the existence of discrete Lorenz attractors. Through a detailed bifurcation analysis, we discuss various formation pathways involving supercritical and subcritical Neimark-Sacker bifurcations, and homoclinic butterflies towards the formation of discrete Lorenz attractors. Using a two-parameter Lyapunov chart analysis, we systematically illustrate chaotic regions and periodic regions in the parameter space, illustrating diverse attractor topologies, including hyperchaotic and wavy chaotic structures. Furthermore, we investigate spatiotemporal patterns in a ring-star network of coupled sinusoidal maps, revealing diverse spatiotemporal patterns such as chimera states, traveling waves, and synchronization transitions, controlled by coupling strengths of ring and star networks respectively. A key contribution is the application of hyperchaotic signals to secure video encryption, where we apply YOLOv10 for object detection and CNN-based cryptographic key generation. This selective encryption approach ensures enhanced security, high computational efficiency, and robustness against differential attacks.