Structure of random Turing-like patterns in discrete-time systems is determined by the initial conditions

Patterns, spatiotemporal ordered structures, are prevalent in diverse systems, arising from the emergence of complexity. Turing proposed a mechanism that involves a short-range activator and a long-range inhibitor to explain the formation of patterns, and patterns that satisfy this mechanism are called Turing patterns. Patterns with similar structures but not caused by the Turing mechanism are referred to as Turing-like patterns. In the absence of external influences, the structure of Turing patterns is generally determined by control parameters. In this study, we revealed that the structure of Turing-like patterns in discrete-time systems is only determined by the ratio of states in the initial conditions. As the ratio changes, the structure of patterns transitions from spots to labyrinth and eventually to inverse spots. We proposed the structure parameter for the quantitative description of the structure of the patterns. And the structure parameter is directly proportional to the ratio in the initial conditions. The mechanism underlying this structure control is attributed to the traversability of multiperiodic states in discrete-time systems, where each local point will go through all states in the periodic orbit. Our findings shed light on the pattern formation for Turing-like patterns in discrete-time systems.