Abstract: Based on the lattice Boltzmann method (LBM), this study explores the nonlinear characteristics of natural convection in a horizontal ring-shaped space, where the topology remains invariant but the Euclidean geometric features vary. Initially, by computing the maximum Lyapunov exponent spectrum, the system’s gradual transition to chaos under high Rayleigh number conditions is mathematically confirmed. The analysis then examines the flow dynamics within the cavity and the variations in dimensionless temperature. As the Rayleigh number increases, the study tracks the progression from stable flow to periodic behavior and, ultimately, to chaos by investigating the phase space trajectories. This sequence a comprehensive depiction of the system's progression toward chaos. The results reveal that as the Rayleigh number (Ra) increases, the geometric properties of the ring-shaped space have a significant influence on the solution's morphology. Among the models considered, the inner-square outer-circle configuration is the most susceptible to instability. The natural convection solutions in the ring-shaped space transition sequentially from a fully stable deterministic state, to a quasi-periodic oscillation solution and ultimately to chaotic behavior, demonstrating intricate dynamic features. The corresponding phase space trajectory evolves from an initial fixed point into a two-dimensional torus, and eventually into more complex structures, effectively illustrating the system’s nonlinear transitions. With further increase in the Rayleigh number, the stable two-dimensional torus becomes increasingly complex, and the system ultimately reaches a chaotic state. In summary, the system exhibits three distinct states: stable solutions, (quasi-)periodic solutions, and chaotic solutions, with clear transitions observed as the Rayleigh number increases.