This paper employs the Lattice Boltzmann Method (LBM) to investigate the nonlinear characteristics of natural convection in toroidal spaces with radius ratios of 2.6, 1.6, 1.4, and 1.2. Based on the maximum Lyapunov exponent, runs test, and phase space trajectory, the transition from steady-state to chaotic state is analyzed. The results show that with increasing Rayleigh number (Ra), the toroidal system successively experiences a steady state, a periodic oscillation state, a quasi-periodic oscillation state, and finally enters a chaotic state. For example, when the radius ratio is 2.6, these transitions occur at Ra values of 5 × 10⁵, 1.5 × 10⁶, 2.1 × 10⁶, and 2.5 × 10⁶, respectively. Furthermore, the study finds that decreasing the radius ratio significantly lowers the critical Rayleigh number, indicating an increased sensitivity of the system to geometry. Moreover, under the same radius ratio and system state, the critical Rayleigh number for a concentric toroidal cavity is consistently higher than that for an eccentric toroidal cavity. These results quantitatively reveal the role of geometric parameters in controlling flow instability and the occurrence of chaos in toroidal convection systems.