The nonlinear properties of natural convection in the space of an eccentric annulus are investigated using the lattice Boltzmann method (LBM). Firstly, the system is mathematically determined to develop into a chaotic state at high Rayleigh numbers through the maximum Lyapunov exponent spectrum and run test. Then, the process of the system transitioning to chaos is characterized based on the characteristics of numerical solution phase diagram and power spectral density (PSD). The results show that with the increase of Rayleigh number Ra, the solution of the eccentric annular system changes from deterministic steady-state solution to periodic oscillation solution through Hopf bifurcation, and the phase diagram trajectory changes from fixed point to limit cycle. With further increase of Rayleigh number, the stable limit cycle bifurcates into a two-dimensional torus, and the system enters a quasi-periodic state. When Rayleigh number Ra reaches a critical value, the phase diagram trajectory of the system exhibits rapid exponential separation, becomes extremely complex, and many incommensurable fundamental frequencies appear in its power spectral density. Chaotic attractors emerge, Hopf bifurcation occurs again, and eventually chaos is reached.