Abstract We investigate the localization properties of a quasi-one-dimensional two-channel system with symmetric and asymmetric onsite energies using the Aubry-Andr'{e} model. By analyzing the Lyapunov exponent and localization length, we characterize the phase transitions and critical behavior of the system. For the symmetric model, we obtain the phase diagram for the entire spectrum, revealing mobility edges between delocalized and localized states. In contrast, for the asymmetric model, we identify a critical line $\lambda_1^c + \lambda_2^c \approx 0.5$ marking the phase transition between delocalized and localized states. We also study the effects of the inter-channel coupling $\tilde{t}$ and observe that increasing $\tilde{t}$ reduces the delocalized phase space, shifting the transition from $\lambda_1 = \lambda_2 = 1$ at $\tilde{t} = 0$ to $\lambda_1^c + \lambda_2^c \approx 0.5$ at larger $\tilde{t}$. Additionally, we explore the influence of energy levels and the incommensurate modulation parameter, demonstrating that while the general phase transition behavior is preserved, subtle differences arise for different incommensurate modulation parameter values. Using the cost function approach, we calculate the critical potential strength $\lambda_c$ and the critical exponent $\nu$, with $\nu \approx 0.5$ for the middle of the spectrum in both symmetric and asymmetric models.
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