This study presents alpha-stable autoregressive (AR) modeling of the dynamics of Chua's circuit in the presence of heavy-tailed noise. The parameters of the AR time series are estimated using the covariation-based Yule-Walker method, and the parameters of alpha-stable distributed residuals are calculated using the regression type method. Visual depictions of the calculated parameters of the AR model and alpha-stable distributions of residuals are presented. The medians of the estimated parameters of the AR model and alpha-stable distributions parameters of residuals are presented for heavy-tailed noise with various stability index parameters. Thus, the impulsive behavior of Chua's circuit can be modeled as alpha-stable AR time series, and the model can provide an alternative approach to describe the chaotic systems driven by heavy-tailed noise.