Respiratory motion (RM) and cardiac motion (CM) degrade the quality and resolution in cardiac PET scans. We have developed non-rigid motion estimation methods to estimate both RM and CM based on 4D cardiac gated PET data alone, and compensate the dual respiratory and cardiac (R&C) motions after (MCAR), during (MCDR), and before (MCBR) image reconstruction. In all three R&C motion correction methods, attenuation-activity mismatch effect was modeled by using transformed attenuation maps using the estimated RM. The difference of using activity preserving and non-activity preserving models in R&C correction was also studied. Realistic Monte Carlo simulated 4D cardiac PET data using the 4D XCAT phantom and accurate models of the scanner design parameters and performance characteristics at different noise levels were employed as the known truth and for method development and evaluation. Results from the simulation study suggested that all three dual R&C motion correction methods provide substantial improvement in the quality of 4D cardiac gated PET images as compared with no motion correction. Specifically, the MCDR method yields the best performance for all different noise levels compared with the MCAR and MCBR methods. While MCBR reduces computational time dramatically but the resultant 4D cardiac gated PET images has overall inferior image quality when compared to that from the MCAR and MCDR approaches in the 'almost' noise free case. Also, the MCBR method has better noise handling properties when compared with MCAR and provides better quantitative results in high noise cases. When the goal is to reduce scan time or patient radiation dose, MCDR and MCBR provide a good compromise between image quality and computational times.