Existing analytical models for acoustic cryocooler vibration analysis assume 1-D axisymmetric behavior. A multiple scales perturbation analysis is used to describe the lateral dynamic response and stability of an axially oscillating Stirling cycle cryocooler core. The model of the oscillating core mass is assumed to possess two degrees of lateral motion freedom and the forcing function is multiple harmonic axial motion. A pair of mechanical springs which support the core mass is assumed to have axial-displacement-dependent lateral stiffness characteristics which results in parametric excitation of the lateral degrees of freedom. The effect of perturbing gas--fluid forces acting on the piston, which disturb the otherwise axisymmetric motion, is investigated. The perturbation analysis predicts the dynamic response in closed form and identifies regions of operation for unstable/chaotic motion. The results of the closed-form dynamic response predictions are compared with a numerical time-step Runga--Kutta method [F. Hausle, J. Acoust. Soc. Am. 94, 1853 (1993)]. Model results are also compared with experimental displacement-time-frequency data obtained with a new design of cryocooler. The insight given by this analysis is expected to lead to improved cryocooler design based on better understanding of the domain of configurational and operational parameters.