Abstract Intercycle variations in the series of 11 yr solar activity cycles have a significant impact on both the space environment and climate. Whether solar cycle variability is dominated by deterministic chaos or stochastic perturbations remains an open question. Distinguishing between the two mechanisms is crucial for predicting solar cycles. Here we reduce the solar dynamo process responsible for the solar cycle to a one-dimensional iterative map, incorporating recent advances in the observed nonlinearity and stochasticity of the cycle. We demonstrate that deterministic chaos is absent in the nonlinear system, regardless of model parameters, if the generation of the poloidal field follows an increase-then-saturate pattern as the cycle strength increases. The synthesized solar cycles generated by the iterative map exhibit a probability density function (PDF) similar to that of observed normal cycles, supporting the dominant role of stochasticity in solar cycle variability. The parameters governing nonlinearity and stochasticity profoundly influence the PDF. The iterative map provides a quick and effective tool for predicting the range, including uncertainty, of the subsequent cycle strength when an ongoing cycle amplitude is known. Due to stochasticity, a solar cycle loses almost all its original information within one or two cycles. Although the simplicity of the iterative map, the behaviors it exhibits are generic for the nonlinear system. Our results provide guidelines for analyzing solar dynamo models in terms of chaos and stochasticity, highlight the limitations in predicting the solar cycle, and motivate further refinement of observational constraints on nonlinear and stochastic processes.