Problem-solving in early childhood has been receiving increasing research attention and recognition for its crucial role in the teaching and learning of arithmetic skills (e.g. Vessonen et al., 2025). When preschoolers engage in problem solving, they can develop key mathematical processes such as reasoning and proof, communication, and the ability to make connections and representations (Chen, 2024; Nafiqoh, 2025). For preschoolers with intellectual disabilities, these key mathematical processes are even more important, as they are closely linked to the acquisition of skills essential for independent living (Charitaki et al., 2024). The present study aimed to investigate the way that individual and contextual factors interact in producing nonlinear transitions in children's problem-solving competencies, to capture discontinuities in children's learning processes with the use of cusp catastrophe modeling (as an optimal model), and to reveal the specific factors that serve as control (asymmetry) and bifurcation (splitting) parameters within early problem-solving dynamics. A total number of 64 preschoolers with intellectual disabilities (min=4, max=8, m=5.16, sd=.73 years) enrolled in this study. We administered children's problem solving based on different domains of early numeracy [Parents' Questionnaire (LeFevre et al., 2009), The Academic Attainment Checklist (Sloper et al., 1990) and Parents' Attitudes Toward Mathematics and Literacy scale (LeFevre et al., 2009). The findings suggest that problem-solving in early childhood unfolds nonlinearly, with the potential for sudden shifts in competence when specific cognitive thresholds are reached. In particular, self-regulated problem-solving acts as a bifurcation variable, potentially pushing children over a developmental "cusp". This aligns with developmental systems theories and contributes to educational psychology by empirically validating a dynamic model for early learning. Moreover, it opens a new methodological pathway for capturing discontinuous changes in behavior and cognition, especially valuable in early education, where such shifts are standard yet hard to model with traditional regression.