This study explores the hypothesis related to a punctuated equilibrium process for conceptual change in science learning, in conjunction with the effects of four cognitive variables: logical thinking and field-dependence/field-independence, divergent and convergent thinking. The participants were fifth and sixth-grade elementary school pupils involved in different tasks, who were asked to describe and interpret chemical phenomena. Latent Class Analysis (LCA) was applied to children's responses, and three clusters or latent classes (LC1, LC2, and LC3) were identified, corresponding to hierarchical levels of conceptual understanding. The ensued LCs align with the theoretical conjecture about a stepwise conceptual change process that might go through various stages or mental models. These levels or stages are conceptualized as attractors, and changes between them were modeled as cusp catastrophes using the four cognitive variables as controls. The analysis showed that logical thinking acted as asymmetry factor, while field-dependence/field-independence, divergent and convergent thinking acted as bifurcation variables. This analytic approach presents a methodology for investigating conceptual change as a punctuated equilibrium process that adds to the nonlinear dynamical research with important implications for theories of conceptual change in science education and psychology as well. Discussion on the new perspective embracing the meta-theoretical framework of complex adaptive systems (CDS) is provided.