Advanced Multivariate Catastrophe Model for Quantitative Analysis of Complex Systems With Case Studies and Validation

Quantitative prediction of state transitions, particularly in complex multivariable-coupled systems, represents a long-standing scientific challenge. Classical catastrophe theory, while conceptually powerful, is severely constrained in practice by its inherently qualitative nature and by limits on dimensionality. To overcome this bottleneck, we introduce the multivariate quantitative catastrophe model (MQCM). This framework, while preserving the core topological architecture of catastrophe theory, incorporates a power-law composite control function to integrate multiple physical parameters, enforcing dimensional homogeneity as a physical constraint. This approach elevates the theory from a paradigm of qualitative classification to one of robust quantitative prediction. The model's predictive capability is verified through two classical problems, blackbody radiation and the heat capacity of solids. In both cases, MQCM starts from a single unified parent formula and, using singularity analysis, independently derives the governing physical laws in the corresponding asymptotic limits. MQCM thus establishes a systematic, mathematically rigorous, and physically insightful framework for the quantitative application of catastrophe theory. The framework is particularly well-suited to complex systems that display distinct scaling laws on opposite sides of a critical point. This work opens a new avenue for understanding critical phenomena and lays a foundation for interdisciplinary applications in materials science, engineering, and beyond.