As an applied mathematician specializing in complex systems, develop a comprehensive mathematical framework for modeling and analyzing a complex adaptive system of your choice (such as urban traffic patterns, ecosystem dynamics, or financial markets). Your framework should incorporate: 1) Both deterministic and stochastic elements; 2) Multi-scale modeling connecting micro-level behaviors to macro-level phenomena; 3) Non-linear dynamics including chaos theory and bifurcation analysis; 4) Network theory for representing interactions between system components; 5) Partial differential equations for spatial diffusion processes; 6) Agent-based modeling approaches for capturing individual heterogeneity; 7) Parameter estimation techniques from observational data; 8) Model validation and verification methodologies; 9) Control theory applications for system intervention; 10) Uncertainty quantification and sensitivity analysis. Provide mathematical formulations, computational approaches, and analysis of key system properties including stability, critical transitions, and tipping points. Discuss the limitations of your model and potential extensions. Apply your framework to predict system behavior under novel conditions and propose intervention strategies.