Quadratization refers to a transformation of arbitrary system of polynomial ordinary differential equations to a system with at most quadratic right-hand side. Such a transformation unveils new variables and model structures that facilitate model analysis, simulation, and control and offers a convenient parameterization for data-driven approaches. Quadratization techniques have found applications in diverse fields, including systems theory, fluid mechanics, chemical reaction modeling, and mathematical analysis.

In this study, we focus on quadratizations that preserve the stability properties of the original model, specifically dissipativity at given equilibria. This preservation is desirable in many applications of quadratization including reachability analysis and synthetic biology. We establish the existence of dissipativity-preserving quadratizations, develop an algorithm for their computation, and demonstrate it on several case studies.

Cities are crucial in societal transformations, and urban development significantly affects living standards and economic growth, particularly as many industrial sector workers reside in urban areas. In this project, we study the accessibility of important amenities (e.g. essential shops and services, schools, etc.) with the influence of public transport through the concept of the X-minute city. We build a statistical system based on 2SFCA and node-place index model to measure the TOD degree of the city based on the railway network of the city.

This project considers specific systems with high-degree nonlinearities and applies the composition of the algorithms described above to solve the reachability problem. The main challenge is to adjust the transformation at most quadratic systems in a way that would make the subsequent Carleman linearization as numerically stable as possible. In the project, we try to first explore 3 typical ODE systems with different characteristics, and then we prove that for any dissipative polynomial ODEs system, there exists a quadratization such that the quadratized system is dissipative as well. Based on this theorem, we later derive that if all the new variables introduced in quadratization can be quadratized by other variables, then the dissipativity of the resulting quadratic system can be ensured.