My fascination with complex systems began not in a classroom, but managing my digital metropolis, “Vectorville,” in the game Cities: Skylines. For weeks, I watched my population swell based on a simple rule: build more, and they will come. Then, a tiny, half-percent tax adjustment triggered a catastrophic "death spiral." Businesses shuttered, a mass exodus began, and my vibrant city collapsed into a ghost town. I was left staring at the ruins as a curious student -- How could a system built on logical, deterministic rules produce such violent, unpredictable chaos? This "death spiral" stimulated my interest -- The collapse of Vectorville is a powerful echo of the warnings found in foundational texts like Limits to Growth, where complex systems—be they cities, economies, or ecosystems—can exhibit behavior that defies simple, linear intuition. The game is a tangible microcosm of these principles, a world where the intricate dance between reinforcing loops (population booms), balancing loops (job availability, pollution), and time delays (construction lag, supply chains) dictates prosperity or ruin. Understanding the mathematical engine that drives these phenomena is a worthy investigation into the fundamental nature of the complex, interconnected systems that govern our own world, from urban planning to climate science. My exploration will therefore be a quest to deconstruct and rebuild the "engine" of my virtual city using the language of mathematics. I will begin by modeling the simple, naive assumption of exponential growth that started my journey. Then, in response to the model’s failures and my own in-game observations, I will systematically introduce layers of complexity by developing and comparing a series of differential equation models. This progression will allow me to explore the stabilizing influence of carrying capacity, the oscillating effects of time delays, and finally, the volatile interdependence of competing factors like economic growth and pollution. Hence, I aim to investigate how the integration of feedback loops and non-linear interactions within a system of differential equations can model the emergence of complex, stable, and chaotic behavior in a simulated city population.