Regularizing Microscopes and Rivers

This paper proposes a generalization of the existing geometric studies of resonance. The Riccati equation associated with any second-order linear equation is extended to any C^∞ first-order equation. The Morse-critical point is generalized to any “generic” critical point. The resonant solution becomes a general canard solution. The paper explains how to find the regularizing blowup, and shows how classical special functions become enlarged in rivers, i.e., some resurgent solutions of polynomial differential equations. The paper shows a matching principle that connects the slow solutions with these rivers. The method to show the existence of canards is applied for some Union-Jack equations, i.e., equations with a critical point where three smooth curves intersect.