Ordinary Differential Equations (7): Stability Theory

Wed, 11 Oct 2023 09:00:00 +0000

A small push hits a system. Does it return to rest, drift away, or break entirely? That single question decides whether bridges survive storms, ecosystems recover from droughts, and economies bounce back from crises. Stability theory answers it — and it does so without ever solving the differential equation. We will learn to read the destiny of a system off the geometry of its phase plane.

Ordinary Differential Equations (6): Linear Systems and the Matrix Exponential

Sun, 24 Sep 2023 09:00:00 +0000

One equation describes one quantity. The world is rarely that obliging. Predator and prey populations push each other up and down. Currents and voltages in an RLC network oscillate together. Chemical species in a reaction network feed into one another. The moment two unknowns share an equation, you have a system, and a single $$y'=ay$$ is no longer enough.

The miracle of the linear case is this: the scalar formula $y(t)=e^{at}y_0$ generalizes verbatim once you learn what $e^{At}$ means for a matrix $$A$$ . Linear algebra and ODEs fuse into one object — the matrix exponential — and its eigenstructure tells you everything about the long-term behavior, the geometry of the flow, and the physics of normal modes and beats.

Phase Space on Chen Kai Blog

Ordinary Differential Equations (7): Stability Theory

Ordinary Differential Equations (6): Linear Systems and the Matrix Exponential