ODE on Chen Kai Blog

Ordinary Differential Equations (9): Chaos Theory and the Lorenz System

Tue, 14 Nov 2023 09:00:00 +0000

In 1961, Edward Lorenz restarted a weather simulation from a rounded-off number — 0.506 instead of 0.506127. Within simulated weeks the forecast was unrecognisable. That single accident gave us the butterfly effect and turned chaos from a metaphor into a science. The lesson is profound and sober: equations that are exactly deterministic can still be practically unpredictable.

Ordinary Differential Equations (8): Nonlinear Systems and Phase Portraits

Sat, 28 Oct 2023 09:00:00 +0000

The real world is nonlinear. Predator-prey cycles, heartbeat rhythms, neuron firing — none of these can be captured by linear equations. When superposition fails, the world acquires new behaviors: limit cycles, multiple equilibria, bistability, hysteresis. This chapter gives you the geometric and analytic tools to read those behaviors directly off a 2D phase portrait.

What You Will Learn#

Why nonlinear systems are fundamentally different from linear ones
Lyapunov stability visualized: level sets, bowls, and basins
Linearization vs. the full nonlinear picture (Hartman-Grobman in action)
Lotka-Volterra predator-prey: closed orbits and conserved quantities
Competition models: four canonical outcomes
Van der Pol oscillator and the geometry of limit cycles
Gradient and Hamiltonian systems
Poincaré-Bendixson: why 2D systems cannot be chaotic

Prerequisites#

Chapter 6 : linear systems, phase portrait classification
Chapter 7 : stability, linearization, Lyapunov functions

From Linear to Nonlinear#

Linear systems obey superposition: if $\mathbf{x}_1$ and $\mathbf{x}_2$ are solutions, so is $c_1\mathbf{x}_1 + c_2\mathbf{x}_2$ . This is the engine that powers the entire toolkit of Chapters 1-6 — exponential ansatz, eigenvectors, fundamental matrices.

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.

Ordinary Differential Equations (5): Power Series and Special Functions

Thu, 07 Sep 2023 09:00:00 +0000

Some ODEs have no solutions in terms of familiar functions. The Bessel equation, the Legendre equation, the Airy equation — all arise naturally in physics (heat conduction in cylinders, gravitational fields of planets, quantum tunneling). Their solutions define entirely new functions. This chapter shows you how to find them using power series, why the Frobenius extension is forced upon us at singular points, and why the same handful of “special functions” keeps appearing across physics and engineering.

Ordinary Differential Equations (4): The Laplace Transform

Mon, 21 Aug 2023 09:00:00 +0000

The Laplace transform turns calculus into algebra. Instead of grinding through integration, guessing trial solutions, and bolting on initial conditions at the end, you transform the entire ODE — equation, forcing, and initial data — into a single polynomial equation in a complex variable $$s$$ . You solve it like a high-school problem, then transform back. Along the way, the shape of the solution becomes geometry: poles in the left half of the complex plane decay, poles on the right blow up, poles on the imaginary axis ring forever. This chapter develops that picture from first principles and connects it to the engineering tools — transfer functions, Bode plots, PID control — that made the Laplace transform the lingua franca of dynamics.

Ordinary Differential Equations (3): Higher-Order Linear Theory

Fri, 04 Aug 2023 09:00:00 +0000

A first-order ODE has memory of one number; a second-order ODE has memory of two. That tiny extra degree of freedom is what lets the same equation describe a plucked guitar string, the suspension of your car, the L-C tank circuit inside an FM radio, and the swaying of a tall building in the wind. In every case the same three regimes appear — oscillate, return-with-a-touch-of-overshoot, or crawl back — and the same algebraic gadget, the characteristic equation, predicts which one happens.

Ordinary Differential Equations (2): First-Order Methods

Tue, 18 Jul 2023 09:00:00 +0000

A bank account, a drug clearing the bloodstream, a tank of brine, a charging capacitor — they all obey the same kind of equation: a first-order ODE. The trick is recognising which of four shapes you are looking at, because each shape has a closed-form move that solves it cleanly. By the end of this chapter you will pattern-match an unfamiliar first-order equation in seconds and know exactly which lever to pull.

Ordinary Differential Equations (1): Origins and Intuition

Sat, 01 Jul 2023 09:00:00 +0000

Everything around you is changing. Coffee cools, populations grow, pendulums swing, viruses spread, stocks oscillate, planets orbit. None of these systems are described by what something equals — they are described by how fast something changes. That second mode of description is what differential equations are for, and learning to read them is, quite literally, learning to read the language physics and biology are written in.

This chapter rebuilds your intuition from scratch. We start with a single cup of coffee, derive the same equation that governs radioactive decay and capacitor discharge, then climb upward to direction fields, classification, and the existence-and-uniqueness theorem that tells you when an ODE has a sensible answer at all.