1
Read a system as a vector field and step it exactly
Say what the arrow at a point is, apply Euler's step to both unknowns, and describe a solution as a curve in the plane rather than a graph against time.
Course · Differential Equations
Systems and the Phase Plane
A single equation gave a curve against time. A pair of equations gives a curve in a plane, and the plane has no time axis: time is the direction you travel along the curve. For a linear system x′ = a·x + b·y, y′ = c·x + d·y the whole picture follows from a two-by-two matrix. Its eigenvalues solve one quadratic, λ² − τ·λ + Δ = 0, whose coefficients are the trace and the determinant, so two exact numbers say whether solutions rush in, rush out, circle or do both. The labs keep that side exact: traces, determinants, eigenvalues, eigenvectors and fitted constants are fractions or surds. What is drawn rather than computed is the curve itself, which the legend says is stepped in floating point. The second half asks what survives when the system is not linear, and the honest answer is: near an equilibrium, most of it.
x′ = A·x: the arrow at a point is A·x
A·v = λ·v gives x = e^(λt)·v
λ² − τ·λ + Δ = 0 τ = trace, Δ = det
Δ < 0 saddle; Δ > 0 node, spiral, centre
stable exactly when Δ > 0 and τ < 0
nonlinear: linearise at an equilibrium
What this course is about
- Subject
- Differential Equations
- Lessons
- 10
- Format
- Interactive
Overview
What you will be able to do
By the end you can read a pair of equations as a field of arrows, find the directions along which a linear system moves in a straight line, write and fit its general solution, classify the origin from two numbers, and carry the classification to the equilibria of a nonlinear system while saying where it stops being valid.
2
Find eigenvalues and eigenvectors and write the straight-line solutions
Solve λ² − τ·λ + Δ = 0, find each direction A·v = λ·v, and write e^(λt)·v.
3
Fit the general solution to a starting point
Write C₁·e^(λ₁t)·v₁ + C₂·e^(λ₂t)·v₂, solve for the constants exactly, and say which term dominates as t grows.
4
Classify the origin and decide its stability
Use τ, Δ and τ² − 4Δ to name a saddle, node, spiral or centre, and state the criterion for asymptotic stability with its borderline cases.
5
Find and linearise the equilibria of a nonlinear system
Solve f = g = 0, compute the Jacobian exactly at each, classify it, and say when the linearisation is inconclusive.
6
Read a model in the phase plane
Locate the equilibria of predator and prey, competing species and an epidemic, classify them, and say what the picture does and does not prove.
Lessons
10 lessons
Linear systems first, because they can be solved completely and because everything after them borrows their language. The first five lessons turn a matrix into a picture, and a picture into a stability verdict. The second half uses that verdict on three models whose equations are not linear.
Systems of Two EquationsA pair of equations sends every point of a plane an arrow, and a solution is a curve that follows the arrows.Not marked
Straight-Line Solutions and EigenvectorsSome directions are only stretched by the matrix, and along each of them the system solves itself.Not marked
The General Solution of a Linear SystemAdd the two straight-line solutions with constants, fit the constants to a start, and see which term wins.Not marked
Saddles, Nodes, Spirals and CentresThe trace and determinant of the matrix sort every linear system in the plane into five pictures.Not marked
Stability of the OriginThe origin attracts every start exactly when the trace is negative and the determinant positive.Not marked
Nonlinear Systems and Their EquilibriaA nonlinear system can rest at several points, and each one is found by making both rates zero at once.Not marked
Linearisation and the JacobianNear an equilibrium a nonlinear system looks like a linear one whose matrix is the Jacobian, except when that matrix is a centre.Not marked
Predator and PreyFoxes and rabbits cycle round a point instead of settling to it, and a conserved quantity, not the Jacobian, shows why.Not marked
Competing SpeciesTwo species that share food have four equilibria, and the Jacobians say whether they coexist or one wins.Not marked
An Epidemic ModelAn outbreak grows while the susceptible fraction exceeds a threshold, peaks there, and ends with susceptibles left over.Not marked
Recommended background
The previous course, Oscillators, Damping and Resonance, for the phase ellipse and for what damping does to a motion, together with Second-Order Linear Equations for the characteristic equation, repeated and complex roots, and an equation rewritten as a system; Euler's steps as exact fractions from Differential Equations and Euler's Method; equilibria and their stability from Equilibria, Stability and Phase Lines; and Algebra's Systems and Matrices for the product of a matrix and a vector and for the determinant.
Practice suggestions
Work with the matrix, not the picture. The trace and the determinant are two exact numbers, and the type of the origin is decided by them before anything is drawn. Compute them by hand first, name the type, and then let the lab confirm it.
Keep the exact and the drawn apart. The tiles are exact fractions or surds. The curves behind them are stepped in floating point so that they can be seen, and the legend says so. Euler's polygon in particular is a recipe, and on some systems it drifts away from the true curve in a way you can compute.
Check a solution by substitution. A pair of formulas is a solution when both differentiate to the right combination of the pair, and the check is short and exact. It catches an eigenvector written the wrong way round, which a picture does not.
Treat a direction as a direction. An eigenvector stands for a whole line, and any non-zero multiple of it is the same answer. The lab prints the smallest whole-number version, with its first non-zero entry positive.
What it does not cover
Systems of three or more equations. The classification by trace and determinant is complete in the plane and nothing like it exists in three dimensions, so every system on the course has exactly two unknowns.
Limit cycles, the Poincaré–Bendixson theorem, Hopf bifurcations and chaos. The nonlinear lessons classify equilibria and draw trajectories; they do not prove that a closed orbit exists, and the predator–prey lesson states its conserved quantity and shows the drawing.
A proof that the linearisation predicts the nonlinear behaviour. The lessons state when it does (a saddle, a node or a spiral, where no eigenvalue has real part zero) and when it does not (a centre), and demonstrate the first case; the theorem that guarantees it belongs to a first course in dynamical systems.
Exact stepping of a nonlinear system. A quadratic right-hand side roughly doubles the digits of every denominator at each step, so the labs draw nonlinear trajectories by floating point and say so; no tile takes its value from a drawn curve.
Modelling as a skill. Predator and prey, competing species and the epidemic are each given with their equations and their assumptions named. Deriving a model from data, fitting its parameters and validating it are not here.