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Nonlinear systems and stability · Differential Equations | Math Infinitum

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Differential Equations
Nonlinear systems and stability

Differential Equations · Chapter 09

Nonlinear systems and stability

Systems whose rates multiply their unknowns together — pendulums, predators and prey, competing species. They rarely have formula solutions, but their equilibria, the Jacobian at each, and a conserved quantity where one exists give the whole phase portrait, with one case the linearisation cannot decide.

Systems whose rates multiply their unknowns together — pendulums, predators and prey, competing species. They rarely have formula solutions, but their equilibria, the Jacobian at each, and a conserved quantity where one exists give the whole phase portrait, with one case the linearisation cannot decide.

x^{'} = f (x)

J (x_{*})

f (x_{*}) = 0

x^{'} = f (x, y)

y^{'} = g (x, y)

det J

tr J

u^{'} = J u

Lessons

Build the foundation

01Nonlinear systemsMost systems that matter are nonlinear: predators and prey, competing species, a pendulum swinging through large angles, reacting chemicals. Their equations rarely have formula solutions, but the phase plane still works: at each point the right-hand sides give a velocity, trajectories follow it, and they still never cross. What changes is that a nonlinear system can have several equilibria, each with its own local picture, and trajectories that run from one to another.Read lesson →
02Equilibria and nullclinesAn equilibrium of a nonlinear system is a point where both rates vanish at once, and finding them all is algebra: solve the two equations together. The curves where each rate vanishes, the nullclines, cross exactly at the equilibria, and they also cut the plane into regions where the direction of motion is roughly known: right or left, up or down. Before anything is solved, the nullclines give a rough map of the flow.Read lesson →
03Linearisation and the JacobianNear an equilibrium a smooth system looks almost linear, just as a smooth curve looks almost straight close to a point. Replace the rates by their best linear approximations and what is left is a linear system in the small displacements from the equilibrium, whose matrix is the Jacobian: the partial derivatives of the rates. Everything the linear chapter learned about that matrix then describes the nonlinear system near that point — with one exception, the subject of the next lesson.Read lesson →
04Stability from the linearisationThe linearisation decides the local picture whenever it can. If every eigenvalue of the Jacobian has negative real part the equilibrium is asymptotically stable, and if one has positive real part it is unstable, whatever the neglected terms do; saddles stay saddles, nodes stay nodes and spirals stay spirals. The one case it cannot decide is the borderline: purely imaginary eigenvalues, a linear centre, which the smallest nonlinear term can turn into a spiral winding either way.

Read lesson →

05Drawing phase portraitsPut the pieces together and a nonlinear portrait can be drawn without solving anything. Find the equilibria, classify each through its Jacobian, draw the nullclines and the direction of motion between them, and connect: trajectories leave one equilibrium along its unstable directions and arrive at another. The curves that separate different long-run fates — the separatrices — run into the saddles along their stable directions.Read lesson →

06Predators and preyRabbits multiply when foxes are scarce, foxes thrive when rabbits are plentiful, and each population drags the other through cycles. The Lotka–Volterra equations make this precise with two products — the encounters between the species — and their phase portrait is a family of closed orbits around a coexistence equilibrium. The linearisation there is a centre, which decides nothing, but the system has a conserved quantity that closes every orbit.Read lesson →

07Competing speciesTwo species eating the same food hold each other back as well as themselves. Each alone would settle at its carrying capacity, as in the logistic model; together, the outcome depends on how strongly each crowds the other compared with how strongly it crowds itself. Weak competition lets them coexist at a stable equilibrium; strong competition makes coexistence a saddle, and one species drives the other out — which one depends on where they start.Read lesson →

08Chapter practiceBring the chapter together: find the equilibria of a nonlinear system, compute and read its Jacobian, decide stability where the linearisation can, recognise when it cannot, and interpret the predator–prey and competition models. No worked example sits above the answers.Read lesson →