10 Non-linear Systems and the Phase Plane
Most systems of interest are non-linear, and most non-linear systems cannot be solved. The geometric approach, introduced by Poincaré in the 1880s, asks different questions: where are the equilibria, which of them are stable, which states end up where, and are there periodic motions? The answers come from the phase portrait, the picture of all orbits in the state space, and the main tool is linearisation: near an equilibrium a smooth system behaves like its linear part, which the previous sections have classified completely. This section develops that tool and applies it to models of competing and interacting populations.
Throughout, \(x' = f(x)\) is an autonomous system on an open set \(U \subseteq \mathbb R^n\) with \(f\) continuously differentiable, so that the flow \(\phi _t\) of Theorem 3.5.1 is defined.
10.2 Linearisation and stability
10.3 Hartman–Grobman and stable manifolds
10.4 Competing species
10.5 Predator and prey
10.6 The phase plane in software
10.7 Practice Problems
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.