Equilibria and linearisation
Nonlinear systems rarely have formulas for their solutions, but near each equilibrium they behave almost like a linear system. Linearisation reveals that local behaviour.
Nonlinear systems
A nonlinear system \(x' = f(x, y)\), \(y' = g(x, y)\) can have several equilibria, limit cycles, and even chaos. Superposition fails, so we study behaviour qualitatively: where the equilibria are, what nearby solutions do, and how trajectories connect.
Finding equilibria
Equilibria satisfy \(f = 0\) and \(g = 0\) simultaneously. The nullclines \(f = 0\) (where \(x' = 0\)) and \(g = 0\) (where \(y' = 0\)) are curves whose intersections are the equilibria. Sketching nullclines and the flow directions between them gives a rough phase portrait quickly.
The Jacobian matrix
Near an equilibrium \((x^*, y^*)\), approximate the system by its linearisation \(\mathbf{u}' = J\mathbf{u}\), where \(J\) is the Jacobian matrix of partial derivatives evaluated at the equilibrium and \(\mathbf{u}\) is the displacement from it.
J = \begin{pmatrix} f_x & f_y \\ g_x & g_y \end{pmatrix}The linearisation theorem
If the Jacobian's eigenvalues all have nonzero real parts, the nonlinear system looks like its linearisation near the equilibrium: nodes, saddles, and spirals carry over. If the linearisation predicts a centre (purely imaginary eigenvalues), the nonlinear terms can change it into a spiral, so further analysis is needed.
Worked example: competing species
\(x' = x(3 - x - 2y)\), \(y' = y(2 - x - y)\). Equilibria: \((0, 0)\), \((3, 0)\), \((0, 2)\), and \((1, 1)\). At \((1, 1)\), \(J = \begin{pmatrix} -1 & -2 \\ -1 & -1 \end{pmatrix}\), with determinant \(-1 < 0\): a saddle. Coexistence is unstable, so one species eventually wins depending on the starting point.
Worked example: the pendulum
The pendulum \(\theta'' + \sin\theta = 0\) becomes \(x' = y\), \(y' = -\sin x\). At \((0, 0)\) (hanging down), \(J = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\): a centre, and energy conservation confirms closed orbits. At \((\pi, 0)\) (balanced upright), \(J = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\): a saddle, unstable as expected.
Basins of attraction
When there are several stable equilibria, each attracts a region of starting points, its basin of attraction. Saddles and their special trajectories (separatrices) form the boundaries between basins. In ecology, basins describe which initial populations lead to which outcomes.
Common mistakes
Evaluating the Jacobian at the wrong point. Trusting a linear centre in a nonlinear system. Missing equilibria by solving \(f = 0\) and \(g = 0\) incompletely, for example forgetting the factor \(x = 0\).
Key takeaways
Equilibria lie where nullclines intersect. The Jacobian at an equilibrium gives a linearisation whose eigenvalues classify nearby behaviour, except in borderline cases like centres. Saddles and separatrices divide the phase plane into basins of attraction.
Flashcard
Curve where x' = 0 or y' = 0
Check your understanding
For the system x' = y, y' = -sin x, the equilibrium (pi, 0) is: