A system is nonlinear when a right-hand side contains a product of unknowns, a power of an unknown, or an unknown inside a function such as \(\sin(x)\) or \(e^{y}\text{.}\) How the independent variable appears never affects this.
Nonlinearity costs us three things: superposition, a single coefficient matrix, and closed-form solutions. The phase plane, direction fields, and Euler’s method all survive.
An equilibrium point \((x^*,y^*)\) needs both derivatives to vanish. Unlike a linear system, a nonlinear system can have several — factor each equation and test every pairing of cases rather than dividing by an unknown.
Nullclines are the curves where one derivative vanishes: motion is vertical across an \(x\)-nullcline and horizontal across a \(y\)-nullcline. Equilibria are the crossings of an \(x\)-nullcline with a \(y\)-nullcline, and one test point per region fixes the direction of the flow throughout that region.
evaluated at an equilibrium is the linearization there. Its eigenvalues classify the equilibrium by the same table used for linear systems — a separate matrix, and often a different type, at each equilibrium.
The linearization can be trusted whenever every eigenvalue has nonzero real part. A purely imaginary pair (or a zero eigenvalue) is inconclusive: the discarded nonlinear terms decide the outcome, so a linearized center may in fact spiral in or out.
Multiple equilibria of different types are what let nonlinear models capture behavior no linear model can — competitive exclusion, epidemic thresholds, and cycles among them.
