Linearising

Linearising is the process of approximating a nonlinear function, equation, or system by a linear one in a neighborhood of a reference point. It is usually done by taking the first-order terms of a Taylor expansion around that point.

For a function f of one variable, the linear approximation at a is L(x) = f(a) + f′(a)(x − a). In multiple dimensions, L(x) = f(a) + ∇f(a)·(x − a). This linear part serves as the best first-order description of the function near the reference point.

In dynamical systems and differential equations, linearising replaces a nonlinear model with its linear part near an equilibrium, obtained from the Jacobian matrix J = ∂f/∂x evaluated at the equilibrium. Stability is inferred from the eigenvalues of J. In control engineering and related fields, linearised (small‑signal) models enable the use of linear techniques for analysis and design, including frequency-domain methods and linear state feedback.

In data analysis, linearising a nonlinear relationship can simplify modeling, for example by applying a logarithm to linearise exponential growth or by other transformations that render a plot approximately linear. However, linear fits are local approximations and can be misleading when applied far from the reference point or when strong nonlinear effects dominate.

Limitations include the local nature of the approximation: linearised models are accurate only near the expansion point and may fail to capture nonlinear phenomena, bifurcations, or large deviations. When necessary, higher-order expansions, piecewise linearisation, or global nonlinear methods may be used to obtain a more faithful representation.