nonlinearity

Nonlinearity refers to properties of systems, equations, or models in which the output is not proportional to the input, and in which the principle of superposition does not apply. In mathematics, a function or equation is nonlinear if it involves terms of degree greater than one or products of variables, or if it fails additivity or homogeneity. By contrast, linear systems satisfy additivity and homogeneity: the response to a sum of inputs is the sum of responses, and scaling an input scales the response by the same factor.

Common nonlinear elements include quadratic or cubic terms, products of variables, exponential or logarithmic relations, and saturation or threshold effects. Nonlinear behavior can arise from feedback, material properties, or complex interactions within a system. Although nonlinear functions can be simple, many nonlinear systems exhibit behaviors that are not possible under linear models.

Consequences of nonlinearity include multiple equilibria, bifurcations, limit cycles, and, in some cases, chaotic dynamics where small changes in initial conditions lead to large differences in outcomes. Because the superposition principle does not apply, many nonlinear problems resist closed-form solutions and require numerical simulation, local linearization, perturbation methods, or qualitative analysis.

Nonlinearity appears across disciplines: in physics and engineering (nonlinear circuits, nonlinear optics), fluid dynamics (turbulence), materials science (nonlinear constitutive relations), biology (population or reaction networks), economics, and climate science. Understanding a nonlinear system often involves selecting an appropriate modeling scale and employing tools such as phase portraits, bifurcation diagrams, and computational methods. It is important to note that a model can be nonlinear with respect to some variables while still being linear with respect to others.