Nonlineaire

Nonlinear (nonlineaire in French) describes relationships, equations, or systems in which the output is not proportional to the input; the superposition principle does not apply. In mathematics, a model is nonlinear if it involves products, powers, or other non-additive functions of the state variables or their derivatives, or if it contains feedback loops. Nonlinear differential equations and difference equations can exhibit behaviors not possible in linear systems, such as multiple equilibria, limit cycles, bifurcations, and chaotic dynamics.

Common examples include the logistic growth model, the pendulum at large amplitudes, reaction-diffusion systems, and the Navier–Stokes equations in fluid dynamics. Nonlinear material laws, such as nonlinear elasticity, and systems with nonlinear control or interactions, also display nonlinearity.

Because nonlinear systems can be highly sensitive to initial conditions and parameters, exact solutions are rare. Analysts rely on numerical simulation, qualitative methods, perturbation theory for weak nonlinearity, or linearization around an operating point to obtain insight.

Historically, the study of nonlinearity predates chaos theory, but the latter brought attention to the rich dynamics of nonlinear systems, including the work of Poincaré and, in the 20th century, Lorenz, Dahlquist, and others.

Nonlinearity is central to many disciplines, including physics, engineering, chemistry, biology, economics, and climate science, where nonlinear models capture feedback, saturation, thresholds, and emergent behavior. Related topics include nonlinear dynamics, chaos theory, and nonlinear optimization.