nonlineært

Nonlineært describes systems and relationships in which the output is not proportional to the input and the principle of superposition does not apply. In nonlinear systems, combining inputs does not yield a simple sum of outputs, and small changes can lead to disproportionately large effects, or the opposite. The term is used across disciplines such as mathematics, physics, engineering, biology, and economics. In Danish, the corresponding adjective is nonlineært.

Mathematically, nonlinearity arises when governing equations involve nonlinear terms in the state or its derivatives, or when coefficients depend on the state itself. Linear systems are governed by linear differential equations or linear algebra, where solutions obey superposition. Nonlinear systems often involve nonlinear differential equations (ODEs or PDEs) and can exhibit complex behavior that resists closed-form solutions.

Common nonlinear phenomena include multiple equilibria, limit cycles, bifurcations, and chaos. Nonlinear dynamics can show sensitive dependence on initial conditions, where nearby states diverge over time, producing unpredictable long-term behavior in some models.

Analysis and modeling rely on qualitative and numerical methods. Dynamical systems theory, phase-space analysis, fixed-point stability, and, when possible, linearization near equilibria are used to gain insight. Perturbation techniques address weak nonlinearity, while numerical simulations (e.g., Runge-Kutta methods) are common for more complex systems.

Applications span many fields: nonlinear optics, fluid dynamics, chemical kinetics, population biology, climate and ecological models, economics, and nonlinear control systems. Studying nonlinearity helps explain phenomena such as turbulence, pattern formation, and complex rhythmic patterns that linear models cannot capture.