Nonlinearly

Nonlinearly is the adverb form of non-linear. It denotes performing an action or evolving in a way that is not proportional or not governed by a linear relationship. In science and mathematics, something that depends nonlinearly on a variable does not have output proportional to input, and linear superposition does not apply.

Nonlinear systems appear throughout science. In mathematics, nonlinear equations involve terms of higher order or products of variables, such as f(x)=x^2 or the logistic growth model p_{t+1}=p_t + r p_t(1-p_t). Nonlinear dynamics can exhibit complex behaviors, including multiple equilibria, limit cycles, and chaotic motion. Nonlinear partial differential equations, like the Navier-Stokes equations for fluid flow, describe phenomena where convective or interactive terms prevent simple linearization.

Modeling nonlinearity in data and systems often requires nonlinear regression, kernel methods, or machine learning approaches. When relationships are nonlinear, simple linear models may misestimate effects; nonlinear models can fit curved relationships but may also require more data and careful interpretation.

Nonlinearity is common in physics, biology, economics, ecology, engineering, and climate science. It underpins threshold effects, saturation, feedback loops, and emergent behavior that cannot be captured by additive linear approximations.

In practice, nonlinearly observed phenomena may require numerical simulation or qualitative analysis to understand system responses under different conditions.