Nonlinearized

Nonlinearized is an adjective used to describe a model, equation, or system that retains nonlinear terms rather than being reduced to a linear approximation. In many branches of science, linearization replaces a nonlinear relationship with its first-order Taylor expansion around a reference point. A nonlinearized model instead includes higher-order terms or nonlinear functional forms, enabling it to describe saturation, thresholds, feedback, and other effects that linear models miss.

In dynamical systems, nonlinearized equations are typically ordinary or partial differential equations that do not admit simple closed-form solutions. They can exhibit a range of behaviors from stable equilibria to complex dynamics such as bifurcations and chaos, depending on parameters and initial conditions.

Analytical and numerical methods are used to study nonlinearized systems. Perturbation theory around a known solution, numerical integration, and simulation are common approaches. Researchers may apply nonlinear optimization, sensitivity analysis, and, in control theory, techniques such as feedback linearization or backstepping to manage nonlinear effects. Higher-order expansions, Volterra series, or spline-based nonlinear approximations are also used.

Applications of nonlinearized models span engineering (aerodynamics, structural dynamics), physics (fluid flow, nonlinear optics), biology (population models, enzyme kinetics), economics, and data science. The term is informal in many disciplines and contrasts with linearized or linear models; its use signals an emphasis on capturing nonlinear behavior rather than simplifying to linearity.