Nonlinear

Nonlinear describes systems, equations, or relationships in which output is not proportional to input and where the principle of superposition does not apply. In mathematics, a model is nonlinear if it cannot be written as a linear combination of its variables and a constant. Linear models satisfy additivity and homogeneity; nonlinear models do not, and small changes can produce disproportionately large or qualitatively different effects.

Common nonlinear expressions include y = x^2, y = sin(x), and products such as u*v, as well as nonlinear differential equations where the governing terms depend on the state in a non-proportional way. Many natural and engineered systems exhibit nonlinearity, and the behavior of nonlinear systems often cannot be inferred from linear intuition alone.

Nonlinear dynamics is the study of systems that can exhibit complex behavior such as bifurcations, chaos, and strange attractors. Nonlinear equations arise across disciplines, including fluid dynamics, nonlinear optics, population biology, structural mechanics, and economics. Phenomena like saturation, threshold effects, hysteresis, and pattern formation are frequently tied to nonlinearities.

Analysis and solution methods for nonlinear models differ from linear ones and often rely on numerical simulation, approximate techniques, or qualitative analysis. Numerical integrators (such as Runge-Kutta methods), iterative solvers for nonlinear algebraic equations, and perturbation theory for weak nonlinearities are common tools. Linearization and local stability analysis provide partial insights near operating points or equilibria, but fully capturing global behavior typically requires specialized nonlinear techniques.

In practice, linear models serve as useful approximations, while nonlinearities are essential for accurately describing many real-world systems and their rich, sometimes unpredictable, dynamics.