nonlinearities

Nonlinearities refer to properties or behaviors of systems in which the output is not proportional to the input, and where the principle of superposition does not hold. In such systems, combining inputs does not produce a simple sum of outputs, and small changes can have large or disproportionate effects.

Mathematically, nonlinearities arise from nonlinear equations or operators. A relation is nonlinear if it cannot be expressed as a linear combination of the inputs. Common examples include quadratic or higher-degree terms, products of variables, or functions with curvature such as sin(x).

In physics and engineering, nonlinearities appear in many contexts: nonlinear optics where the response of a material depends on the intensity of light leading to harmonic generation and self-phase modulation; nonlinear acoustics; nonlinear elasticity; and in electrical circuits with diodes or transistors, where the current–voltage characteristic is not linear.

Nonlinear systems in control and signal processing cannot be analyzed with linear time-invariant tools; they may exhibit complexity such as bifurcations, chaotic behavior, and harmonics or intermodulation effects. Analysis methods include perturbation techniques for weak nonlinearities, numerical integration of differential equations, and nonlinear optimization.

Nonlinearities also occur in natural and social systems, including population dynamics, climate models, and economic interactions, where responses depend on the state of the system and can be highly context-dependent.