Linearizare

Linearizare (linearization) is the process of approximating a nonlinear model by a linear one in a small neighborhood of a reference point. It is widely used across mathematics, physics, engineering, and economics to simplify analysis and design. The idea is to retain the first-order terms of a nonlinear relationship while neglecting higher-order contributions.

In one-variable calculus, the linear approximation to a differentiable function f at a point x0 is f(x) ≈ f(x0) + f′(x0)(x − x0). In several variables, the linearization uses the Jacobian matrix: f(x) ≈ f(x0) + Jf(x0)(x − x0). This yields a first-order model that matches the function’s value and tangent behavior at the reference point.

In dynamical systems and differential equations, linearization about an equilibrium x* yields a system dx/dt ≈ DF(x*) (x − x*). The eigenvalues of the Jacobian DF(x*) determine local stability. The Hartman–Grobman theorem states that near hyperbolic fixed points, the nonlinear flow is qualitatively similar to the linearized system.

In control theory and signal processing, linearization provides a tractable model around an operating point to design controllers or observers. For nonlinear controllers, linearization facilitates analysis and implementation by reducing complexity.

Limitations include restricted validity: linearized models accurately describe behavior only close to the reference point. They may misrepresent dynamics far from that point, near bifurcations, or in strongly nonlinear regimes; higher-order methods or piecewise-linear models may be required. See also Taylor expansion, Jacobian, Newton’s method, and stability analysis.