mathematics//dynamical systems//linearization

How to replace a nonlinear dynamics locally by a matrix without confusing the two. A linear map respects **superposition**: transforming a linear combination equals combining the transformations. \(f(x)=Fx\) is linear; \(Fx+b\) is affine if \(b\ne0\). A curved trajectory can belong to a linear system: "linear" does not mean "straight in time".

How to replace a nonlinear dynamics locally by a matrix without confusing the two. A linear map respects superposition: transforming a linear combination equals combining the transformations. f(x)=Fxf(x)=Fxf(x)=Fx is linear; Fx+bFx+bFx+b is affine if b≠0b\ne0b=0. A curved trajectory can belong to a linear system: "linear" does not mean "straight in time".

The Jacobian collects sensitivities,

Jij(x)=∂fi∂xj.J_{ij}(x)=\frac{\partial f_i}{\partial x_j}.Jij​(x)=∂xj​∂fi​​.

Diagonal entries are first-order self-effects; the others are couplings. The units of JijJ_{ij}Jij​ are [xi]/([xj][t])[x_i]/([x_j][t])[xi​]/([xj​][t]), not necessarily the same across entries when the states have different units.

Linearizing around an equilibrium. If f(x∗,u∗)=0f(x^,u^)=0f(x∗,u∗)=0, define δx=x−x∗\delta x=x-x^δx=x−x∗ and δu=u−u∗\delta u=u-u^δu=u−u∗. First order gives

δx˙≈Fδx+Bδu,F=∂f∂x∣∗,B=∂f∂u∣∗.\delta\dot x\approx F\delta x+B\delta u,\qquad F=\left.\frac{\partial f}{\partial x}\right|_*,\quad B=\left.\frac{\partial f}{\partial u}\right|_*.δx˙≈Fδx+Bδu,F=∂x∂f​​∗​,B=∂u∂f​​∗​.

If the point is not an equilibrium, f(x∗,u∗)f(x^,u^)f(x∗,u∗) appears as well. Around a nominal trajectory that satisfies the model, its evolution is subtracted and the matrices can depend on time. The matrix adds no forces: it reorganizes the sensitivity of the mechanisms already modeled.

What eigenvalues allow you to conclude. For an equilibrium and a sufficiently regular field, if every eigenvalue of FFF has negative real part there is local asymptotic stability; if one has positive real part, instability. With no positive real parts but some zero real parts the test can be inconclusive: x˙=−x3\dot x=-x^3x˙=−x3 and x˙=x3\dot x=x^3x˙=x3 both have a zero Jacobian at the origin, one attracts and the other repels. The discarded term decides. See equilibrium and stability.

From component to collective pattern. The Jacobian shows how each rate changes when each state changes. Its modes show which combinations evolve together. Two readings of the same approximation.

The Jacobian is a function, not a number to compute once. In a linear field it is the same matrix everywhere, which is what makes the field one global pattern; in a nonlinear field it changes with the position, and it can be taken at any point, not only at equilibria. Linearizing at a point recovers the local linear pattern and lets the linear toolkit be applied there (linear field).

In a drone, sin⁡θ≈θ\sin\theta\approx\thetasinθ≈θ near level flight. That approximation can fail at large tilt or under saturation. A local result does not become global by writing the matrix precisely.