mathematics//dynamical systems//linear field

In a linear system, \(\dot x=Fx\), the whole field is **one global coherent pattern** built by a single matrix: a stretching, a contraction, a rotation, a shear, or a combination of them, applied identically everywhere. Move to a point twice as far from the origin and the arrow there is exactly twice as long in the same direction: \(f(2x)=2f(x)\), \(f(x+y)=f(x)+f(y)\).

In a linear system, x˙=Fx\dot x=Fxx˙=Fx, the whole field is one global coherent pattern built by a single matrix: a stretching, a contraction, a rotation, a shear, or a combination of them, applied identically everywhere. Move to a point twice as far from the origin and the arrow there is exactly twice as long in the same direction: f(2x)=2f(x)f(2x)=2f(x)f(2x)=2f(x), f(x+y)=f(x)+f(y)f(x+y)=f(x)+f(y)f(x+y)=f(x)+f(y).

Nonlinearity breaks that proportion. With f(x)=x3f(x)=x^3f(x)=x3, f(2x)=8x3f(2x)=8x^3f(2x)=8x3, not 2f(x)2f(x)2f(x); with x2x^2x2 the sign stops flipping with xxx. The arrow at a proportionally displaced point is no longer proportional, and the field deforms depending on where you are. Nothing needs to be cut, cornered or discontinuous: the field can be perfectly smooth. What is lost is a single geometric rule valid everywhere; the effective rule now depends on the region of the state space.

The Jacobian makes this exact. In a linear field it is the same matrix FFF at every point: constant Jacobian is what "linear transformation" means. In a nonlinear field the Jacobian is a function of position, J(x)J(x)J(x), and it changes as you move across the state space, not only at equilibria. Computing it at any point recovers the local linear pattern, the stretch-rotate-shear that best matches the field there, which is why linearizing at a point allows the local behavior to be studied with the linear toolkit (modes).

The product terms of population models are the everyday source of the deformation: doubling both species quadruples the encounter term. An equilibrium is where the field vanishes, and the Jacobian there decides what the local pattern is, but the field keeps its own varying Jacobian everywhere else (phase portrait).