mathematics//dynamical systems//coupling
Two variables are **coupled** when the dynamics of at least one depends on the other. In \(\dot x_i=f_i(x,u,t)\) the first-order instantaneous sensitivity is
Two variables are coupled when the dynamics of at least one depends on the other. In x˙i=fi(x,u,t)\dot x_i=f_i(x,u,t)x˙i=fi(x,u,t) the first-order instantaneous sensitivity is
Jij(x)=∂fi∂xj.J_{ij}(x)=\frac{\partial f_i}{\partial x_j}.Jij(x)=∂xj∂fi.
If it is positive, a small increase of xjx_jxj raises x˙i\dot x_ix˙i with everything else fixed; if negative, it lowers it. That alone does not fix the final sign of x˙i\dot x_ix˙i: the other terms may dominate.
Direct does not mean causally proven. In x˙1=−ax1−kx2+u\dot x_1=-a x_1-kx_2+ux˙1=−ax1−kx2+u with k>0k>0k>0, x2x_2x2 exerts direct cross inhibition. The equation postulates that dependence; it does not prove a real causal mechanism exists. Observed correlation and modeled coupling are not the same thing either.
A zero of the Jacobian at a point is not always structural absence. If f1=x22f_1=x_2^2f1=x22, then ∂f1/∂x2=0\partial f_1/\partial x_2=0∂f1/∂x2=0 at x2=0x_2=0x2=0 although the dependence exists. The zero removes the first-order sensitivity there, not every possible effect.
An effect can be indirect. With
x˙2=ax1−bx2,x˙3=cx2−dx3,\dot x_2=ax_1-bx_2,\qquad \dot x_3=cx_2-dx_3,x˙2=ax1−bx2,x˙3=cx2−dx3,
x1x_1x1 does not appear in the equation of x3x_3x3 but can affect it through x2x_2x2. Perturb only x1x_1x1 by a small ε\varepsilonε: at first x˙2\dot x_2x˙2 changes, and for small times s>0s>0s>0 an approximate contribution δx3(s)≈acεs2/2\delta x_3(s)\approx ac\varepsilon s^2/2δx3(s)≈acεs2/2 appears. The propagation is gradual, with no mandatory dead time. "First one variable changes" does not require the next ones to stay exactly still for a finite interval; see delay and lag.
Reciprocity and loops. Coupling is unilateral if only one direction is present and reciprocal if both are. Two reciprocal inhibitions form a loop of positive sign, since the product of two negatives is positive. Its consequences depend on magnitudes and on the rest of the dynamics: that is the subject of feedback loop.
Physical example. In a drone, torque changes angular velocity, angular velocity changes tilt, and tilt changes horizontal acceleration. The effect is mediated by several states; it is not an independent horizontal force.