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. When one variable changes first, the next ones are not required 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.
Coupling scales up to networks of agents. When each agent has its own dynamics and interacts only with some others, the whole system is written x˙i=f(xi)+∑jaij g(xi,xj)\dot x_i=f(x_i)+\sum_j a_{ij},g(x_i,x_j)x˙i=f(xi)+∑jaijg(xi,xj), where aija_{ij}aij is the adjacency matrix of who is coupled to whom (adjacency and degree) and ggg the rule of interaction: drones aligning with their neighbours, generators synchronizing on a grid, robots sharing aisles. The pattern of couplings, more than any single agent, then decides the collective behaviour (complex system).
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.