mathematics//differential equations//external forcing

An external forcing is a term in a rate equation that depends on time, or on an input chosen from outside, but not on the state: the world pushing on the system on a schedule of its own.

An external forcing is a term in a rate equation that depends on time, or on an input chosen from outside, but not on the state: the world pushing on the system on a schedule of its own.

x˙=f(x)+g(t).\dot x=f(x)+g(t).x˙=f(x)+g(t).

Here g(t)g(t)g(t) is the forcing: it enters the rate whatever the state xxx is.

A seasonal temperature acting on a tank, a pump that switches on at fixed hours and the sin⁡t\sin tsint in x˙=−x+sin⁡t\dot x=-x+\sin tx˙=−x+sint are all forcings. The forcing makes the system non-autonomous, since the rule now contains ttt explicitly (time, autonomous system).

A forcing differs from feedback in what it depends on. A feedback term depends on the state and returns to it; a forcing term does not care what the state is. A linear system's response to a forcing is a convolution with its impulse response, the memory kernel of delay and lag: each push leaves a decaying trace.

The same term is called an input when it is chosen rather than suffered, and feedback control is about choosing it as a function of the state, which turns a forcing back into feedback. In control, a feedforward action is an external forcing designed to cancel a known disturbance before feedback has to react (feedback control). A periodic forcing can also produce a periodic response that is neither an equilibrium nor a limit cycle (equilibrium and stability).

equationẋ = −x + 1.0 sin(1.0 t) startx(0) = 1.50 One equation, a family of solutions: every curve follows the slopes of the field, and the starting point picks which curve is the solution.

Move the start of x˙=−x+Asin⁡ωt\dot x=-x+A\sin\omega tx˙=−x+Asinωt anywhere: after a few time units every curve joins the same swing, and a faster ω\omegaω makes that swing smaller.