mathematics//differential equations//external forcing

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.

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).

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

Forcing versus feedback. 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. 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).