mathematics//differential equations//time

In a differential equation, time is normally the **independent variable that parametrizes the trajectory**. The state \(x(t)\) is a point moving through the state space; \(t\) is the clock that says where along the path the state is.

In a differential equation, time is normally the independent variable that parametrizes the trajectory. The state x(t)x(t)x(t) is a point moving through the state space; ttt is the clock that says where along the path the state is.

Every trajectory is parametrized by ttt, in autonomous systems too. The difference is whether ttt appears in the rule that generates the field. In an autonomous system, x˙=f(x)\dot x=f(x)x˙=f(x), it does not: the clock tells where the state is, it does not modify the rules of the terrain. In a non-autonomous system it does:

x˙=−x+sin⁡t.\dot x=-x+\sin t.x˙=−x+sint.

Here the law that determines x˙\dot xx˙ changes with time. The same state at a different instant has a different velocity.

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.

Drag the start and compare the copper curve, the same start two units later, with the dashed copy of the green one: here they part, while in the logistic equation, which has no ttt in its rule, they lie on top of each other.

That changes the picture. With three state variables the field is a sea of arrows in a three-dimensional space and time is what moves the state through it. If the field is non-autonomous, time is effectively a fourth dimension, and the arrows themselves change as it passes (phase portrait).

Time is not the same kind of dimension as a fourth state variable. If a field on three coordinates is extended with a fourth coordinate, moving along it changes the arrows too, but reversibly: moving forward changes them one way, moving back changes them back. Time is not something the state moves along; it passes, the arrows change with it, and there is no going back.

In a non-autonomous system the state without the clock is no longer a sufficient state: knowing it is not enough to continue, so time has to be carried as part of the state or the memory. The standard trick hides the difference by appending ttt to the state with t˙=1\dot t=1t˙=1. The system becomes autonomous in one more dimension, and the non-autonomous example above is an autonomous system in (x,t)(x,t)(x,t). An external forcing is the usual way ttt enters a rule.