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 you are.

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 you are.

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 you where you are, 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. Same state, different instant, different velocity.

What that does to the picture. With three state variables the field is a sea of arrows in a three-dimensional space and time is what moves you 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: crawl forward and they change one way, crawl back and they change back. Time is not something you move along; it passes, the arrows change with it, and you cannot go back.

The standard trick to hide the difference: append 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). This is also why a sufficient state of a non-autonomous system has to include the clock. An external forcing is the usual way ttt enters a rule.