an autonomous planar system
An autonomous planar system is the kind of system the whole phase-plane theory is built for: two quantities changing in time, x and y, whose rates of change depend only on the present values of x and y — never directly on the clock. 'Planar' because there are two variables (it lives in a plane); 'autonomous' because the rules are self-governing, the same today as tomorrow. Most physical laws are like this: the force on a pendulum depends on its current angle, not on what year it is.
In symbols it is a pair x' = f(x, y), y' = g(x, y), where the right sides hold no explicit t. This single feature has a powerful geometric consequence: the velocity arrow at each point of the plane is fixed once and for all, so the state has a unique direction to follow at every location, and — as a result — trajectories can never cross. The picture is frozen; only the moving point changes. A second-order autonomous equation like x'' = h(x, x') becomes one of these by setting y = x', giving x' = y, y' = h(x, y).
This matters because the constancy of the field is exactly what makes phase portraits, equilibrium classification, and the no-crossing rule work. Drop autonomy — let f or g depend on t — and the arrows themselves shift with time, trajectories can apparently intersect, and the clean planar geometry breaks down. The phase-plane method is, at bottom, the geometry of autonomous two-variable systems.
x' = x(1 - x - y), y' = y(0.75 - y - 0.5x) is autonomous and planar: two species whose growth rates depend on the current populations, not on the date. By contrast x' = y, y' = -x + cos(t) is planar but NOT autonomous, because t appears.
No explicit t on the right-hand sides is the test for autonomy.
Autonomy is about t appearing EXPLICITLY, not about being constant. f(x, y) certainly changes as x and y move; what matters is that it carries no separate dependence on time. A forced system with cos(t) on the right is non-autonomous and falls outside the basic planar theory.