Invariant integral and the transition to steady states in separable dynamical systems

We show that the transition between fixed points in a separable dynamical s ystem is fully described by an invariant integral. We discuss in detail the case of a system with two temporal variables with bilinear coupling. where the new stable state is attained asymptotically through spiraling into the fixed point. Through the invariance, it is possible to establish condition s for the control parameter that permit a (targeted) transition in finite t ime and without relaxation oscillations.