You're looking for the theory of linear (or nonlinear) dynamical systems. Unfortunately it's not one kernel of intuition backed by consistent notation, it's man

You're looking for the theory of linear (or nonlinear) dynamical systems. Unfortunately it's not one kernel of intuition backed by consistent notation, it's many with no consistency. A good course on controls and signals/systems will beat those intuitions into you and you learn the math/parlance without getting attached to any one notational convention. The real intuition is "everything is a filter." Everything else is about analysis and synthesis of that idea.