A method is disclosed for controlling the air-fuel ratio in a four-stroke gasoline direct-injection controlled auto-ignition combustion. The engine is operated with two sequential pairs of expansion and contraction strokes during two revolutions of the engine crank, the two revolutions defining a combustion cycle. A system is employed for variably actuating the intake and exhaust valves and adjusting the flow of air and burned gases entering the combustion chamber. Adjusting the flow affects the resulting air-fuel ratio in the combustion chamber. The valve actuating system is employable to operate the intake and exhaust valves with an exhaust re-compression or an exhaust re-breathing valve strategy. Either valve strategy affects the air-fuel ratio in the cylinder and causes a proportion of burned gases from previous combustion cycles to be retained in the cylinder to provide the necessary conditions for auto-ignition of the air-fuel mixture.