A new engine cycle, called the Bajulaz cycle, is introduced. The cycle has the unique characteristic that it utilizes two chambers, an air-heating chamber and combustion chamber. By allowing heat transfer from the combustion chamber to the air-heating chamber, the heat of combustion is used more efficiently leading to a higher thermal efficiency.Utilizing an ideal thermodynamic cycle analysis, the thermal efficiency of the Bajulaz cycle is shown to depend only on the compression ratios based on the two chamber volumes and the fractional heat transfer between the two chambers. Based on predictions of peak temperature and pressure generated from the ideal cycle analysis, the Bajulaz engine is shown to be thermally more efficient and has lower NOx emission than the corresponding Otto engine. Combustion is shown to occur over a full 360° of crankshaft rotation in the Bajulaz engine. The Bajulaz engine thus does not have any restrictive requirement on ignition timing (as in an Otto engine) and on the timing and duration of fuel injection (as in a Diesel engine). It can also utilize multiple fuels including those with slow burning rate.