An internal combustion engine comprising: an in-cylinder fuel injector for injecting fuel to an inside of a combustion chamber; an exhaust purification catalyst which is arranged in an engine exhaust passage; a secondary air feed device for feeding air at an upstream side of the exhaust purification catalyst in the engine exhaust passage; a control device for controlling the in-cylinder fuel injector and the secondary air feed device; and an ignition device for igniting an air-fuel mixture of fuel and air in the combustion chamber; wherein the control device is formed to be able to perform first catalyst warm-up control and second catalyst warm-up control which promote a temperature rise of the exhaust purification catalyst, the first catalyst warm-up control includes control which injects fuel from the in-cylinder fuel injector in a compression stroke to form a high concentration region where a concentration of fuel at part of the combustion chamber rises and a low concentration region where the concentration of the fuel is lower than the high concentration region, and control which retards an ignition timing to make a temperature of an exhaust gas which flows out from the combustion chamber rise, the second catalyst warm-up control includes control which feeds air into the engine exhaust passage to make components which are contained in the exhaust gas oxidize to make the temperature of the exhaust gas rise, the control device is configured to perform the first catalyst warm-up control after start-up of the internal combustion engine in the state not performing the second catalyst warm-up control and perform control for simultaneously performing the first catalyst warm-up control and the second catalyst warm-up control after performing the first catalyst warm-up control, the control which performs the first catalyst warm-up control in the state not performing the second catalyst warm-up control is a control which forms a first stratified state where the air-fuel ratio of the combustion chamber as a whole becomes lean and where the air-fuel ratio of the high concentration region becomes rich, and the control for simultaneously performing the first catalyst warm-up control and the second catalyst warm-up control is a control which forms a second stratified state where the air-fuel ratio of the combustion chamber as a whole becomes rich at the time of ignition and where the stratification degree is weaker than the first stratified state. an in-cylinder fuel injector for injecting fuel to an inside of a combustion chamber; an exhaust purification catalyst which is arranged in an engine exhaust passage; a secondary air feed device for feeding air at an upstream side of the exhaust purification catalyst in the engine exhaust passage; a control device for controlling the in-cylinder fuel injector and the secondary air feed device; and an ignition device for igniting an air-fuel mixture of fuel and air in the combustion chamber; wherein the control device is formed to be able to perform first catalyst warm-up control and second catalyst warm-up control which promote a temperature rise of the exhaust purification catalyst, the first catalyst warm-up control includes control which injects fuel from the in-cylinder fuel injector in a compression stroke to form a high concentration region where a concentration of fuel at part of the combustion chamber rises and a low concentration region where the concentration of the fuel is lower than the high concentration region, and control which retards an ignition timing to make a temperature of an exhaust gas which flows out from the combustion chamber rise, the second catalyst warm-up control includes control which feeds air into the engine exhaust passage to make components which are contained in the exhaust gas oxidize to make the temperature of the exhaust gas rise, the control device is configured to perform the first catalyst warm-up control after start-up of the internal combustion engine in the state not performing the second catalyst warm-up control and perform control for simultaneously performing the first catalyst warm-up control and the second catalyst warm-up control after performing the first catalyst warm-up control, the control which performs the first catalyst warm-up control in the state not performing the second catalyst warm-up control is a control which forms a first stratified state where the air-fuel ratio of the combustion chamber as a whole becomes lean and where the air-fuel ratio of the high concentration region becomes rich, and the control for simultaneously performing the first catalyst warm-up control and the second catalyst warm-up control is a control which forms a second stratified state where the air-fuel ratio of the combustion chamber as a whole becomes rich at the time of ignition and where the stratification degree is weaker than the first stratified state.