General existence conditions of homoclinic trajectories in dissipative systems. Lorenz, Shimizu-Morioka, Lu and Chen systems

► General method for proof of existence of homoclinic trajectories is developed. ► This method applied to Lorenz, Shimizu-Morioka, Lu and Chen systems. ► New existence conditions of homoclinic trajectories for these systems are obtained. ► New numerical algorithm of homoclinic point approximation is given.