Existence criterion of homoclinic trajectories in the Glukhovsky–Dolzhansky system

• New applications of Fishing principles are developed.
• General nonexistence criterion for homoclinic trajectories is proved.
• Existence of homoclinic trajectories in Glukhovsky–Dolzhansky system is shown.

Existence criterion of homoclinic trajectories in the Glukhovsky–Dolzhansky system, describing three-mode model of rotating fluid convection, is obtained. New applications of the Fishing principle are developed.