The order of coexistence of homoclinic trajectories for interval maps - Kuznietsov, M. V. (Ukrains’kyi Matematychnyi Zhurnal, 2019)
UDC 517.9 А nonperiodic trajectory of a discrete dynamical system is called n-homoclinic if its α- and ω-limit sets coincide and form the same cycle of period n. We prove the statement formulated in that the ordering 1 3 5 7 … 2 · 1 2 · 32 · 5 … 2^2 · 1 2^2 · 3 2^2 · 5 … determines the coexistence of homoclinic trajectories of one-dimensional systems: If a one-dimensional dynamical system possesses an n-homoclinic trajectory, then it also has an m-homoclinic trajectory for each m such that n m . It is also proved that every one-dimensional dynamical system with a cycle of period n eq 2^i also possesses an n-homoclinic trajectory.