The order of coexistence of homoclinic trajectories for interval maps

  • M. V. Kuznietsov


UDC 517.9
А nonperiodic trajectory of a discrete dynamical system is called $n$-homoclinic if its $\alpha$- and $\omega$-limit sets coincide and form the same cycle of period $n.$ We prove the statement formulated in that the ordering $1 \triangleright 3 \triangleright 5 \triangleright 7 \triangleright \ldots \triangleright 2 \cdot 1 \triangleright 2 \cdot 3\triangleright 2 \cdot 5 \triangleright \ldots \triangleright 2^2 \cdot 1 \triangleright 2^2 \cdot 3 \triangleright 2^2 \cdot 5 \triangleright \ldots $ 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 \triangleright m .$ It is also proved that every one-dimensional dynamical system with a cycle of period $ n \neq 2^i $ also possesses an $n$-homoclinic trajectory.
How to Cite
Kuznietsov, M. V. “The Order of Coexistence of Homoclinic Trajectories for Interval Maps”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, no. 7, July 2019, pp. 1003-8,
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