The order of coexistence of homoclinic trajectories for interval maps

  • M. V. Kuznietsov

Abstract

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.
Published
25.07.2019
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, http://umj.imath.kiev.ua/index.php/umj/article/view/1493.
Section
Short communications