The order of coexistence of homoclinic trajectories for interval maps
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 \triangle...
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| Date: | 2019 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian |
| Published: |
Institute of Mathematics, NAS of Ukraine
2019
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/1493 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | 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. |
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