Behavior of solutions of a second order rational difference equation
In this paper, we solve the difference equation xn+1 = α xnxn-1 - 1 , n = 0, 1, . . . , where α > 0 and the initial values x-1, x0 are real numbers. We find some invariant sets and discuss the global behavior of the solutions of that equation. We show that when α > 2 3 √ 3 , under certain cond...
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| Format: | Article |
| Language: | English |
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University of Kragujevac, Faculty of Technical Sciences Čačak, Serbia
2019-01-01
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| Series: | Mathematica Moravica |
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| Online Access: | https://scindeks-clanci.ceon.rs/data/pdf/1450-5932/2019/1450-59321901011A.pdf |
| Summary: | In this paper, we solve the difference equation xn+1 = α xnxn-1 - 1 , n = 0, 1, . . . , where α > 0 and the initial values x-1, x0 are real numbers. We find some invariant sets and discuss the global behavior of the solutions of that equation. We show that when α > 2 3 √ 3 , under certain conditions there exist solutions, that are either periodic or converging to periodic solutions. We show also the existence of dense solutions in the real line. Finally, we show that when α < 2 3 √ 3 , one of the negative equilibrium points attracts all orbits with initials outside a set of Lebesgue measure zero. |
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| ISSN: | 1450-5932 2560-5542 |