Caro, A.; Linero, A. General cycles of potential form. (English) Zbl 1202.39002 Int. J. Bifurcation Chaos Appl. Sci. Eng. 20, No. 9, 2735-2749 (2010). Summary: We consider a general difference equation of order \(k\) of the form \[ x_{n+k} = F (x_{n+k-1}, \cdots, x_{n+2}, x_{n+1}) f_0 (x_n) \] with \(f_{0} : (0, \infty ) \rightarrow (0, \infty ), F : (0, \infty )^{k-1} \rightarrow (0, \infty )\) continuous maps and initial conditions \(x_{i} \in (0, \infty ), i = 0, 1, \cdots , k-1\). Under the hypothesis that \(f_{x}(y) := F(x, x, \cdots , x)f_{0}(y)\) is an involution, we provide all the \((k + 1)\)-cycles of the above-mentioned type, and we prove that if \(F\) separates variables, then all the \((k + 2)\)-cycles have a potential form \[ x_{n+k} = cx^{\alpha_{k-1}}_{n+k-1} \cdots x^{\alpha_1}_{n+1} x^{\alpha_0}_n, \quad n= 0,1, \dots, \] where \(c > 0, \alpha _{i} \in \mathbb R, x_{i} \in (0, \infty ), i = 0, 1, \cdots , k-1\). For this reason, we also describe all the \(p\)-cycles of order \(k \geq 2\) having the above potential form. Moreover, taking the particular case \(k = 3\) and \(p = 8\) disproves Conjecture 2.1 by E. A. Grove and G. Ladas [Advances in Discrete Mathematics and Applications 4. Boca Raton, FL: Chapman & Hall/CRC. xiii, 379 p. $ 99.00; £ 56.99 (2005; Zbl 1078.39009)]. Cited in 1 Document MSC: 39A10 Additive difference equations 39A23 Periodic solutions of difference equations Keywords:difference equations; periodic solution; \(p\)-cycle; potential cycles; topological conjugation; linear difference equations; functional equations Citations:Zbl 1078.39009 PDFBibTeX XMLCite \textit{A. Caro} and \textit{A. Linero}, Int. J. Bifurcation Chaos Appl. Sci. Eng. 20, No. 9, 2735--2749 (2010; Zbl 1202.39002) Full Text: DOI References: [1] DOI: 10.1016/S0022-247X(03)00272-5 · Zbl 1031.39012 · doi:10.1016/S0022-247X(03)00272-5 [2] DOI: 10.1017/CBO9781139086578 · doi:10.1017/CBO9781139086578 [3] DOI: 10.1080/10236190701388518 · Zbl 1130.39003 · doi:10.1080/10236190701388518 [4] DOI: 10.1080/10236190701351144 · Zbl 1127.39004 · doi:10.1080/10236190701351144 [5] DOI: 10.1016/j.na.2009.06.070 · Zbl 1185.39010 · doi:10.1016/j.na.2009.06.070 [6] DOI: 10.1080/10236190802162739 · Zbl 1187.39020 · doi:10.1080/10236190802162739 [7] DOI: 10.1080/10236190410001667977 · Zbl 1053.39008 · doi:10.1080/10236190410001667977 [8] Cima A., J. Diff. Eqs. Appl. 12 pp 696– [9] DOI: 10.2307/3615344 · Zbl 0285.05028 · doi:10.2307/3615344 [10] Coxeter H. S. M., Acta Arith. 18 pp 297– [11] Csörnyei M., Monatsh. Math. 132 pp 215– [12] Grove E. A., Advances in Discrete Mathematics and Applications 4, in: Periodicities in Nonlinear Difference Equations (2005) [13] DOI: 10.2307/3609570 · doi:10.2307/3609570 [14] Kelley W., Difference Equations: An Introduction with Applications (2001) · Zbl 0970.39001 [15] DOI: 10.1007/978-94-017-1703-8 · doi:10.1007/978-94-017-1703-8 [16] Lang S., Analysis I (1976) [17] DOI: 10.2307/3606036 · doi:10.2307/3606036 [18] DOI: 10.2307/3609268 · doi:10.2307/3609268 [19] DOI: 10.2307/3612778 · doi:10.2307/3612778 [20] DOI: 10.1080/1023619031000061061 · Zbl 1030.39012 · doi:10.1080/1023619031000061061 [21] Mickens R. E., Difference Equations. Theory and Applications (1990) [22] Stević S., Rostock. Math. Kolloq. 61 pp 21– [23] Zheng Yu., Diff. Eqs. Dyn. Syst. 6 pp 319– This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.