On the existence of 6-cycles for some families of difference equations of third order. (English) Zbl 1540.39010
The authors prove the non-existence of 6-cycles for a specific class of third-order difference equations involving continuous symmetric bivariate functions. In the absence of the symmetric property, they determine the potential 6-cycles and describe avenues for further studies concerning cycles of other lengths.
Reviewer: Jonathan Hoseana (Bandung)
MSC:
| 39A23 | Periodic solutions of difference equations |
| 39A30 | Stability theory for difference equations |
Keywords:
\(p\)-cycle; symmetric function; potential cycles; equilibrium point; iteration; homeomorphismReferences:
| [1] | Abu-Saris RM, Al-Hassan QM. On global periodicity of difference equations. Journal of Mathematical Analysis and Applications 2003; 282(2): 468-477. https://doi.org/10.1016/S0022-247X(03)00272-5 · Zbl 1031.39012 · doi:10.1016/S0022-247X(03)00272-5 |
| [2] | Agarwal RP, Romanenko EY. Stable periodic solutions of difference equations. Applied Mathematics Letters 1998; 11 (4): 81-84. https://doi.org/10.1016/S0893-9659(98)00060-3 · Zbl 0941.39004 · doi:10.1016/S0893-9659(98)00060-3 |
| [3] | Agarwal RP, Zhang W. Periodic solutions of difference equations with general periodicity. Computers and Mathe-matics with Applications 2001; 42: 719-727. https://doi.org/10.1016/S0898-1221(01)00191-2 · Zbl 1005.39006 · doi:10.1016/S0898-1221(01)00191-2 |
| [4] | Balibrea F, Linero A. On the global periodicity of some difference equations of third order. Journal of Difference Equations and Applications 2007; 13: 1011-1027. https://doi.org/10.1080/10236190701388518 · Zbl 1130.39003 · doi:10.1080/10236190701388518 |
| [5] | Baumol WJ, Wolff EN. Feedback between R & D and productivity growth: a chaos model. In: Benhabib J (ed.). Cycles and chaos in Economic Equilibrium. Princeton: Princeton University Press 1992; 355-373. |
| [6] | Camouzis E, Ladas G. Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjec-tures. Boca Raton FL: Advances in Discrete Mathematics and Applications, Chapman & Hall/CRC, 2008. · Zbl 1129.39002 |
| [7] | Cánovas JS, Linero Bas A, Soler López G. On global periodicity of difference equations, Taiwanese Journal of Mathematics 2009; 13: 1963-1983. https://doi.org/10.11650/twjm/1500405651 · Zbl 1257.39010 · doi:10.11650/twjm/1500405651 |
| [8] | Cánovas JS, Linero Bas A, Soler López G. A characterization of k -cycles, Nonlinear Analysis 2010; 72: 364-372. https://doi.org/10.1016/j.na.2009.06.070 · Zbl 1185.39010 · doi:10.1016/j.na.2009.06.070 |
| [9] | Caro A, Linero A. Existence and uniqueness of p -cycles of second and third order. Journal of Difference Equations and Applications 2009; 15: 489-500. https://doi.org/10.1080/10236190802162739 · Zbl 1187.39020 · doi:10.1080/10236190802162739 |
| [10] | Caro A, Linero A. General cycles of potential form. International Journal of Bifurcation and Chaos in Applied Science and Engineering 2010; 20(9): 2735-2749. https://doi.org/10.1142/S0218127410027325 · Zbl 1202.39002 · doi:10.1142/S0218127410027325 |
| [11] | Cima A, Gasull A, Mañosa V, Mañosas F. Different approaches to the global periodicity problem. In: Alsedà Ll et al. (eds.). Difference equations, discrete dynamical systems and applications. Berlin: Springer Proceedings in Mathematics and Statistics, Springer 2016; 180: 85-106. · Zbl 1355.39021 |
| [12] | Cima A, Gasull A, Mañosas F. On periodic rational difference equations of order k . Journal of Difference Equations and Applications 2004; 10: 549-559. https://doi.org/10.1080/10236190410001667977 · Zbl 1053.39008 · doi:10.1080/10236190410001667977 |
| [13] | Cima A, Gasull A, Mañosa V. Global periodicity and complete integrability of discrete dynamical systems. Journal of Difference Equations and Applications 2006; 12: 696-716. https://doi.org/10.1080/10236190600703031 · Zbl 1127.39010 · doi:10.1080/10236190600703031 |
| [14] | Csörnyei M, Laczkovich M. Some periodic and non-periodic recursions. Monatshefte für Mathematik 2001; 132: 215-236. https://doi.org/10.1007/s006050170042 · Zbl 1036.11002 · doi:10.1007/s006050170042 |
| [15] | Grove EA, Ladas G. Periodicities in nonlinear difference equations. Boca Raton: Chapman & Hall/CRC, 2005. · Zbl 1078.39009 |
| [16] | Haynes R, Kwasik S, Mast J, Schultz R. Periodic maps on R 7 without fixed points. Mathematical Proceedings of the Cambridge Philosophical Society 2002; 132: 131-136. https://doi.org/10.1017/S0305004101005345 · Zbl 0997.55004 · doi:10.1017/S0305004101005345 |
| [17] | Linero A. Some results on periodicity of difference equations. In: Liz E, Mañosa V (eds.). Proceedings of the Workshop Future Directions in Difference Equations. Universidad de Vigo, Vigo 2011; 121-143. · Zbl 1382.39018 |
| [18] | Mestel BD. On globally periodic solutions of the difference equation xn+1 = f (xn) xn-1 . Journal of Difference Equations and Applications 2003; 9: 201-209. https://doi.org/10.1080/1023619031000061061 · Zbl 1030.39012 · doi:10.1080/1023619031000061061 |
| [19] | Montgomery D. Pointwise periodic homeomorphisms. American Journal of Mathematics 1937; 59: 118-120. https://doi.org/10.2307/2371565 · JFM 63.0565.05 · doi:10.2307/2371565 |
| [20] | Pielou EC. An introduction to Mathematical Ecology. New York: Wiley-Interscience, 1969. · Zbl 0259.92001 |
| [21] | Rubio-Massegú J. On the global periodicity of discrete dynamical systems and applications to ratio-nal difference equations. Journal of Mathematical Analysis and Applications 2008; 343(1): 182-189. https://doi.org/10.1016/j.jmaa.2008.01.048 · Zbl 1152.39008 · doi:10.1016/j.jmaa.2008.01.048 |
| [22] | Sedaghat H. Nonlinear difference equations. Theory with applications to social science models, Mathematical Modelling: Theory and Applications. Dordrecht: Kluwer Academic Publishers, 2003. · Zbl 1020.39007 |
| [23] | Zheng Y. Existence of nonperiodic solutions of the Lyness Equation xn+1 = α+xn x n-1 . Journal of Mathematical Analysis and Applications 1997; 209: 94-102. DOI: https//doi.org/10.1006/jmaa.1997.5340 · Zbl 0876.39001 · doi:10.1006/jmaa.1997.5340 |
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