Abstract
Our intent in this chapter is to study the periodicity properties of second order nonlinear difference equations. We will focus on examining the periodic traits of rational difference equations and Max-Type difference equations.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
A.M. Amleh, D.A. Georgiou, E.A. Grove, and G. Ladas, On the recursive sequence \(x_{n+1}=\alpha +\frac {x_{n-1}}{x_{n}}\), J. Math. Anal. Appl.; (233)(1999), 790–798.
A.M. Amleh, J. Hoag, and G. Ladas, A Difference Equation with Eventually Periodic Solutions, Computer Math. Applic. 36, (1998), 401–404.
W.J. Briden, E.A. Grove, C.M. Kent, and G. Ladas, Eventually Periodic Solutions of \(x_{n+1}=max\{\frac {1}{x_{n}}, \frac {A_{n}}{x_{n-1}}\}\), Commun. Appl. Nonlinear Anal. 6 (1999), no. 4.
W.J. Briden, G. Ladas, and T. Nesemann, On the Recursive Sequence \(x_{n+1}=max\left \{\frac {1}{x_{n}}, \frac {A_{n}}{x_{n-1}}\right \}\), J. Differ. Equations. Appl. 5 (1999), 491–494.
E. Camouzis, G. Ladas, I.W. Rodriques, and S. Northsfield. On the rational recursive sequences \(x_{n+1}=\frac {\beta {x_{n}}^{2}}{1+{x_{n}}^{2}}\mbox{.}\) Computers Math. Appl., 28:37–43, 1994.
E. Chatterjee, E.A. Grove, Y. Kostrov, and G. Ladas, On the Trichotomy character of \(x_{n+1}=\frac {\alpha +\gamma x_{n-1}}{A+Bx_{n}+x_{n-2}}\), J. Difference Equa. Appl.,(9)(2003), 1113–1128.
C. Gibbons, M.R.S. Kulenovic, and G. Ladas, On the Recursive Sequence \(x_{n+1} = \frac {\alpha + \beta x_{n-1}}{\gamma + x_{n}} ,\) Math. Sci. Res. Hot-Line 4 (2) (2000), 1–11.
E.A. Grove, G. Ladas, M. Predescu, and M. Radin, On the global character of the difference equation \(x_{n+1}=\frac {\alpha + \gamma x_{n-(2k+1)}+\delta x_{n-2l}}{A+x_{n-2l}}\), J. Difference Equa. Appl., (9)(2003), 171–200.
G.L. Karakostas and S. Stevic, On the recursive sequence \(x_{n+1}=B + \frac {x_{n-k}}{a_{0}x_{n}+\dots +a_{k-1}x_{n-k+1}+\gamma }\), J. Difference Equa. Appl., (10)(2004), 809–815.
C.M. Kent, E.A. Grove, G. Ladas, and M.A. Radin; On \(x_{n+1}=max\left \{\frac {1}{x_{n}}, \frac {A_{n}}{x_{n-1}}\right \}\) with a Period 3 Parameter; Fields Institute Communications, Volume 29, March 2001.
C.M. Kent, M. Kustesky, A.Q. Nguyen, B.V. Nguyen, Eventually Periodic Solutions of \(x_{n+1}=max\left \{\frac {A_{n}}{x_{n}}, \frac {B_{n}}{x_{n-1}}\right \}\), With Period Two Cycle Parameters.
M.R.S. Kulenovic, G. Ladas, and N.R. Prokup, On the Recursive Sequence \(x_{n+1}=\frac {ax_{n} + bx_{n-1}}{A + x_{n}}.\) J. Differ. Equations Appl. 6(5) (2000), 563–576.
M.R.S. Kulenovic, G. Ladas, and W.S. Sizer, On the Recursive Sequence \(x_{n+1} = \frac {\alpha x_{n} + \beta x_{n-1}}{\gamma x_{n} + \delta x_{n-1}}\) Math. Sci. Res. Hot-Line 2 (5) (1998), 1–16.
C.M. Kent, M.A. Radin; On the Boundedness of Positive Solutions of a Reciprocal Max-Type Difference Equation with Periodic Parameters; International Journal of Difference Equations, (8)(2)(2013), 195–213.
G. Ladas, On the recursive sequence \(x_{n+1}=max\left \{\frac {A_{0}}{x_{n}},\ldots ,\frac {A_{k}}{x_{n-k}}\right \}\), J. Diff. Equa. Appl., 2 (2) (1996), 339–341.
S. Stević, On the recursive sequence \(x_{n+1}=\frac {\alpha +\sum _{i=1}^{k} \alpha _{i} x_{n-p_{i}}}{1+\sum _{j=1}^{m}\beta _{j}x_{n-q_{j}}}\), J. Difference Equa. Appl., (13)(2007), 41–46.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Radin, M.A. (2018). Periodic Traits of Second Order Nonlinear Difference Equations. In: Periodic Character and Patterns of Recursive Sequences. Springer, Cham. https://doi.org/10.1007/978-3-030-01780-4_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-01780-4_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-01779-8
Online ISBN: 978-3-030-01780-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)