Period-Two Trichotomies of a Difference Equation of Order Higher Than Two
DOI:
https://doi.org/10.5644/SJM.04.1.07Keywords:
Attractivity, difference equation, invariant intervals, periodtwo solution, unboundedAbstract
We investigate the period-two trichotomies of solutions of the equation $$x_{n+1} = f(x_{n}, x_{n-1},x_{n-2}), \quad n=0, 1, \ldots $$ where the function $f$ satisfies certain monotonicity conditions. We give fairly general conditions for period-two trichotomies to occur and illustrate the results with numerous examples.
1991 Mathematics Subject Classification. 39A10, 39A11
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References
A. M. Amleh, D. A. Georgiou, E. A. Grove, and G. Ladas, On the recursive sequence yn+1 = α + xn−1 xn , J. Math. Anal. Appl., 233 (1999), 790–798.
Dˇz. Burgi´c, S. Kalabuˇsi´c, and M. R. S. Kulenovi´c, Non-hyperbolic dynamics for competitive systems in the plane and global period-doubling bifurcations, Adv. Dyn. Syst. Appl., 3 (2008), (to appear).
E. Camouzis and G. Ladas, Dynamics of third order rational difference equations with open problems and conjectures, Chapman & Hall/CRC Press, Boca Raton, 2007.
E. Camouzis, G. Ladas, and H.D. Voulov, On the dynamics xn+1 = α+γxn−1+δxn−2 A+xn−2, J. Difference Equ. Appl., 9 (2003), 731–738.
E. Chatterjee, E. A. Grove, Y. Kostrov, and G. Ladas, On the trichotomy character xn+1 = α+γxn−1 A+Bxn+xn−2 , J. Difference Equ. Appl., 9 (2003), 1113–1128.
S. Elaydi, An Introduction to Difference Equations, 3rd ed., Springer-Verlag, New York, 2005.
E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, CRC Press, 2005.
C. Gibbons, M. R. S. Kulenovi´c, and G. Ladas, On the recursive sequence yn+1 = α+βyn−1 γ+yn , Math. Sci. Res. Hot-Line, 4 (2) (2000), 1–11.
J. Hale and H. Kocak, Dynamics and Bifurcation, Springer, New York, 1991.
V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993.
M. R. S. Kulenovi´c and G. Ladas, Dynamics of Second Order Rational Difference Equations, Open Problems and Conjectures, Chapman& Hall/CRC Press, Boca Raton, 2001.
M. R. S. Kulenovi´c and O. Merino, Disrete Dynamical Systems and Difference Equations with Mathematica, Chapman& Hall/CRC Press, Boca Raton, 2002.
M. R. S. Kulenovi´c and O. Merino, A global attractivity result for maps with invariant boxes, Discrete Cont. Dyn. Syst. Ser. B, 6 (2006), 97–110.
M. R. S. Kulenovi´c and O. Merino, Competitive-exclusion versus competitivecoexistence for systems in the plane, Discrete Cont. Dyn. Syst. Ser. B, 6 (2006), 1141–1156.
M. R. S. Kulenovi´c and O. Merino, A note on unbounded solutions of a class of second
order rational difference equations, J. Difference Equ. Appl., 12 (2006), 777–781.
M. R. S. Kulenovi´c and O. Merino, Global Bifurcation of Competitive Systems in the Plane, (to appear).
G. Ladas, On third-order rational difference equations, Part 1, J. Difference Equ. Appl., 10 (2004), 869–879.
J. F. Selgrade and J. H. Roberds, Period-doubling bifurcations for systems of difference equations and aplications to models in population biology, Nonlinear Anal. TMA, 29 (1997), 185–199.
H. L. Smith, The discrete dynamics of monotonically decomposable maps, J. Math. Biology, (2006).
