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Articles

Global stability and bifurcations of perturbed Gumowski–Mira difference equation

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Pages 774-790 | Received 01 Feb 2015, Accepted 18 Apr 2015, Published online: 02 Sep 2015

References

  • G. Bastien, and M. Rogalski, On the algebraic difference equations un+2+un=ψ(un+1) in R related to a family of elliptic quartics in the plane, J. Math. Anal. Appl. 326(2) (2007), pp. 822–844.
  • G. Bastien, and M. Rogalski, Results and problems about solutions of perturbed Lyness' type order k difference equations in R*+un+k(un+λ)=f(un+k−1,…,un+1), with examples, and test of the efficiency of a quasi-Lyapunov function method, J. Differ. Equ. Appl. 19(8) (2013), pp. 1331–1352. doi:10.1080/10236198.2012.748758.
  • S. Basu, and O. Merino, Global behavior of solutions to two classes of second-order rational difference equations, Adv. Differ. Equ. 2009(1) (2009), p. 128602, article no. 128602.10.1155/2009/128602.
  • A. Cima, A. Gasull, and V. Mañosa, Nonautonomous two-periodic Gumovski–Mira difference equations, Int. J. Bifurcat. Chaos 22(11) (2012), p. 1250264. doi:10.1142/S0218127412502641.
  • C.A. Clark, E.J. Janowski, and M.R.S. Kulenović, Stability of the Gumowski–Mira equation with period-two coefficient, J. Math. Anal. Appl. 307(1) (2005), pp. 292–304. doi:10.1016/j.jmaa.2004.10.046.
  • G.F. Deng, Q.Y. Lu, and N.Z. Liu, A general method for studying quadratic perturbations of the third-order Lyness difference equation, Adv. Differ. Equ. 2013(1) (2013), p. 193, article no. 193.10.1186/1687-1847-2013-193.
  • J.P. England, B. Krauskopf, and H.M. Osinga, Bifurcations of stable sets in noninvertible planar maps, Int. J. Bifurcat. Chaos 15(3) (2005), pp. 891–904. doi:10.1142/S0218127405012466.
  • I. Gumowski, and C. Mira, Recurrences and Discrete Dynamic Systems, Lecture Notes in Mathematics 809 Springer, Berlin, 1980.
  • V.L. Kocic, G. Ladas, G. Tzanetopoulos, and E. Thomas, On the stability of Lyness' equation, Dyn. Contin. Discrete Impuls. Syst 1 (1995), pp. 245–254.
  • M.R.S. Kulenović, Invariants and related Liapunov functions for difference equations, Appl. Math. Lett. 13(7) (2000), pp. 1–8. doi:10.1016/S0893-9659(00)00068-9.
  • S.P. Li, and W.N. Zhang, Bifurcations in a second-order difference equation from macroeconomics† supported by NSFC, trapoyt and moe research grants, J. Differ. Equ. Appl. 14(1) (2008), pp. 91–104. doi:10.1080/10236190701483145.
  • X.Y. Li, and D.M. Zhu, Global asymptotic stability for two recursive difference equations, Appl. Math. Comput. 150(2) (2004), pp. 481–492. doi:10.1016/S0096-3003(03)00286-8.
  • O. Merino, Global attractivity of the equilibrium of a difference equation: An elementary proof assisted by computer algebra system, J. Differ. Equ. Appl. 17(1) (2011), pp. 33–41. doi:10.1080/10236190902932718.
  • J.F. Selgrade, and J.H. Roberds, Period-doubling bifurcations for systems of difference equations and applications to models in population biology, Nonlinear Anal. Ser. A 29(2) (1997), pp. 185–199. doi:10.1016/S0362-546X(96)00041-7.

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