×

Asymptotic dynamics of a difference equation with a parabolic equilibrium. (English) Zbl 1454.39025

Authors’ abstract: The aim of this work is the study of the asymptotic dynamical behaviour, of solutions that approach parabolic fixed points in difference equations. In one dimensional difference equations, we present the asymptotic development for positive solutions tending to the fixed point. For higher dimensions, through the study of two families of difference equations in the two and three dimensional case, we take a look at the asymptotic dynamic behaviour. To show the existence of solutions we rely on the parametrization method.

MSC:

39A30 Stability theory for difference equations
39A10 Additive difference equations
39A22 Growth, boundedness, comparison of solutions to difference equations
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Baldomà, I.; Fontich, E., Stable manifolds associated to fixed points with linear part equal to identity, J. Differ. Equ., 197, 1, 45-72 (2004) · Zbl 1036.37011
[2] Baldomà, I.; Fontich, E.; de la Llave, R.; Martín, P., The parameterization method for one-dimensional invariant manifolds of higher dimensional parabolic fixed points, Discrete Contin. Dyn. Syst., 17, 4, 835-865 (2007) · Zbl 1123.37011
[3] Baldomà, I., Fontich, E., Martín, P.: Invariant manifolds of parabolic fixed points (I). Existence and depence on parameters (2016). arXiv:1603.02533v1 [math.DS] · Zbl 1437.37022
[4] Baldomà, I., Fontich, E., Martín, P.: Invariant manifolds of parabolic fixed points (II). Approximations by sums of homogeneous functions (2016). arXiv:1603.02535v1 [math.DS] · Zbl 1437.37023
[5] Berg, L., Asymptotische Darstellungen und Entwicklungen, Hoch-schulbücher für Mathematik (1968), Berlin: VEB Deutscher Verlag der Wissenschaften, Berlin · Zbl 0165.36901
[6] Berg, L., On the asymptotics of nonlinear difference equations, J. Anal. Appl., 21, 4, 1061-1074 (2002) · Zbl 1030.39006
[7] Berg, L., Inclusion theorems for non-linear difference equations with applications, J. Differ. Equ. Appl., 10, 4, 399-408 (2004) · Zbl 1056.39003
[8] Berg, L., On the asymptotics of the difference equation \(x_{n-3}=x_n(1+x_{n-1}x_{n-2})\), J. Differ. Equ. Appl., 14, 1, 105-108 (2008) · Zbl 1138.39003
[9] Berg, L.; Stević, S., On the asymptotics of the difference equation \(y_n(1+y_{n-1}\dots y_{n-k+1})=y_{n-k}\), J. Differ. Equ. Appl., 17, 4, 577-586 (2011) · Zbl 1220.39011
[10] Beverton, RJ; Holt, SJ, On the Dynamics of Exploited Fish Populations (1957), London: Fish. Invest, London
[11] Cabré, X.; Fontich, E.; de la Llave, R., The parameterization method for invariant manifolds. I. Manifolds associated to non-resonant subspaces, Indiana Univ. Math. J., 52, 2, 283-328 (2003) · Zbl 1034.37016
[12] Cabré, X.; Fontich, E.; de la Llave, R., The parameterization method for invariant manifolds. II. Regularity with respect to parameters, Indiana Univ. Math. J., 52, 2, 329-360 (2003) · Zbl 1034.37017
[13] Cabré, X.; Fontich, E.; de la Llave, R., The parameterization method for invariant manifolds. III. Overview and applications, J. Differ. Equ., 218, 2, 444-515 (2005) · Zbl 1101.37019
[14] Carleson, L., Gamelin, T.: Complex Dynamics Universitext: Tracts in Mathematics. Springer, New York, Inc (1993). ISBN 13: 978-0-387-97942-7 · Zbl 0782.30022
[15] Casasayas, J.; Fontich, E.; Nunes, A., Invariant manifolds for a class of parabolic points, Nonlinearity, 5, 1193-1210 (1992) · Zbl 0783.58068
[16] Easton, RW, Parabolic orbits in the planar three-body problem, J. Differ. Equ., 52, 116-134 (1984) · Zbl 0487.70014
[17] Fontich, E., Stable curves asymptotic to a degenerate fixed point, Nonlinear Anal., 35, 711-733 (1999) · Zbl 0918.58064
[18] Grove, E. A., Kent, C. M., Ladas, G., Valicenti, S., Levins R.: Global stability in some population models. Communications in difference equations. In: Proceedings of the 4th International Conference on Difference Equations (Poznan, 1998), Gordon and Breach, Amsterdam, pp. 149-176 (2000) · Zbl 0988.39018
[19] Gutnik, L., Stević, S.: On the Behaviour of the Solutions of a Second-Order Difference Equation. Discrete Dyn. Nat. Soc. ID 27562 (2007) · Zbl 1180.39002
[20] Haro, A.; de la Llave, R., A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: rigorous results, J. Differ. Equ., 228-2, 530-579 (2006) · Zbl 1102.37017
[21] Haro, A.; de la Llave, R., A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: explorations and mechanisms for the breakdown of hyperbolicity, SIAM J. Appl. Dyn. Syst., 6-1, 142-207 (2007) · Zbl 1210.37062
[22] Huo, HF; Li, WT, Permanence and global stability of positive solutions of a nonautonomous discrete ratio-dependent predator-prey model, Discrete Dyn. Nat. Soc., 2005, 2, 135-144 (2005) · Zbl 1111.39007
[23] Kuang, Y.; Cushing, JM, Global stability in a nonlinear difference-delay equation model of flour beetle population growth, J. Differ. Equ. Appl., 2, 1, 31-37 (1996) · Zbl 0862.39005
[24] Martínez, R.; Pinyol, C., Parabolic orbits in the elliptic restricted three body problem, J. Differ. Equ., 111, 299-339 (1994) · Zbl 0804.70009
[25] McGehee, R., A stable manifold theorem for degenerate fixed points with applications to celestial mechanics, J. Differ. Equ., 14, 70-88 (1973) · Zbl 0264.70007
[26] Milnor, J., Dynamics in One Complex Variable (1991), Stony Brook: Institute for Mathematical Sciences, SUNY, Stony Brook
[27] Resman, M., \( \varepsilon \)-Neighborhoods of orbits and formal classification of parabolic diffeomorphisms, Discrete Contin. Dyn. A, 33-8, 3767-3790 (2013) · Zbl 1278.37030
[28] Robinson, C., Homoclinic orbits and oscillation for the planar three-body problem, J. Differ. Equ., 52, 356-377 (1984) · Zbl 0495.70025
[29] Simó, C.: Stability of degenerate fixed points of analytic area preserving mappings. Bifurcation, Ergodic Theory and Applications (Dijon, 1981) pp. 184-194, Astérisque, vol. 98. Soc. Math. France, Paris (1982) · Zbl 0516.58028
[30] Slotnick, D.L.: Asymptotic behavior of solutions of canonical systems near a closed, unstable orbit. Contributions to The Theory of Non-linear Oscillations, vol. IV pp. 85-110, Annals of Mathematics Studies, no. 41, Princeton University Press, Princeton (1958) · Zbl 0088.06602
[31] Stević, S., Asymptotic behaviour of a sequence defined by iteration, Mat. Vesnik (3-4), 48, 99-105 (1996) · Zbl 1032.40002
[32] Stević, S., Behavior of the positive solutions of the generalized Beddington-Holt equation, Panam. Math. J., 10, 4, 77-85 (2000) · Zbl 1039.39005
[33] Stević, S., On the recursive sequence \(x_{n+1}=x_{n-1}/g(x_n)\), Taiwan. J. Math., 6-3, 405-414 (2002) · Zbl 1019.39010
[34] Stević, S., Asymptotic behavior of a nonlinear difference equation, Indian J. Pure Appl. Math., 34, 12, 1681-1687 (2003) · Zbl 1049.39012
[35] Stević, S.: Asymptotic behaviour of a class of nonlinear difference equations. Discrete Dyn. Nat. Soc. ID 47156 (2006) · Zbl 1121.39006
[36] Stević, S., On positive solutions of a (k+1)th order difference equation, Appl. Math. Lett., 19-5, 427-431 (2006) · Zbl 1095.39010
[37] Stević, S.: On monotone solutions of some classes of difference equations. Discrete Dyn. Nat. Soc. ID 53890 (2006) · Zbl 1109.39013
[38] Stević, S.: On a discrete epidemic model. Discrete Dyn. Nat. Soc. ID 87519 (2007) · Zbl 1180.39004
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.