×

Transversal homoclinic orbits for higher dimensional difference equations. (English) Zbl 1002.39026

The authors deal with the difference equations \[ x_{n+1}= g(x_n) +\varepsilon h(u,x_n,\varepsilon), \] where \(x_n\in\mathbb{R}^p\) and \(\varepsilon\in \mathbb{R}\) is a small parameter. The main goal of this paper is to study the homoclinic bifurcations of difference equations in a degenerate case. They obtain a Melnikov vector mapping for difference equations with the help of which the existence of transversal homoclinic orbits can be detected.

MSC:

39A11 Stability of difference equations (MSC2000)
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
37C29 Homoclinic and heteroclinic orbits for dynamical systems
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1080/10236199608808060 · Zbl 0880.39009 · doi:10.1080/10236199608808060
[2] DOI: 10.1016/0022-0396(90)90034-M · Zbl 0713.34055 · doi:10.1016/0022-0396(90)90034-M
[3] DOI: 10.1016/0375-9601(95)00587-S · Zbl 1020.37506 · doi:10.1016/0375-9601(95)00587-S
[4] DOI: 10.1016/0022-0396(80)90104-7 · Zbl 0439.34035 · doi:10.1016/0022-0396(80)90104-7
[5] DOI: 10.1007/BFb0065310 · doi:10.1007/BFb0065310
[6] Fečkan M., J. Differential Equations
[7] Fečkan M., Appl. Math. 36 pp 355– (1991)
[8] Fečkan M., Appl. Math. 38 pp 101– (1993)
[9] Fečkan M., Boll. U.M.I. 10 pp 175– (1996)
[10] DOI: 10.1137/0149040 · Zbl 0687.58023 · doi:10.1137/0149040
[11] DOI: 10.1006/jdeq.1995.1136 · Zbl 0840.34045 · doi:10.1006/jdeq.1995.1136
[12] DOI: 10.1088/0951-7715/8/6/014 · Zbl 0841.58050 · doi:10.1088/0951-7715/8/6/014
[13] Melnikov V.K., Trans Moscow Math. Soc 12 pp 1– (1964)
[14] Meyer K.R., Trans. Amer. Math 314 pp 63– (1989)
[15] Palmer K.J., Dynamics Reported 1 pp 265– (1988) · doi:10.1007/978-3-322-96656-8_5
[16] DOI: 10.1016/0022-0396(84)90082-2 · Zbl 0508.58035 · doi:10.1016/0022-0396(84)90082-2
[17] DOI: 10.1216/rmjm/1181073065 · Zbl 0724.34055 · doi:10.1216/rmjm/1181073065
[18] DOI: 10.1007/BFb0098592 · doi:10.1007/BFb0098592
[19] DOI: 10.1007/BF00945119 · Zbl 0692.58032 · doi:10.1007/BF00945119
[20] Vanderbauwhede A., Results in Mathematics 21 pp 211– (1992) · Zbl 0762.34022 · doi:10.1007/BF03323080
[21] Wiggins S., Global Bifurcations and Chaos”, Appl. Math.Sci., 73 (1988) · Zbl 0661.58001 · doi:10.1007/978-1-4612-1042-9
[22] DOI: 10.1007/BF02218723 · Zbl 0844.34043 · doi:10.1007/BF02218723
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.