×

Plane maps with denominator. II: Noninvertible maps with simple focal points. (English) Zbl 1057.37042

Summary: This paper is the second part of an earlier work devoted to the properties specific to maps of the plane characterized by the presence of a vanishing denominator, which gives rise to the generation of new types of singularities, called set of nondefinition, focal points and prefocal curves. A prefocal curve is a set of points which are mapped (or “focalized”) into a single point, called focal point, by the inverse map when it is invertible, or by at least one of the inverses when it is noninvertible. In the case of noninvertible maps, the previous text dealt with the simplest geometrical situation, which is nongeneric. To be more precise, this situation occurs when several focal points are associated with a given prefocal curve. The present paper defines the generic case for which only one focal point is associated with a given prefocal curve. This is due to the fact that only one inverse of the map has the property of focalization, but with properties different from those of invertible maps. Then the noninvertible maps of the previous part I [Int. J. Bifurcation Chaos Appl. Sci. Eng. 9, No. 1, 119—153 (1999; Zbl 0996.37052)] appear as resulting from a bifurcation leading to the merging of two prefocal curves, without merging of two focal points.

MSC:

37E99 Low-dimensional dynamical systems
37C20 Generic properties, structural stability of dynamical systems
37G10 Bifurcations of singular points in dynamical systems
37M20 Computational methods for bifurcation problems in dynamical systems

Citations:

Zbl 0996.37052
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1007/978-1-4612-1936-1 · doi:10.1007/978-1-4612-1936-1
[2] DOI: 10.1063/1.166158 · Zbl 1055.37567 · doi:10.1063/1.166158
[3] DOI: 10.1103/PhysRevLett.79.1018 · doi:10.1103/PhysRevLett.79.1018
[4] DOI: 10.1142/S0218127497001229 · Zbl 0899.58045 · doi:10.1142/S0218127497001229
[5] Bischi G. I., Econ. Notes 26 pp 143–
[6] DOI: 10.1142/S0218127499000079 · Zbl 0996.37052 · doi:10.1142/S0218127499000079
[7] DOI: 10.1016/S0362-546X(01)00343-1 · Zbl 1042.37515 · doi:10.1016/S0362-546X(01)00343-1
[8] DOI: 10.1016/S0362-546X(01)00637-X · Zbl 1042.91512 · doi:10.1016/S0362-546X(01)00637-X
[9] DOI: 10.2307/2171879 · Zbl 0898.90042 · doi:10.2307/2171879
[10] DOI: 10.1063/1.166414 · Zbl 0982.37033 · doi:10.1063/1.166414
[11] Gumowski I., Dynamique Chaotique (1980)
[12] DOI: 10.1142/S0218127494000241 · Zbl 0818.58032 · doi:10.1142/S0218127494000241
[13] DOI: 10.1142/9789812798732 · doi:10.1142/9789812798732
[14] Mira C., Int. J. Bifurcation and Chaos 6 pp 1439–
[15] DOI: 10.1142/S0218127494001210 · Zbl 0872.65078 · doi:10.1142/S0218127494001210
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.