Knot Points in Two-Dimensional Maps and Related Properties

We consider the class of two-dimensional maps of the plane for which there exists a whole one-dimensional singular set (for example, a straight line) that is mapped into one point, called a "knot point" of the map. The special character of this kind of point has been already observed in maps of this class with at least one of the inverses having a vanishing denominator. In that framework, a knot is the so-called focal point of the inverse map (it is the same point). In this paper, we show that knots may also exist in other families of maps, not related to an inverse having values going to infinity. Some particular properties related to focal points persist, such as the existence of a "point to slope" correspondence between the points of the singular line and the slopes in the knot, lobes issuing from the knot point and loops in infinitely many points of an attracting set or in invariant stable and unstable sets.

[1]  H. C. Yee,et al.  GLOBAL ASYMPTOTIC BEHAVIOR OF ITERATIVE IMPLICIT SCHEMES , 1994 .

[2]  Christian Mira,et al.  Plane Maps with Denominator: I. Some Generic Properties , 1999 .

[3]  P. Fischer,et al.  Plane Maps, Singularities and Quasi-Fixed Points , 2006, Int. J. Bifurc. Chaos.

[4]  Laura Gardini,et al.  Basin Fractalization Due to Focal Points in a Class of Triangular Maps , 1997 .

[5]  En-Guo Gu,et al.  On the global analysis of dynamics in a delayed regulation model with an external interference , 2007 .

[6]  Lora Billings,et al.  On noninvertible mappings of the plane: Eruptions. , 1996, Chaos.

[7]  EN-GUO GU.  The Feasible Domains and their bifurcations in an Extended logistic Model with an External Interference , 2007, Int. J. Bifurc. Chaos.

[8]  William A. Brock,et al.  A rational route to randomness , 1997 .

[9]  Gian Italo Bischi,et al.  On a Rent-Seeking Game Described by a Non-Invertible Iterated Map with Denominator , 2001 .

[10]  Christian Mira,et al.  On Some Properties of Invariant Sets of Two-Dimensional Noninvertible Maps , 1997 .

[11]  DAVID MUMFORD,et al.  Global Analysis , 2003 .

[12]  Lora Billings,et al.  Lyapunov Exponents, Singularities, and a Riddling Bifurcation , 1997 .

[13]  J. Yorke,et al.  CHAOTIC ATTRACTORS IN CRISIS , 1982 .

[14]  Christian Mira,et al.  Plane Maps with denominator. Part III: Nonsimple Focal Points and Related bifurcations , 2005, Int. J. Bifurc. Chaos.

[15]  Laura Gardini,et al.  Basin boundaries and focal points in a map coming from Bairstow's method. , 1999, Chaos.

[16]  Christian Mira,et al.  Chaotic Dynamics in Two-Dimensional Noninvertible Maps , 1996 .

[17]  Gian Italo Bischi,et al.  Global Analysis of a Non-Linear Model with Learning , 1997 .

[18]  Christian Mira,et al.  Maps with vanishing denominators , 2007, Scholarpedia.

[19]  C. Mira,et al.  Chaotic Dynamics: From the One-Dimensional Endomorphism to the Two-Dimensional Diffeomorphism , 1987 .

[20]  E Mosekilde,et al.  Torus breakdown in noninvertible maps. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  En-Guo Gu,et al.  Global bifurcations of Domains of Feasible Trajectories: an Analysis of a Discrete Predator-prey Model , 2006, Int. J. Bifurc. Chaos.

[22]  Christian Mira,et al.  Plane Maps with denominator. Part II: Noninvertible Maps with Simple Focal Points , 2003, Int. J. Bifurc. Chaos.

[23]  FRACTALIZATION OF BASIN BOUNDARY IN TWO-DIMENSIONAL NONINVERTIBLE MAPS , 1999 .