PLANE MAPS WITH DENOMINATOR I: SOME GENERIC PROPERTIES

This paper is devoted to the study of some global dynamical properties and bifurcations of two-dimensional maps related to the presence, in the map or in one of its inverses, of a vanishing denominator. The new concepts of focal points and of prefocal curves are introduced in order to characterize some new kinds of contact bifurcations specific to maps with denominator. The occurrence of such bifurcations gives rise to new dynamic phenomena, and new structures of basin boundaries and invariant sets, whose presence can only be observed if a map (or some of its inverses) has a vanishing denominator.

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Attractors for maps with fractional inverse

Discrete Dynamics in Nature and Society ◽

10.1155/ddns.2005.343 ◽

2005 ◽

Vol 2005 (3) ◽

pp. 343-355 ◽

Cited By ~ 1

Author(s):

M. R. Ferchichi ◽

I. Djellit

Keyword(s):

Two Dimensional ◽

Focal Points ◽

New Concepts ◽

Specific Contact ◽

Dynamical Properties ◽

Dynamic Phenomena

In this work, we consider some dynamical properties and specific contact bifurcations of two-dimensional maps having an inverse with vanishing denominator. We introduce new concepts and notions of focal points and prefocal curves which may cause and generate new dynamic phenomena. We put in evidence a link existing between basin bifurcation of a map with fractional inverse and the prefocal curve of this inverse.

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On Some Properties of Focal Points

Discrete Dynamics in Nature and Society ◽

10.1155/2009/646258 ◽

2009 ◽

Vol 2009 ◽

pp. 1-11

Author(s):

M. R. Ferchichi ◽

I. Djellit

Keyword(s):

Fixed Point ◽

Focal Point ◽

Two Dimensional ◽

Focal Points ◽

Dynamical Properties

We consider some dynamical properties of two-dimensional maps having an inverse with vanishing denominator. We put in evidence a link between a fixed point of a map with fractional inverse and a focal point of this inverse.

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On the notion of the dimension with respect to a dynamical system

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700002546 ◽

1984 ◽

Vol 4 (3) ◽

pp. 405-420 ◽

Cited By ~ 2

Author(s):

Ya. B. Pesin

Keyword(s):

Dynamical System ◽

Hausdorff Dimension ◽

Invariant Sets ◽

Dimensional Case ◽

Two Dimensional ◽

One Dimensional ◽

Common Features ◽

Dynamical Properties ◽

The One ◽

Hyperbolic Sets

AbstractFor the invariant sets of dynamical systems a new notion of dimension-the so-called dimension with respect to a dynamical system-is introduced. It has some common features with the general topological notion of the dimension, but it also reflects the dynamical properties of the system. In the one-dimensional case it coincides with the Hausdorff dimension. For multi-dimensional hyperbolic sets formulae for the calculation of our dimension are obtained. These results are generalizations of Manning's results obtained by him for the Hausdorff dimension in the two-dimensional case.

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Basin Fractalization Due to Focal Points in a Class of Triangular Maps

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127497001229 ◽

1997 ◽

Vol 07 (07) ◽

pp. 1555-1577 ◽

Cited By ~ 17

Author(s):

Gian-Italo Bischi ◽

Laura Gardini

Keyword(s):

Focal Point ◽

Numerical Study ◽

Basins Of Attraction ◽

Focal Points ◽

Stable Sets ◽

Bifurcation Mechanism ◽

New Concepts ◽

New Type ◽

Basin Boundaries ◽

Triangular Maps

For a class of rational triangular maps of a plane, characterized by the presence of points in which a component assumes the form [Formula: see text], a new type of bifurcation is evidenced which creates loops in the boundaries of the basins of attraction. In order to explain such bifurcation mechanism, new concepts of focal point and line of focal values are defined, and their effects on the geometric behavior of the map and of its inverses are studied in detail. We prove that the creation of loops, which generally constitute the boundaries of lobes of the basins issuing from the focal points, is determined by contacts between basin boundaries and the line of focal values. A particular map is proposed for which the sequence of such contact bifurcations occurs, causing a fractalization of basin boundaries. Through the analytical and the numerical study of this example new structures of the basins of attraction are evidenced, characterized by fans of stable sets issuing from the focal points, assuming the shape of lobes and arcs, the latter created by the merging of lobes due to contacts between the basin boundaries and the critical curve LC.

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Action-Minimizing Curves for Tonelli Lagrangians

10.23943/princeton/9780691164502.003.0004 ◽

2017 ◽

Author(s):

Alfonso Sorrentino

Keyword(s):

Critical Value ◽

The Other ◽

Invariant Sets ◽

Simple Pendulum ◽

New Concepts ◽

Peierls Barrier ◽

Average Action

This chapter discusses the notion of action-minimizing orbits. In particular, it defines the other two families of invariant sets, the so-called Aubry and Mañé sets. It explains their main dynamical and symplectic properties, comparing them with the results obtained in the preceding chapter for the Mather sets. The relation between these new invariant sets and the Mather sets is described. As a by-product, the chapter introduces the Mañé's potential, Peierls' barrier, and Mañé's critical value. It discusses their properties thoroughly. In particular, it highlights how this critical value is related to the minimal average action and describes these new concepts in the case of the simple pendulum.

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KNOT POINTS IN TWO-DIMENSIONAL MAPS AND RELATED PROPERTIES

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127409023184 ◽

2009 ◽

Vol 19 (02) ◽

pp. 545-555 ◽

Cited By ~ 1

Author(s):

F. TRAMONTANA ◽

L. GARDINI ◽

D. FOURNIER-PRUNARET ◽

P. CHARGE

Keyword(s):

Focal Point ◽

Singular Line ◽

Two Dimensional ◽

Focal Points ◽

Singular Set ◽

One Dimensional ◽

Attracting Set ◽

Straight Line ◽

Special Character ◽

Stable And Unstable Sets

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.

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