On Some Properties of Focal Points

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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PLANE MAPS WITH DENOMINATOR I: SOME GENERIC PROPERTIES

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127499000079 ◽

1999 ◽

Vol 09 (01) ◽

pp. 119-153 ◽

Cited By ~ 41

Author(s):

GIAN-ITALO BISCHI ◽

LAURA GARDINI ◽

CHRISTIAN MIRA

Keyword(s):

Invariant Sets ◽

Two Dimensional ◽

Focal Points ◽

Generic Properties ◽

New Concepts ◽

Dynamical Properties ◽

Basin Boundaries ◽

Dynamic Phenomena

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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TERBIUM MOLYBDATE: GROUP THEORY AND FLUCTUATIONS

Modern Physics Letters B ◽

10.1142/s0217984987000338 ◽

1987 ◽

Vol 01 (05n06) ◽

pp. 239-244

Author(s):

SERGE GALAM

Keyword(s):

Fixed Point ◽

Group Theory ◽

Parameter Space ◽

Landau Theory ◽

Two Dimensional ◽

Stable Fixed Point ◽

Theory Analysis ◽

Continuous Transition ◽

Dimensional Parameter ◽

First Order

A new mechanism to explain the first order ferroelastic—ferroelectric transition in Terbium Molybdate (TMO) is presented. From group theory analysis it is shown that in the two-dimensional parameter space ordering along either an axis or a diagonal is forbidden. These symmetry-imposed singularities are found to make the unique stable fixed point not accessible for TMO. A continuous transition even if allowed within Landau theory is thus impossible once fluctuations are included. The TMO transition is therefore always first order. This explanation is supported by experimental results.

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Chaotic Itinerancy Observed in Mutually Coupled Gaussian Maps

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127418300112 ◽

2018 ◽

Vol 28 (04) ◽

pp. 1830011

Author(s):

Mio Kobayashi ◽

Tetsuya Yoshinaga

Keyword(s):

Dynamical System ◽

Fixed Point ◽

Chaotic Behavior ◽

Periodic Points ◽

Two Dimensional ◽

Stable Fixed Point ◽

Bifurcation Structure ◽

Unstable Fixed Point ◽

Gaussian Map ◽

Gaussian Maps

A one-dimensional Gaussian map defined by a Gaussian function describes a discrete-time dynamical system. Chaotic behavior can be observed in both Gaussian and logistic maps. This study analyzes the bifurcation structure corresponding to the fixed and periodic points of a coupled system comprising two Gaussian maps. The bifurcation structure of a mutually coupled Gaussian map is more complex than that of a mutually coupled logistic map. In a coupled Gaussian map, it was confirmed that after a stable fixed point or stable periodic points became unstable through the bifurcation, the points were able to recover their stability while the system parameters were changing. Moreover, we investigated a parameter region in which symmetric and asymmetric stable fixed points coexisted. Asymmetric unstable fixed point was generated by the [Formula: see text]-type branching of a symmetric stable fixed point. The stability of the unstable fixed point could be recovered through period-doubling and tangent bifurcations. Furthermore, a homoclinic structure related to the occurrence of chaotic behavior and invariant closed curves caused by two-periodic points was observed. The mutually coupled Gaussian map was merely a two-dimensional dynamical system; however, chaotic itinerancy, known to be a characteristic property associated with high-dimensional dynamical systems, was observed. The bifurcation structure of the mutually coupled Gaussian map clearly elucidates the mechanism of chaotic itinerancy generation in the two-dimensional coupled map. We discussed this mechanism by comparing the bifurcation structures of the Gaussian and logistic maps.

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Dynamical properties of ring dark soliton in quasi-two dimensional Bose-Einstein condensates

Acta Physica Sinica ◽

10.7498/aps.57.6786 ◽

2008 ◽

Vol 57 (11) ◽

pp. 6786

Author(s):

Zhang Wei-Xi ◽

Wang Deng-Long ◽

Ding Jian-Wen

Keyword(s):

Dark Soliton ◽

Two Dimensional ◽

Dynamical Properties ◽

Bose Einstein

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Two-dimensional motion of a rolling non-circular cylinder on a flat horizontal surface

Canadian Journal of Physics ◽

10.1139/cjp-2017-0423 ◽

2018 ◽

Vol 96 (6) ◽

pp. 627-632

Author(s):

Amir Aghamohammadi ◽

Mohammad Khorrami

Keyword(s):

Circular Cylinder ◽

Energy Conservation ◽

Cross Section ◽

Equilibrium Configuration ◽

Focal Point ◽

Center Of Mass ◽

Horizontal Surface ◽

Two Dimensional ◽

Small Oscillations ◽

Equilibrium Configurations

The two dimensional motion of a generally non-circular non-uniform cylinder on a flat horizontal surface is investigated. Assuming that the cylinder does not slip, energy conservation is used to study the motion in general. Points of returns, and small oscillations around equilibrium configuration are studied. As examples, cylinders are studied for which the cross section is an ellipse, with the center of mass at the center of the ellipse or at a focal point, and the frequencies of small oscillations around their equilibrium configurations are found. The conditions for losing contact or sliding are also investigated. Finally, the motion is studied in more detail for the case of a nearly circular cylinder.

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Dimension and Dimensional Reduction in Quantum Gravity

Universe ◽

10.3390/universe5030083 ◽

2019 ◽

Vol 5 (3) ◽

pp. 83 ◽

Cited By ~ 9

Author(s):

Steven Carlip

Keyword(s):

Fixed Point ◽

Quantum Gravity ◽

Dimensional Reduction ◽

Short Distance ◽

Evidence Based ◽

Two Dimensional ◽

Anomalous Scaling ◽

Lines Of Evidence ◽

Quantum Spacetime

If gravity is asymptotically safe, operators will exhibit anomalous scaling at the ultraviolet fixed point in a way that makes the theory effectively two-dimensional. A number of independent lines of evidence, based on different approaches to quantization, indicate a similar short-distance dimensional reduction. I will review the evidence for this behavior, emphasizing the physical question of what one means by “dimension” in a quantum spacetime, and will discuss possible mechanisms that could explain the universality of this phenomenon.

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On the Measure-Preserving Mappings with Three-Dimensions

International Astronomical Union Colloquium ◽

10.1017/s0252921100065957 ◽

1993 ◽

Vol 132 ◽

pp. 73-89

Author(s):

Yi-Sui Sun

Keyword(s):

Fixed Point ◽

Invariant Manifolds ◽

Three Dimensional ◽

Invariant Curve ◽

Three Dimensions ◽

Two Dimensional ◽

Kolmogorov Entropy ◽

Numerical Exploration ◽

Area Preserving ◽

Dimensional Mapping

AbstractWe have systematically made the numerical exploration about the perturbation extension of area-preserving mappings to three-dimensional ones, in which the fixed points of area preserving are elliptic, parabolic or hyperbolic respectively. It has been observed that: (i) the invariant manifolds in the vicinity of the fixed point generally don’t exist (ii) when the invariant curve of original two-dimensional mapping exists the invariant tubes do also in the neighbourhood of the invariant curve (iii) for the perturbation extension of area-preserving mapping the invariant manifolds can only be generated in the subset of the invariant manifolds of original two-dimensional mapping, (iv) for the perturbation extension of area preserving mappings with hyperbolic or parabolic fixed point the ordered region near and far from the invariant curve will be destroyed by perturbation more easily than the other one, This is a result different from the case with the elliptic fixed point. In the latter the ordered region near invariant curve is solid. Some of the results have been demonstrated exactly.Finally we have discussed the Kolmogorov Entropy of the mappings and studied some applications.

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Global Dynamics of Some Exponential Type Systems

Discrete Dynamics in Nature and Society ◽

10.1155/2020/1515730 ◽

2020 ◽

Vol 2020 ◽

pp. 1-24

Author(s):

A. Q. Khan ◽

H. M. Arshad ◽

B. A. Younis ◽

KH. I. Osman ◽

Tarek F. Ibrahim ◽

...

Keyword(s):

Fixed Point ◽

Lyapunov Function ◽

Exponential Type ◽

Type Systems ◽

Global Dynamics ◽

Dynamical Properties ◽

Boundedness And Persistence ◽

Invariant Rectangle ◽

Theoretical Results ◽

Positive Fixed Point

We explore the boundedness and persistence, existence of an invariant rectangle, local dynamical properties about the unique positive fixed point, global dynamics by the discrete-time Lyapunov function, and the rate of convergence of some 2,3-type exponential systems of difference equations. Finally, theoretical results are numerically verified.

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