Non-accessible critical points of Cremer polynomials

This paper researches the dynamic behavior of a general form of the Fibonacci function, which is a quasi-sine Fibonacci function. It analyses the fixed points of the quasi-sine Fibonacci function on the real axis and the complex plane, and then constructs the Julia set of it using the escape-time method, discovering that the Julia set is fractal and it is on the x-axis symmetry. Using the conception of critical point, the quasi-sine Fibonacci function is generalized. Later the paper examines the dynamic behavior of the generalized quasi-sine Fibonacci function on critical points, and finds that the Mandelbrot set is also on the x-axis symmetry. Finally, it is discovered that there is a jumping phenomenon on the critical points.

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Bifurcations of Heteroclinic Loop with Twisted Conditions

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127417501206 ◽

2017 ◽

Vol 27 (08) ◽

pp. 1750120

Author(s):

Yinlai Jin ◽

Suoling Yang ◽

Yuanyuan Liu ◽

Dandan Xie ◽

Nana Zhang

Keyword(s):

Periodic Orbit ◽

Critical Point ◽

Critical Points ◽

Unperturbed System ◽

Homoclinic Loop ◽

Heteroclinic Loop ◽

Principal Eigenvalues ◽

Bifurcation Problems

The bifurcation problems of twisted heteroclinic loop with two hyperbolic critical points are studied for the case [Formula: see text], [Formula: see text], [Formula: see text], where [Formula: see text] and [Formula: see text] are the pair of principal eigenvalues of unperturbed system at the critical point [Formula: see text], [Formula: see text]. Under the transversal conditions, the authors obtained some results of the existence and the number of 1-homoclinic loop, 1-periodic orbit, double 1-periodic orbit, 2-homoclinic loop and 2-periodic orbit. Moreover, the relative bifurcation surfaces and the existence regions are given, and the corresponding bifurcation graphs are drawn.

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On the critical points of a polynomial

Bulletin of the Australian Mathematical Society ◽

10.1017/s000497270003152x ◽

1998 ◽

Vol 57 (1) ◽

pp. 173-174 ◽

Cited By ~ 2

Author(s):

Abdul Aziz ◽

B.A. Zargar

Keyword(s):

Convex Hull ◽

Critical Point ◽

Critical Points ◽

Complex Polynomial ◽

Open Ball

Let p be a complex polynomial, of the form , where |zk| ≥ 1 when 1 ≤ k ≤ n − 1. Then p′(z) ≠ 0 if |z| /n.Let B(z, r) denote the open ball in with centre z and radius r, and denote its closure. The Gauss-Lucas theorem states that every critical point of a complex polynomial p of degree at least 2 lies in the convex hull of its zeros. This theorem has been further investigated and developed. B. Sendov conjectured that, if all the zeros of p lie in then, for any zero ζ of p, the disc contains at least one zero of p′; see [3, Problem 4.1]. This conjecture has attracted much attention-see, for example, [1], and the papers cited there. In connection with this conjecture, Brown [2] posed the following problem.

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Phase space correspondence between Jordan and Einstein frames

International Journal of Modern Physics D ◽

10.1142/s0218271821500875 ◽

2021 ◽

pp. 2150087

Author(s):

Amin Salehi

Keyword(s):

Phase Space ◽

Critical Point ◽

Critical Points ◽

Dynamical Behavior ◽

Cosmological Parameters ◽

Jordan Frame ◽

Field Equations ◽

Theories Of Gravity ◽

The Stability ◽

Jordan And Einstein Frames

Scalar–tensor theories of gravity can be formulated in the Einstein frame or in the Jordan frame (JF) which are related with each other by conformal transformations. Although the two frames describe the same physics and are equivalent, the stability of the field equations in the two frames is not the same. Here, we implement dynamical system and phase space approach as a robustness tool to investigate this issue. We concentrate on the Brans–Dicke theory in a Friedmann–Lemaitre–Robertson–Walker universe, but the results can easily be generalized. Our analysis shows that while there is a one-to-one correspondence between critical points in two frames and each critical point in one frame is mapped to its corresponds in another frame, however, stability of a critical point in one frame does not guarantee the stability in another frame. Hence, an unstable point in one frame may be mapped to a stable point in another frame. All trajectories between two critical points in phase space in one frame are different from their corresponding in other ones. This indicates that the dynamical behavior of variables and cosmological parameters is different in two frames. Hence, for those features of the study, which focus on observational measurements, we must use the JF where experimental data have their usual interpretation.

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UNIMODAL INTERVAL MAPS OBTAINED FROM THE MODIFIED CHUA EQUATIONS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127493000246 ◽

1993 ◽

Vol 03 (02) ◽

pp. 323-332 ◽

Cited By ~ 4

Author(s):

MICHAŁ MISIUREWICZ

Keyword(s):

Periodic Orbit ◽

Critical Point ◽

Real Line ◽

Positive Measure ◽

Schwarzian Derivative ◽

Large Range ◽

Unimodal Maps ◽

Interval Maps ◽

Central Piece ◽

Piecewise Continuous

Following Brown [1992, 1993] we study maps of the real line into itself obtained from the modified Chua equations. We fix our attention on a one-parameter family of such maps, which seems to be typical. For a large range of parameters, invariant intervals exist. In such an invariant interval, the map is piecewise continuous, with most of pieces of continuity mapped in a monotone way onto the whole interval. However, on the central piece there is a critical point. This allows us to find sometimes a smaller invariant interval on which the map is unimodal. In such a way, we get one-parameter families of smooth unimodal maps, very similar to the well-known family of logistic maps x ↦ ax(1−x). We study more closely one of those and show that these maps have negative Schwarzian derivative. This implies the existence of at most one attracting periodic orbit. Moreover, there is a set of parameters of positive measure for which chaos occurs.

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Global Phase Portrait and Bifurcation Diagram for a Multi-Parametric Linear System

Journal of Southwest Jiaotong University ◽

10.35741/issn.0258-2724.55.4.28 ◽

2020 ◽

Vol 55 (4) ◽

Author(s):

Jorge Rodríguez Contreras ◽

Alberto Reyes Linero ◽

Juliana Vargas Sánchez

Keyword(s):

Linear System ◽

Critical Point ◽

Phase Portrait ◽

Bifurcation Diagram ◽

Critical Points ◽

Global Dynamics ◽

Parametric Surfaces ◽

Critical Points At Infinity ◽

Global Phase Portrait ◽

The Stability

The goal of this article is to conduct a global dynamics study of a linear multiparameter system (real parameters (a,b,c) in R^3); for this, we take the different changes that these parameters present. First, we find the different parametric surfaces in which the space is divided, where the stability of the critical point is defined; we then create a bifurcation diagram to classify the different bifurcations that appear in the system. Finally, we determine and classify the critical points at infinity, considering the canonical shape of the Poincaré sphere, and thus, obtain a global phase portrait of the multiparametric linear system.

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A method to estimate statistical errors of properties derived from charge-density modelling

Acta Crystallographica Section A Foundations and Advances ◽

10.1107/s2053273318004308 ◽

2018 ◽

Vol 74 (3) ◽

pp. 170-183 ◽

Cited By ~ 11

Author(s):

Bertrand Fournier ◽

Benoît Guillot ◽

Claude Lecomte ◽

Eduardo C. Escudero-Adán ◽

Christian Jelsch

Keyword(s):

Charge Density ◽

Least Squares ◽

Critical Point ◽

Critical Points ◽

Property Values ◽

Density Model ◽

Interaction Energies ◽

Topological Properties ◽

A Charge ◽

Density Modelling

Estimating uncertainties of property values derived from a charge-density model is not straightforward. A methodology, based on calculation of sample standard deviations (SSD) of properties using randomly deviating charge-density models, is proposed with theMoProsoftware. The parameter shifts applied in the deviating models are generated in order to respect the variance–covariance matrix issued from the least-squares refinement. This `SSD methodology' procedure can be applied to estimate uncertainties ofanyproperty related to a charge-density model obtained by least-squares fitting. This includes topological properties such as critical point coordinates, electron density, Laplacian and ellipticity at critical points and charges integrated over atomic basins. Errors on electrostatic potentials and interaction energies are also available now through this procedure. The method is exemplified with the charge density of compound (E)-5-phenylpent-1-enylboronic acid, refined at 0.45 Å resolution. The procedure is implemented in the freely availableMoProprogram dedicated to charge-density refinement and modelling.

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II. On the critical state of gases

Proceedings of the Royal Society of London ◽

10.1098/rspl.1879.0126 ◽

1880 ◽

Vol 30 (200-205) ◽

pp. 323-329 ◽

Cited By ~ 6

Keyword(s):

Critical Point ◽

Critical Points ◽

Critical State ◽

Atmospheric Pressure ◽

Boiling Point ◽

Chemical Society ◽

Boiling Points ◽

Intermediate Condition ◽

Specific Volumes

In a paper read before the Chemical Society, in May, 1879, I gave an account of a method of determining what is termed by Kopp the “specific volumes” of liquids; that was shown to be the volume of liquid at its boiling-point, at ordinary atmospheric pressure, obtainable from 22,326 volumes of its gas, supposed to exist at 0°. Being desirous of extending these researches, with the view of ascertaining such relations at higher temperatures, since April, 1879, I have made numerous experiments, the results of, and deductions from which I hope to publish before long. The temperatures observed vary from the boiling-points of the liquids examined, to about 50° above their critical points; and in course of these experiments I have noticed some curious facts, which may not be unworthy of the attention of the Society. It is well known that at temperatures above that which produces what is termed by Dr. Andrews the “critical point” of a liquid, the substance is supposed to exist in a peculiar condition, and Dr Andrews purposely abstained from speculating on the nature of the matter, whether it be liquid or gaseous, or in an intermediate condition, to which no name has been given. As my observations bear directly on this point, it may be advisable first to describe the experiments I have made, and then to draw the deductions which appear to follow from them.

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Meromorphic multifunctions in complex dynamics

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700006568 ◽

1992 ◽

Vol 12 (1) ◽

pp. 39-52 ◽

Cited By ~ 9

Author(s):

L. Baribeau ◽

T. J. Ransford

Keyword(s):

Critical Points ◽

Complex Dynamics ◽

Julia Set ◽

Kleinian Groups ◽

Rational Maps ◽

Analytic Family ◽

Limit Sets ◽

Upper Semicontinuous ◽

Open Set ◽

Similar Vein

AbstractLet {RA} be an analytic family of rational maps and denote by j(λ) the Julia set of Rλ. We prove that the upper semicontinuous regularization j(λ) of j(λ) (which coincides with j(λ) for all λ in a dense open set) is a meromorphic multifunction, and give applications that illustrate the instability of Julia sets. In a similar vein, we also consider forward orbits of critical points and limit sets of Kleinian groups.

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Lagrangean conditions and quasiduality

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700023431 ◽

1977 ◽

Vol 16 (3) ◽

pp. 325-339 ◽

Cited By ~ 89

Author(s):

B.D. Craven

Keyword(s):

Critical Point ◽

Critical Points ◽

Local Minimum ◽

Constraint Qualification ◽

Necessary Conditions ◽

Weak Duality ◽

Partially Ordered Vector Space ◽

Counter Example ◽

Cone Constraints ◽

Constrained Minimization Problem

For a constrained minimization problem with cone constraints, lagrangean necessary conditions for a minimum are well known, but are subject to certain hypotheses concerning cones. These hypotheses are now substantially weakened, but a counter example shows that they cannot be omitted altogether. The theorem extends to minimization in a partially ordered vector space, and to a weaker kind of critical point (a quasimin) than a local minimum. Such critical points are related to Kuhn-Tucker conditions, assuming a constraint qualification; in certain circumstances, relevant to optimal control, such a critical point must be a minimum. Using these generalized critical points, a theorem analogous to duality is proved, but neither assuming convexity, nor implying weak duality.

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