Physical measures for infinitely renormalizable Lorenz maps

A physical measure on the attractor of a system describes the statistical behavior of typical orbits. An example occurs in unimodal dynamics: namely, all infinitely renormalizable unimodal maps have a physical measure. For Lorenz dynamics, even in the simple case of infinitely renormalizable systems, the existence of physical measures is more delicate. In this article, we construct examples of infinitely renormalizable Lorenz maps which do not have a physical measure. A priori bounds on the geometry play a crucial role in (unimodal) dynamics. There are infinitely renormalizable Lorenz maps which do not have a priori bounds. This phenomenon is related to the position of the critical point of the consecutive renormalizations. The crucial technical ingredient used to obtain these examples without a physical measure is the control of the position of these critical points.

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Renormalization for Lorenz maps of monotone combinatorial types

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2017.12 ◽

2017 ◽

Vol 39 (1) ◽

pp. 132-158

Author(s):

DENIS GAIDASHEV

Keyword(s):

Critical Point ◽

A Priori ◽

Periodic Points ◽

Unit Interval ◽

Return Maps ◽

Lorenz Maps

Lorenz maps are maps of the unit interval with one critical point of order $\unicode[STIX]{x1D70C}>1$ and a discontinuity at that point. They appear as return maps of sections of the geometric Lorenz flow. We construct real a priori bounds for renormalizable Lorenz maps with certain monotone combinatorics and a sufficiently flat critical point, and use these bounds to show existence of periodic points of renormalization, as well as existence of Cantor attractors for dynamics of infinitely renormalizable Lorenz maps.

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Benhabib on Democratic Iterations in a Global Order

Law & Ethics of Human Rights ◽

10.2202/1938-2545.1015 ◽

2008 ◽

Vol 2 (1) ◽

pp. 1-14

Author(s):

Yossi Dahan ◽

Yossi Yonah

Keyword(s):

Critical Point ◽

Critical Points ◽

Crucial Role ◽

Critical Assessment ◽

Popular Sovereignty ◽

Global Order ◽

Democratic Ideal ◽

Normative Approach ◽

The Third ◽

Structural Failures

Seyla Benhabibs article, Twilight of Sovereignty or the Emergence of Cosmopolitan Norms offers a penetrating analysis of the contemporary global order and suggests a normative approach by which to mend its structural failuresviewed from the democratic ideal of popular sovereignty and guided by what she calls cosmopolitan norms.The authors take issue with Benhabib's position on both the descriptive and the normative grounds, and make three critical points in this matter: the first two points concern Benhabib's descriptive portrayal of the global order. The third critical point concerns her normative position, i.e., her ideal of the good (global) polity, displayed through her idea of democratic iteration operating through global civil society.The critical assessment of Benhabib's views ensues from the authors endorsement of the transformationalist positionthe state, although somewhat undermined by global processes, still possesses considerable power and maintains a crucial role in determining the trajectory of these processes.

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Regular or stochastic dynamics in families of higher-degree unimodal maps

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2013.10 ◽

2013 ◽

Vol 34 (5) ◽

pp. 1538-1566 ◽

Cited By ~ 2

Author(s):

TREVOR CLARK

Keyword(s):

Phase Space ◽

Parameter Space ◽

Critical Points ◽

Stochastic Dynamics ◽

A Priori ◽

Fixed Degree ◽

Unimodal Maps ◽

Hybrid Classes ◽

Real Analytic

AbstractWe construct a lamination of the space of unimodal maps with critical points of fixed degree $d\geq 2$ by the hybrid classes. The structure of the lamination yields a partition of the parameter space for one-parameter real analytic families of unimodal maps and allows us to transfer a priori bounds in the phase space to the parameter space. This implies that almost every map in such a family is either regular or stochastic.

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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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Chaotic period doubling

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385708080371 ◽

2009 ◽

Vol 29 (2) ◽

pp. 381-418 ◽

Cited By ~ 6

Author(s):

V. V. M. S. CHANDRAMOULI ◽

M. MARTENS ◽

W. DE MELO ◽

C. P. TRESSER

Keyword(s):

Fixed Point ◽

Asymptotic Behavior ◽

Chaotic Dynamics ◽

A Priori ◽

Period Doubling ◽

Small Scale ◽

Unimodal Maps ◽

One Dimensional ◽

Renormalization Operator ◽

Renormalization Theory

AbstractThe period doubling renormalization operator was introduced by Feigenbaum and by Coullet and Tresser in the 1970s to study the asymptotic small-scale geometry of the attractor of one-dimensional systems that are at the transition from simple to chaotic dynamics. This geometry turns out not to depend on the choice of the map under rather mild smoothness conditions. The existence of a unique renormalization fixed point that is also hyperbolic among generic smooth-enough maps plays a crucial role in the corresponding renormalization theory. The uniqueness and hyperbolicity of the renormalization fixed point were first shown in the holomorphic context, by means that generalize to other renormalization operators. It was then proved that, in the space ofC2+αunimodal maps, forα>0, the period doubling renormalization fixed point is hyperbolic as well. In this paper we study what happens when one approaches from below the minimal smoothness thresholds for the uniqueness and for the hyperbolicity of the period doubling renormalization generic fixed point. Indeed, our main result states that in the space ofC2unimodal maps the analytic fixed point is not hyperbolic and that the same remains true when adding enough smoothness to geta prioribounds. In this smoother class, calledC2+∣⋅∣, the failure of hyperbolicity is tamer than inC2. Things get much worse with just a bit less smoothness thanC2, as then even the uniqueness is lost and other asymptotic behavior becomes possible. We show that the period doubling renormalization operator acting on the space ofC1+Lipunimodal maps has infinite topological entropy.

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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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On α-Limit Sets in Lorenz Maps

Entropy ◽

10.3390/e23091153 ◽

2021 ◽

Vol 23 (9) ◽

pp. 1153

Author(s):

Łukasz Cholewa ◽

Piotr Oprocha

Keyword(s):

Topological Entropy ◽

Dynamical Behavior ◽

Unimodal Maps ◽

Limit Sets ◽

Lorenz Maps ◽

Provided Examples

The aim of this paper is to show that α-limit sets in Lorenz maps do not have to be completely invariant. This highlights unexpected dynamical behavior in these maps, showing gaps existing in the literature. Similar result is obtained for unimodal maps on [0,1]. On the basis of provided examples, we also present how the performed study on the structure of α-limit sets is closely connected with the calculation of the topological entropy.

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