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

2013 ◽  
Vol 34 (5) ◽  
pp. 1538-1566 ◽  
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.

scholarly journals Physical measures for infinitely renormalizable Lorenz maps

2016 ◽  
Vol 38 (2) ◽  
pp. 717-738 ◽  
Author(s):  
M. MARTENS ◽  
B. WINCKLER
Keyword(s):  
Critical Point ◽  
Critical Points ◽  
Crucial Role ◽  
A Priori ◽  
Physical Measure ◽  
Unimodal Maps ◽  
Statistical Behavior ◽  
Physical Measures ◽  
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.

scholarly journals Regular or stochastic dynamics in real analytic families of unimodal maps

2003 ◽  
Vol 154 (3) ◽  
pp. 451-550 ◽  
Author(s):  
Artur Avila ◽  
Mikhail Lyubich ◽  
Welington de Melo
Keyword(s):  
Stochastic Dynamics ◽  
Unimodal Maps ◽  
Real Analytic

Phase space correspondence between Jordan and Einstein frames

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.

scholarly journals Chaotic period doubling

2009 ◽  
Vol 29 (2) ◽  
pp. 381-418 ◽  
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.

The Stages of Team Venture Formation: A Decision-making Model

1993 ◽  
Vol 17 (2) ◽  
pp. 17-27 ◽  
Author(s):  
Judith B. Kamm ◽  
Aaron J. Nurick
Keyword(s):  
Decision Making ◽  
Social Networks ◽  
Critical Points ◽  
A Priori ◽  
Feedback Loops ◽  
Second Stage ◽  
Or Groups ◽  
Decision Making Model

This model of multi-founder organizational formation assumes that organizations emerge In stages, following an a priori sequence of transitions. The idea stage comes first. In it, individuals or groups within the context of their social networks make decisions about the business concept and what Is needed to implement it. The second stage consists of implementation decisions, Including who will supply resources, what Inducements will be used to attract more partners if necessary, and how the team will be kept together. Feedback loops Indicate that the process may return to the concept and implementation needs decisions, depending upon choices made at certain critical points.

scholarly journals EQUILIBRIUM KAWASAKI DYNAMICS OF CONTINUOUS PARTICLE SYSTEMS

2007 ◽  
Vol 10 (02) ◽  
pp. 185-209 ◽  
Author(s):  
YURI KONDRATIEV ◽  
EUGENE LYTVYNOV ◽  
MICHAEL RÖCKNER
Keyword(s):  
Stochastic Dynamics ◽  
A Priori ◽  
Scaling Limit ◽  
Glauber Dynamics ◽  
Particle Systems ◽  
Death Process ◽  
Kawasaki Dynamics ◽  
Interacting Particles ◽  
Lattice Spin Systems ◽  
Diffusion Dynamics

We construct a new equilibrium dynamics of infinite particle systems in a Riemannian manifold X. This dynamics is an analog of the Kawasaki dynamics of lattice spin systems. The Kawasaki dynamics now is a process where interacting particles randomly hop over X. We establish conditions on the a priori explicitly given symmetrizing measure and the generator of this dynamics, under which a corresponding conservative Markov processes exists. We also outline two types of scaling limit of the equilibrium Kawasaki dynamics: one leading to an equilibrium Glauber dynamics in continuum (a birth-and-death process), and the other leading to a diffusion dynamics of interacting particles (in particular, the gradient stochastic dynamics).

BUILDING ELECTRONIC BURSTERS WITH THE MORRIS–LECAR NEURON MODEL

2006 ◽  
Vol 16 (12) ◽  
pp. 3617-3630 ◽  
Author(s):  
ALEXANDRE WAGEMAKERS ◽  
MIGUEL A. F. SANJUÁN ◽  
JOSÉ M. CASADO ◽  
KAZUYUKI AIHARA
Keyword(s):  
Phase Space ◽  
Parameter Space ◽  
Neuron Model ◽  
Electronic Circuit ◽  
Initial Point ◽  
Bifurcation Diagrams ◽  
Design And Implementation ◽  
Bursting Neurons ◽  
Fast Spiking ◽  
Dynamical Features

We propose a method for the design of electronic bursting neurons, based on a simple conductance neuron model. A burster is a particular class of neuron that displays fast spiking regimes alternating with resting periods. Our method is based on the use of an electronic circuit that implements the well-known Morris–Lecar neuron model. We use this circuit as a tool of analysis to explore some regions of the parameter space and to contruct several bifurcation diagrams displaying the basic dynamical features of that system. These bifurcation diagrams provide the initial point for the design and implementation of electronic bursting neurons. By extending the phase space with the introduction of a slow driving current, our method allows to exploit the bistabilities which are present in the Morris–Lecar system to the building of different bursting models.

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