Unimodal mappings and Li-Yorke chaos

1998 ◽  
Vol 63 (5) ◽  
pp. 598-607
Author(s):  
V. A. Dobrynskii
Keyword(s):  
Unimodal Mappings

scholarly journals Dynamics and Universality of Unimodal Mappings with Infinite Criticality

2005 ◽  
Vol 258 (1) ◽  
pp. 103-133 ◽  
Author(s):  
Genadi Levin ◽  
Grzegorz Światek
Keyword(s):  
Unimodal Mappings

Some properties of one-dimensional unimodal mappings

2010 ◽  
Vol 81 (1) ◽  
pp. 16-21
Author(s):  
E. V. Nedostupov ◽  
D. A. Sarancha ◽  
E. N. Chigerev ◽  
Yu. S. Yurezanskaya
Keyword(s):  
One Dimensional ◽  
Unimodal Mappings

Asymptotic rigidity of scaling ratios for critical circle mappings

1999 ◽  
Vol 19 (4) ◽  
pp. 995-1035 ◽  
Author(s):  
EDSON DE FARIA
Keyword(s):  
Critical Point ◽  
Surgical Techniques ◽  
Careful Analysis ◽  
The Family ◽  
Critical Circle ◽  
Renormalization Operator ◽  
Irrational Rotation Number ◽  
Circle Mappings ◽  
Commuting Pairs ◽  
Unimodal Mappings

Let $f$ be a smooth homeomorphism of the circle having one cubic-exponent critical point and irrational rotation number of bounded combinatorial type. Using certain pull-back and quasiconformal surgical techniques, we prove that the scaling ratios of $f$ about the critical point are asymptotically independent of $f$. This settles in particular the golden mean universality conjecture. We introduce the notion of a holomorphic commuting pair, a complex dynamical system that, in the analytic case, represents an extension of $f$ to the complex plane and behaves somewhat as a quadratic-like mapping. We define a suitable renormalization operator that acts on such objects. Through careful analysis of the family of entire mappings given by $z\mapsto z+\theta -(1/2\pi)\sin{2\pi z}$, $\theta$ real, we construct examples of holomorphic commuting pairs, from which certain necessary limit set pre-rigidity results are extracted. The rigidity problem for $f$ is thereby reduced to one of renormalization convergence. We handle this last problem by means of Teichmüller extremal methods made available through the recent work of Sullivan on Riemann surface laminations and renormalization of unimodal mappings.

On the absence of cycles for unimodal mappings of the square

1998 ◽  
Vol 63 (3) ◽  
pp. 325-332
Author(s):  
V. A. Dobrynskii
Keyword(s):  
Unimodal Mappings

О КАЛИБРОВКЕ АВТОНОМНОЙ МОДЕЛИ ТУНДРОВОЙ БИОЛОГИЧЕСКОЙ ПОПУЛЯЦИИ ЛЕММИНГОВ

2020 ◽  
Vol 65 (6) ◽  
pp. 1184-1195
Author(s):  
Г.К. Каменев ◽  
◽  
Д.А. Саранча ◽  
В.О. Поляновский ◽  
◽  
...  
Keyword(s):  
Error Function ◽  
Model Identification ◽  
Taimyr Peninsula ◽  
One Dimensional ◽  
Identification Criterion ◽  
Identification Sets Method ◽  
Using Data ◽  
Parameter Values ◽  
Unimodal Mappings

The paper presents an autonomous model of the biological population of lemmings of a phenomenological type, designed in complex studies of tundra communities. In the model, the dynamics of a population is described through a difference equation relating the population in two neighboring years that depends on three parameters of the biological and ecological genesis. The combination of parameter values included in the equation under consideration determines a class of one-dimensional unimodal mappings of a dynamical system in which the bifurcation properties, asymptotics, and stability of trajectories were analytically and numerically studied. In this paper the main focus is made on model identification criterion. The method of identification sets is proposed to be used for calibration of the model. The identification sets method is based on the approximation and visualization of small-dimensional projections of a multidimensional graph of the error function given in the space of three environmental and two population parameters. This paper describes a case study of model identification using data on the tundra lemming population on the Taimyr Peninsula. It is shown that in this case two biological and ecological parameters allow for stable location distribution.

Investigation of the Class of One-Dimensional Unimodal Mappings Obtained in the Modeling of the Lemming Population

BIOPHYSICS ◽  
2018 ◽  
Vol 63 (4) ◽  
pp. 596-610 ◽  
Author(s):  
G. K. Kamenev ◽  
D. A. Sarancha ◽  
V. O. Polyanovsky
Keyword(s):  
One Dimensional ◽  
Unimodal Mappings
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