Deterministic chaos in one-dimensional maps—the period doubling and intermittency routes

Pramana ◽  
1992 ◽  
Vol 39 (3) ◽  
pp. 193-252 ◽  
Author(s):  
G Ambika ◽  
K Babu Joseph
Keyword(s):  
Deterministic Chaos ◽  
Period Doubling ◽  
One Dimensional ◽  
One Dimensional Maps

ROUTE TO CHAOS VIA INVERSE CASCADE OF CONTINUOUS BIFURCATIONS

1995 ◽  
Vol 05 (01) ◽  
pp. 123-132 ◽  
Author(s):  
M. GUTMAN ◽  
V. GONTAR
Keyword(s):  
Period Doubling ◽  
Scaling Behavior ◽  
One Dimensional ◽  
Inverse Cascade ◽  
Discontinuous Bifurcation ◽  
Universal Properties ◽  
Route To Chaos ◽  
Piecewise Continuous ◽  
One Dimensional Maps

A route to chaos via an inverse cascade of continuous bifurcations that arithmetically reduce the period of successive orbits has been obtained for piecewise continuous one-dimensional maps. We have studied the mechanism of these bifurcations and established that their scaling behavior is governed by constants with new universal properties. The possibility of obtaining discontinuous bifurcation from any selected orbit of a cascade of period-doubling to any orbit of inverse cascade has been demonstrated.

scholarly journals INVESTIGATION OF DIFFERENCE EQUATIONS WITH A RATIONAL RIGHT-HAND SIDES

2020 ◽  
Vol 8 (2) ◽  
Author(s):  
I. Klevchuk
Keyword(s):  
Difference Equations ◽  
Symbolic Dynamics ◽  
Piecewise Linear ◽  
Discrete Dynamical System ◽  
Deterministic Chaos ◽  
Period Doubling ◽  
Rational Functions ◽  
Invariant Set ◽  
One Dimensional ◽  
Universal Properties

The aim of the present article is to investigate of some properties of solutions of nonli- near difference equations. A period doubling bifurcation in a discrete dynamical system leads to the appearance of deterministic chaos. We use permutable rational functions for study of some classes of one-dimensional mappings. Also n-dimensional generalizations of permutable polynomials may be obtained. We investigate polynomial and rational mappings with invariant measure and construct equivalent piecewise linear mappings. These mappings have countably many cycles. We applied the methods of symbolic dynamics to the theory of unimodal mappi- ngs. We use whole p-adic numbers for study the invariant set of some mapping in the theory of universal properties of one-parameter families. Feigenbaum constants play an important role in this theory.

scholarly journals Type-II intermittency in a class of two coupled one-dimensional maps

2000 ◽  
Vol 5 (3) ◽  
pp. 233-245 ◽  
Author(s):  
J. Laugesen ◽  
E. Mosekilde ◽  
T. Bountis ◽  
S. P. Kuznetsov
Keyword(s):  
Period Doubling ◽  
Coupled System ◽  
Two Dimensions ◽  
Type Ii ◽  
One Dimensional ◽  
System A ◽  
Subcritical Hopf Bifurcation ◽  
Theoretical Predictions ◽  
The Individual ◽  
One Dimensional Maps

The paper shows how intermittency behavior of type-II can arise from the coupling of two one-dimensional maps, each exhibiting type-III intermittency. This change in dynamics occurs through the replacement of a subcritical period-doubling bifurcation in the individual map by a subcritical Hopf bifurcation in the coupled system. A variety of different parameter combinations are considered, and the statistics for the distribution of laminar phases is worked out. The results comply well with theoretical predictions. Provided that the reinjection process is reasonably uniform in two dimensions, the transition to type-II intermittency leads directly to higher order chaos. Hence, this transition represents a universal route to hyperchaos.

scholarly journals Universal Features of Tangent Bifurcation

1985 ◽  
Vol 38 (1) ◽  
pp. 1 ◽  
Author(s):  
R Delbourgo ◽  
BG Kenny
Keyword(s):  
Renormalization Group ◽  
Limit Cycles ◽  
Period Doubling ◽  
Accumulation Point ◽  
Universal Functions ◽  
One Dimensional ◽  
Chaotic Region ◽  
Renormalization Group Equations ◽  
Tangent Bifurcation ◽  
One Dimensional Maps

We exhibit certain universal characteristics of limit cycles pertaining to one-dimensional maps in the 'chaotic' region beyond the point of accumulation connected with period doubling. Universal, Feigenbaum-type numbers emerge for different sequences, such as triplication. More significantly we have established the existence of different classes of universal functions which satisfy the same renormalization group equations, with the same parameters, as the appropriate accumulation point is reached.

scholarly journals On the finiteness of attractors for one-dimensional maps with discontinuities

2021 ◽  
Vol 389 ◽  
pp. 107891
Author(s):  
P. Brandão ◽  
J. Palis ◽  
V. Pinheiro
Keyword(s):  
One Dimensional ◽  
One Dimensional Maps

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.

scholarly journals Weighted kneading theory of one-dimensional maps with a hole

2004 ◽  
Vol 2004 (38) ◽  
pp. 2019-2038 ◽  
Author(s):  
J. Leonel Rocha ◽  
J. Sousa Ramos
Keyword(s):  
Hausdorff Dimension ◽  
Power Series ◽  
Formal Power Series ◽  
Topological Entropy ◽  
Escape Rate ◽  
One Dimensional ◽  
One Dimensional Maps ◽  
Formal Power ◽  
Kneading Theory

The purpose of this paper is to present a weighted kneading theory for one-dimensional maps with a hole. We consider extensions of the kneading theory of Milnor and Thurston to expanding discontinuous maps with a hole and introduce weights in the formal power series. This method allows us to derive techniques to compute explicitly the topological entropy, the Hausdorff dimension, and the escape rate.

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