scholarly journals The attractor of piecewise expanding maps of the interval

2019 ◽  
Vol 20 (02) ◽  
pp. 2050009 ◽  
Author(s):  
Gianluigi Del Magno ◽  
João Lopes Dias ◽  
Pedro Duarte ◽  
José Pedro Gaivão
Keyword(s):  
Periodic Orbits ◽  
Elementary Proof ◽  
Expanding Maps ◽  
Small Perturbations ◽  
Topological Description ◽  
Piecewise Expanding Maps

We consider piecewise expanding maps of the interval with finitely many branches of monotonicity and show that they are generically combinatorially stable, i.e. the number of ergodic attractors and their corresponding mixing components do not change under small perturbations of the map. Our methods provide a topological description of the attractor and give an elementary proof of the density of periodic orbits.

scholarly journals Coupled-expanding maps under small perturbations

2011 ◽  
Vol 29 (3) ◽  
pp. 1291-1307 ◽  
Author(s):  
Xu Zhang ◽  
◽  
Yuming Shi ◽  
Guanrong Chen ◽  
Keyword(s):  
Expanding Maps ◽  
Small Perturbations

BIFURCATIONS OF PERIODIC ORBITS IN A JOSEPHSON EQUATION WITH A PHASE SHIFT

2002 ◽  
Vol 12 (07) ◽  
pp. 1515-1530 ◽  
Author(s):  
ZHUJUN JING ◽  
HONGJUN CAO
Keyword(s):  
Numerical Simulation ◽  
Phase Shift ◽  
Periodic Orbits ◽  
Averaging Method ◽  
Melnikov Function ◽  
Second Order ◽  
Constant Current ◽  
Small Perturbations ◽  
Theoretical Predictions ◽  
Simulation Results

The Josephson equation with constant current and sinusoidal forcings and a phase shift is investigated in detail: the existence and the bifurcations of harmonics and subharmonics under small perturbations are given, by using the second-order averaging method and Melnikov function; the influence on bifurcations of periodic or subharmonics as the phase shift varies is considered; some numerical simulation results are reported in order to prove our theoretical predictions.

scholarly journals Shrinking targets in parametrised families

2017 ◽  
Vol 166 (2) ◽  
pp. 265-295 ◽  
Author(s):  
MAGNUS ASPENBERG ◽  
TOMAS PERSSON
Keyword(s):  
Fixed Point ◽  
Hausdorff Dimension ◽  
Expanding Maps ◽  
Target Property ◽  
Non Linear ◽  
Piecewise Expanding Maps

AbstractWe consider certain parametrised families of piecewise expanding maps on the interval, and estimate and sometimes calculate the Hausdorff dimension of the set of parameters for which the orbit of a fixed point has a certain shrinking target property. This generalises several similar results for β-transformations to more general non-linear families. The proofs are based on a result by Schnellmann on typicality in parametrised families.

scholarly journals On the mixing properties of piecewise expanding maps under composition with permutations, II: Maps of non-constant orientation

2016 ◽  
Vol 16 (03) ◽  
pp. 1660013 ◽  
Author(s):  
Nigel P. Byott ◽  
Congping Lin ◽  
Yiwei Zhang
Keyword(s):  
Piecewise Linear ◽  
Unit Interval ◽  
Interval Exchange ◽  
Expanding Maps ◽  
Mixing Properties ◽  
Linear Maps ◽  
Piecewise Linear Maps ◽  
Fold Map ◽  
Constant Slope ◽  
Piecewise Expanding Maps

For an integer [Formula: see text], let [Formula: see text] be the partition of the unit interval [Formula: see text] into [Formula: see text] equal subintervals, and let [Formula: see text] be the class of piecewise linear maps on [Formula: see text] with constant slope [Formula: see text] on each element of [Formula: see text]. We investigate the effect on mixing properties when [Formula: see text] is composed with the interval exchange map given by a permutation [Formula: see text] interchanging the [Formula: see text] subintervals of [Formula: see text]. This extends the work in a previous paper [N. P. Byott, M. Holland and Y. Zhang, DCDS 33 (2013) 3365–3390], where we considered only the “stretch-and-fold” map [Formula: see text].

DYNAMICS OF THE MUTHUSWAMY–CHUA SYSTEM

2013 ◽  
Vol 23 (08) ◽  
pp. 1350136 ◽  
Author(s):  
YUANFAN ZHANG ◽  
XIANG ZHANG
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Chaotic Attractor ◽  
Electronic Circuit ◽  
Small Perturbations ◽  
Chaotic Phenomena ◽  
Invariant Surfaces

The Muthuswamy–Chua system [Formula: see text] describes the simplest electronic circuit which can have chaotic phenomena. In this paper, we first prove the existence of three families of consecutive periodic orbits of the system when α = 0, two of which are located on consecutive invariant surfaces and form two invariant topological cylinders. Then we prove that for α > 0 if the system has a periodic orbit or a chaotic attractor, it must intersect both of the planes z = 0 and z = -1 infinitely many times as t tends to infinity. As a byproduct, we get an example of unstable invariant topological cylinders which are not normally hyperbolic and which are also destroyed under small perturbations.

Generation of Periodic Orbits from Chaotic Attractors of Weakly Coupled Diode Resonators

1997 ◽  
Vol 07 (04) ◽  
pp. 897-902
Author(s):  
Jong Cheol Shin ◽  
Sook-Il Kwun ◽  
Youngtae Kim
Keyword(s):  
Periodic Orbits ◽  
Chaotic Dynamics ◽  
Chaotic Attractors ◽  
Unstable Periodic Orbits ◽  
Chaotic Motions ◽  
Small Perturbations ◽  
Controlling Chaos ◽  
Weakly Coupled ◽  
Dynamical Properties

We have designed coupled diode resonators to study the effect of small perturbations due to weak symmetric coupling on chaotic dynamics. Our experiment clearly demonstrated that chaos of the diode resonators was suppressed so that chaotic motions were converted into periodic ones with small modifications to the attractor when an appropriate coupling signal perturbed the diode resonators. Many unstable periodic orbits were stabilized and they were very stable depending on the dynamical properties of the coupling signals. Our results suggest that coupling of signals belonging to the same class is effective in controlling chaos.

scholarly journals Piecewise expanding maps on the plane with singular ergodic properties

2000 ◽  
Vol 20 (6) ◽  
pp. 1851-1857 ◽  
Author(s):  
MASATO TSUJII
Keyword(s):  
Empirical Measure ◽  
Expanding Maps ◽  
Ergodic Properties ◽  
Expanding Map ◽  
Open Set ◽  
Point Measure ◽  
Piecewise Expanding Maps

For $1\le r<\infty$, we construct a piecewise $C^{r}$ expanding map $F:D\to D$ on the domain $D=(0,1)\times (-1,1)\subset\mathbb{R}^{2}$ with the following property: there exists an open set $B$ in $D$ such that the diameter of $F^{n}(B)$ converges to $0$ as $n\to\infty$ and the empirical measure $n^{-1}\sum_{k=0}^{n-1}\delta_{F^{k}(x)}$ converges to the point measure $\delta_{p}$ at $p=(0,0)$ as $n \to\infty$ for any point $x\in B$.

scholarly journals Harmonic Averages and New Explicit Constants for Invariant Densities of Piecewise Expanding Maps of the Interval

2012 ◽  
Vol 146 (4) ◽  
pp. 850-863 ◽  
Author(s):  
Paweł Góra ◽  
Zhenyang Li ◽  
Abraham Boyarsky ◽  
Harald Proppe
Keyword(s):  
Expanding Maps ◽  
Invariant Densities ◽  
Piecewise Expanding Maps
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