STABLE OSCILLATIONS AND DEVIL'S STAIRCASE IN THE VAN DER POL OSCILLATOR

2000 ◽  
Vol 10 (01) ◽  
pp. 155-164 ◽  
Author(s):  
T. GILBERT ◽  
R. W. GAMMON
Keyword(s):  
Rotation Number ◽  
Variable Parameter ◽  
Relaxation Oscillator ◽  
Van Der Pol Oscillator ◽  
Driving Frequency ◽  
Devil's Staircase ◽  
Van Der Pol ◽  
Devil’S Staircase ◽  
Staircase Structure ◽  
Parameter Values

A forced van der Pol relaxation oscillator is studied experimentally in the regime of stable oscillations. The variable parameter is chosen to be the driving frequency. For a range of parameter values, we show that the rotation number varies continuously from 0 to 1. This work provides experimental evidence that period-adding bifurcations to chaos previously reported by Kennedy and Chua are intimately connected to the existence of a regime of stable oscillations where the rotation number shows a Devil's-staircase structure.

scholarly journals Frequency locking in an injection-locked frequency divider equation

2008 ◽  
Vol 465 (2101) ◽  
pp. 283-306 ◽  
Author(s):  
Michele V Bartuccelli ◽  
Jonathan H.B Deane ◽  
Guido Gentile
Keyword(s):  
Output Signal ◽  
Numerical Results ◽  
Frequency Divider ◽  
Driving Frequency ◽  
Devil's Staircase ◽  
Frequency Locking ◽  
Devil’S Staircase ◽  
Explicit Formulae ◽  
Staircase Structure ◽  
Driving Signal

We consider a model for a resonant injection-locked frequency divider, and study analytically the locking onto rational multiples of the driving frequency. We provide explicit formulae for the width of the plateaux appearing in the devil's staircase structure of the lockings, and in particular show that the largest plateaux correspond to even integer values for the ratio of the frequency of the driving signal to the frequency of the output signal. Our results prove the experimental and numerical results available in the literature.

On the Existence of Limit Cycles and Relaxation Oscillations in a 3D van der Pol-like Memristor Oscillator

2017 ◽  
Vol 27 (07) ◽  
pp. 1750102
Author(s):  
Marcelo Messias ◽  
Anderson L. Maciel
Keyword(s):  
Phase Space ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Relaxation Oscillator ◽  
Van Der Pol Oscillator ◽  
Relaxation Oscillations ◽  
Van Der Pol ◽  
Piecewise Linear System ◽  
Parameter Values

We study a van der Pol-like memristor oscillator, obtained by substituting a Chua’s diode with an active controlled memristor in a van der Pol oscillator with Chua’s diode. The mathematical model for the studied circuit is given by a three-dimensional piecewise linear system of ordinary differential equations, depending on five parameters. We show that this system has a line of equilibria given by the [Formula: see text]-axis and the phase space [Formula: see text] is foliated by invariant planes transverse to this line, which implies that the dynamics is essentially two-dimensional. We also show that in each of these invariant planes may occur limit cycles and relaxation oscillations (that is, nonsinusoidal repetitive (periodic) solutions), depending on the parameter values. Hence, the oscillator studied here, constructed with a memristor, is also a relaxation oscillator, as the original van der Pol oscillator, although with a main difference: in the case of the memristor oscillator, an infinity of oscillations are produced, one in each invariant plane, depending on the initial condition considered. We also give conditions for the nonexistence of oscillations, depending on the position of the invariant planes in the phase space.

ENTROPY OF MAPS WITH HORIZONTAL GAPS

2004 ◽  
Vol 14 (04) ◽  
pp. 1489-1492 ◽  
Author(s):  
MICHAŁ MISIUREWICZ
Keyword(s):  
Topological Entropy ◽  
Entropy Function ◽  
Devil's Staircase ◽  
Piecewise Monotone ◽  
Interval Maps ◽  
Continuous Map ◽  
Devil’S Staircase ◽  
Staircase Structure

We study the behavior of topological entropy in one-parameter families of interval maps obtained from a continuous map f by truncating it at the level depending on the parameter. When f is piecewise monotone, the entropy function has the devil's staircase structure.

scholarly journals Asymptotic Limit-cycle Analysis of Oscillating Chemical Reactions

2021 ◽  
Author(s):  
Alain Brizard ◽  
Samuel Berry
Keyword(s):  
Limit Cycle ◽  
Chemical Reactions ◽  
Relaxation Oscillation ◽  
Three Dimensional ◽  
Van Der Pol Oscillator ◽  
Asymptotic Limit ◽  
Two Dimensional ◽  
Van Der Pol ◽  
Parameter Values ◽  
Analytical Expressions

Abstract The asymptotic limit-cycle analysis of mathematical models for oscillating chemical reactions is presented. In this work, after a brief presentation of mathematical preliminaries applied to the biased Van der Pol oscillator, we consider a two-dimensional model of the Chlorine dioxide Iodine Malonic-Acid (CIMA) reactions and the three-dimensional and two-dimensional Oregonator models of the Belousov-Zhabotinsky (BZ) reactions. Explicit analytical expressions are given for the relaxation-oscillation periods of these chemical reactions that are accurate within 5% of their numerical values. In the two-dimensional CIMA and Oregonator models, we also derive critical parameter values leading to canard explosions and implosions in their associated limit cycles.

DEVIL'S STAIRCASE OF GAP MAPS

2003 ◽  
Vol 13 (01) ◽  
pp. 115-122 ◽  
Author(s):  
JUNG-CHAO BAN ◽  
CHENG-HSIUNG HSU ◽  
SONG-SUN LIN
Keyword(s):  
Topological Entropy ◽  
Devil's Staircase ◽  
Unimodal Maps ◽  
One Dimensional ◽  
Devil’S Staircase ◽  
Staircase Structure ◽  
Entropy Functions ◽  
Kneading Theory

This study demonstrates the devil's staircase structure of topological entropy functions for one-dimensional symmetric unimodal maps with a gap inside. The results are obtained by using kneading theory and are helpful in investigating the communication of chaos.

scholarly journals The Staircase Structure of the Southern Brazilian Continental Shelf

2009 ◽  
Vol 2009 ◽  
pp. 1-17
Author(s):  
M. S. Baptista ◽  
L. A. Conti
Keyword(s):  
Continental Shelf ◽  
Sea Level ◽  
Sea Level Changes ◽  
Devil's Staircase ◽  
The Earth ◽  
Devil’S Staircase ◽  
The World ◽  
Sea Level Variations ◽  
Staircase Structure ◽  
Global Sea Level

We show some evidences that the Southeastern Brazilian Continental Shelf (SBCS) has a devil's staircase structure, with a sequence of scarps and terraces with widths that obey fractal formation rules. Since the formation of these features is linked with the sea-level variations, we say that the sea level changes in an organized pulsating way. Although the proposed approach was applied in a particular region of the Earth, it is suitable to be applied in an integrated way to other shelves around the world, since the analyses favor the revelation of the global sea-level variations.

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