On the possibility for an autooscillatory system under external periodic drive action to exhibit universal behavior characteristic of the transition to chaos via period-doubling bifurcations in conservative systems

2008 ◽  
Vol 34 (11) ◽  
pp. 985-988 ◽  
Author(s):  
A. P. Kuznetsov ◽  
S. P. Kuznetsov ◽  
A. V. Savin ◽  
D. V. Savin
Keyword(s):  
Period Doubling ◽  
Universal Behavior ◽  
Transition To Chaos ◽  
Conservative Systems ◽  
Behavior Characteristic ◽  
Period Doubling Bifurcations

Multiple Scenarios of Transition to Chaos in the Alternative Splicing Model

2017 ◽  
Vol 27 (02) ◽  
pp. 1730006 ◽  
Author(s):  
Vladislav V. Kogai ◽  
Vitaly A. Likhoshvai ◽  
Stanislav I. Fadeev ◽  
Tamara M. Khlebodarova
Keyword(s):  
Alternative Splicing ◽  
Period Doubling ◽  
Genetic System ◽  
Protein Isoforms ◽  
Type I ◽  
Transition To Chaos ◽  
Delayed Arguments ◽  
The Mathematical Model ◽  
Period Doubling Bifurcations ◽  
Type Of Transition

We have investigated the scenarios of transition to chaos in the mathematical model of a genetic system constituted by a single transcription factor-encoding gene, the expression of which is self-regulated by a feedback loop that involves protein isoforms. Alternative splicing results in the synthesis of protein isoforms providing opposite regulatory outcomes — activation or repression. The model is represented by a differential equation with two delayed arguments. The possibility of transition to chaos dynamics via all classical scenarios: a cascade of period-doubling bifurcations, quasiperiodicity and type-I, type-II and type-III intermittencies, has been numerically demonstrated. The parametric features of each type of transition to chaos have been described.

BIFURCATION PHENOMENA IN THE 1:1 RESONANT HORN FOR THE FORCED VAN DER POL—DUFFING EQUATION

1992 ◽  
Vol 02 (01) ◽  
pp. 93-100 ◽  
Author(s):  
A.S. DMITRIEV ◽  
U.A. KOMLEV ◽  
D.V. TURAEV
Keyword(s):  
Period Doubling ◽  
Duffing Equation ◽  
Van Der Pol ◽  
Parallel Processes ◽  
Transition To Chaos ◽  
Bifurcation Phenomena ◽  
Small Dissipation ◽  
Period Doubling Bifurcations

This paper presents the 1:1 resonant horn bifurcation phenomena for the forced van der Pol—Duffing equation. It is shown that the transition to chaos in the case of small dissipation evolves in two parallel processes: a sequence of period-doubling bifurcations and the birth, growth and merging of homoclinic structures.

Robust Estimation-Based Control of Chaotic Behavior in an Oscillator with Inertial and Elastic Symmetric Nonlinearities

2003 ◽  
Vol 9 (6) ◽  
pp. 665-684 ◽  
Author(s):  
A. A. Al-Qaisia ◽  
A. M. Harb ◽  
A. A. Zaher ◽  
M. A. Zohdy
Keyword(s):  
Robust Estimation ◽  
Nonlinear Oscillator ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Physical Parameters ◽  
Bifurcation And Chaos ◽  
Transition To Chaos ◽  
Simulation Results ◽  
Forced Nonlinear Oscillator ◽  
Period Doubling Bifurcations

In this paper, we study the dynamics of a forced nonlinear oscillator with inertial and elastic symmetric nonlinearities using modern nonlinear, bifurcation and chaos theories. The results for the response have shown that, for a certain combination of physical parameters, this oscillator exhibits a chaotic behavior or a transition to chaos through a sequence of period doubling bifurcations. The main objective of this paper is to control the chaotic behavior for this type of oscillator. A nonlinear estimation-based controller is proposed and the transient performance is investigated. The design of the parameter update mechanism is analyzed while discussing ways to extend its performance to further account for other types of uncertainties. We examine robustness problems as well as ways to tune the controller parameters. Simulation results are presented for the uncontrolled and controlled cases, verifying the effectiveness and the capability of the proposed controller. Finally, a discussion and conclusions are given with possible future extensions.

HIERARCHY OF INSTABILITIES AND SOLITONS IN AN ACOUSTIC DISTRIBUTED SYSTEM WITH FEEDBACK

1994 ◽  
Vol 04 (06) ◽  
pp. 1525-1534
Author(s):  
S.G. DOLINCHUK ◽  
V.I. ZADOROZHNII ◽  
A.M. FEDORCHENKO
Keyword(s):  
Steady State ◽  
Distributed System ◽  
Negative Feedback ◽  
Positive Feedback ◽  
Period Doubling ◽  
Physical Parameters ◽  
Transition To Chaos ◽  
Hard Excitation ◽  
Additional Feedback ◽  
Period Doubling Bifurcations

We have considered the hierarchy of instabilities, the transition to chaos, and the periodic generation of three-wave solitons in an acoustic distributed system with additional feedback and pumping. At positive feedback only steady-state conditions are shown to be stable. Self-oscillations, period-doubling bifurcations, and transition from quasiperiodicity to chaos of parametrically coupled waves have been found at negative feedback. “Hard” excitation of the generator leads to the formation of solitons. Established effects have been studied on dependence with physical parameters of a TeO 2 crystal.

ANALYSIS OF STOCHASTIC CYCLES IN THE CHEN SYSTEM

2010 ◽  
Vol 20 (05) ◽  
pp. 1439-1450 ◽  
Author(s):  
IRINA BASHKIRTSEVA ◽  
GUANRONG CHEN ◽  
LEV RYASHKO
Keyword(s):  
Period Doubling ◽  
Spatial Arrangement ◽  
Sensitivity Function ◽  
Function Technique ◽  
Chen System ◽  
Poincaré Sections ◽  
Transition To Chaos ◽  
Stochastic Sensitivity ◽  
Poincare Sections ◽  
Period Doubling Bifurcations

We study the stochastically forced Chen system in its parameter zone under the transition to chaos via period-doubling bifurcations. We suggest a stochastic sensitivity function technique for the analysis of stochastic cycles. We show that this approach allows to construct the dispersion ellipses of random trajectories for any Poincaré sections, and these ellipses reflect the essential features of a spatial arrangement of random trajectories near deterministic cycles. For the Chen system, we demonstrate a growth of stochastic sensitivity of the forced cycles under transition to chaos.

scholarly journals Can short time delays influence the variability of the solar cycle?

2010 ◽  
Vol 6 (S271) ◽  
pp. 288-296
Author(s):  
Laurène Jouve ◽  
Michael R. E. Proctor ◽  
Geoffroy Lesur
Keyword(s):  
Solar Cycle ◽  
Convection Zone ◽  
Mean Field ◽  
Period Doubling ◽  
Temporal Modulation ◽  
Solar Convection Zone ◽  
Flux Tubes ◽  
Solar Convection ◽  
Short Time ◽  
Period Doubling Bifurcations

AbstractWe present the effects of introducing results of 3D MHD simulations of buoyant magnetic fields in the solar convection zone in 2D mean-field Babcock-Leighton models. In particular, we take into account the time delay introduced by the rise time of the toroidal structures from the base of the convection zone to the solar surface. We find that the delays produce large temporal modulation of the cycle amplitude even when strong and thus rapidly rising flux tubes are considered. The study of a reduced model reveals that aperiodic modulations of the solar cycle appear after a sequence of period doubling bifurcations typical of non-linear systems. We also discuss the memory of such systems and the conclusions which may be drawn concerning the actual solar cycle variability.

Bifurcations and Chaos in a Minimal Neural Network: Forced Coupled Neurons

2021 ◽  
Vol 31 (10) ◽  
pp. 2150147
Author(s):  
Yo Horikawa
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Period Doubling ◽  
Output Function ◽  
Piecewise Constant ◽  
Border Collision Bifurcations ◽  
Saddle Node ◽  
Periodic Input ◽  
Symmetric Periodic Solution ◽  
Period Doubling Bifurcations

The bifurcations and chaos in a system of two coupled sigmoidal neurons with periodic input are revisited. The system has no self-coupling and no inherent limit cycles in contrast to the previous studies and shows simple bifurcations qualitatively different from the previous results. A symmetric periodic solution generated by the periodic input underdoes a pitchfork bifurcation so that a pair of asymmetric periodic solutions is generated. A chaotic attractor is generated through a cascade of period-doubling bifurcations of the asymmetric periodic solutions. However, a symmetric periodic solution repeats saddle-node bifurcations many times and the bifurcations of periodic solutions become complicated as the output gain of neurons is increasing. Then, the analysis of border collision bifurcations is carried out by using a piecewise constant output function of neurons and a rectangular wave as periodic input. The saddle-node, the pitchfork and the period-doubling bifurcations in the coupled sigmoidal neurons are replaced by various kinds of border collision bifurcations in the coupled piecewise constant neurons. Qualitatively the same structure of the bifurcations of periodic solutions in the coupled sigmoidal neurons is derived analytically. Further, it is shown that another period-doubling route to chaos exists when the output function of neurons is asymmetric.

Two Groups of Chaotic Vibrations in a Single-Degree-of-Freedom Maglev System

1997 ◽  
Author(s):  
Zhixiang Xu ◽  
Hideyuki Tamura
Keyword(s):  
Magnetic Levitation ◽  
Excitation Frequency ◽  
Power Spectra ◽  
Period Doubling ◽  
Excitation Amplitude ◽  
Degree Of Freedom ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Moving Base ◽  
Period Doubling Bifurcations

Abstract In this paper, a single-degree-of-freedom magnetic levitation dynamic system, whose spring is composed of a magnetic repulsive force, is numerically analyzed. The numerical results indicate that a body levitated by magnetic force shows many kinds of vibrations upon adjusting the system parameters (viz., damping, excitation amplitude and excitation frequency) when the system is excited by the harmonically moving base. For a suitable combination of parameters, an aperiodic vibration occurs after a sequence of period-doubling bifurcations. Typical aperiodic vibrations that occurred after period-doubling bifurcations from several initial states are identified as chaotic vibration and classified into two groups by examining their power spectra, Poincare maps, fractal dimension analyses, etc.

scholarly journals Chaotic dynamics of a pulse modulated control system for a heating unit

2018 ◽  
Vol 224 ◽  
pp. 02055
Author(s):  
Yuriy A. Gol’tsov ◽  
Alexander S. Kizhuk ◽  
Vasiliy G. Rubanov
Keyword(s):  
Control System ◽  
Differential Equations ◽  
Chaotic Dynamics ◽  
Period Doubling ◽  
Classical Period ◽  
Nonlinear Phenomena ◽  
Pulse Control ◽  
Heating Unit ◽  
Border Collision Bifurcation ◽  
Period Doubling Bifurcations

The dynamic modes and bifurcations in a pulse control system of a heating unit, the condition of which is described through differential equations with discontinuous right–hand sides, have been studied. It has been shown that the system under research can demonstrate a great variety of nonlinear phenomena and bifurcation transitions, such as quasiperiodicity, multistable behaviour, chaotization of oscillations through a classical period–doubling bifurcations cascade and border–collision bifurcation.

Sign in / Sign up

Export Citation Format

Share Document