UNCERTAINTY IN CHAOS SYNCHRONIZATION

2001 ◽  
Vol 11 (06) ◽  
pp. 1723-1735 ◽  
Author(s):  
GUO-QUN ZHONG ◽  
KIM-FUNG MAN ◽  
KING-TIM KO
Keyword(s):  
Coupling Strength ◽  
Chaos Synchronization ◽  
Phase Synchronization ◽  
Initial Conditions ◽  
Coupled Systems ◽  
Sensitive Dependence ◽  
Initial States ◽  
Synchronization Regime

In this paper a variety of uncertainty phenomena in chaos synchronization, which are caused by the sensitive dependence on initial conditions and coupling strength, are numerically investigated. Two identical Chua's circuits are considered for both mutually- and unidirectionally-coupled systems. It is found that initial states of the system play an important role in chaos synchronization. Depending on initial conditions, distinct behaviors, such as in-phase synchronization, anti-phase synchronization, oscillation-quenching, and bubbling of attractors, may occur. Based on the findings, we clarify that the systems, which satisfy the standard synchronization criterion, do not necessarily operate in a synchronization regime.

SYNCHRONIZING CHAOTIC ATTRACTORS OF CHUA'S CANONICAL CIRCUIT: THE CASE OF UNCERTAINTY IN CHAOS SYNCHRONIZATION

2006 ◽  
Vol 16 (07) ◽  
pp. 1961-1976 ◽  
Author(s):  
I. M. KYPRIANIDIS ◽  
A. N. BOGIATZI ◽  
M. PAPADOPOULOU ◽  
I. N. STOUBOULOS ◽  
G. N. BOGIATZIS ◽  
...  
Keyword(s):  
Coupling Strength ◽  
Chaos Synchronization ◽  
Phase Synchronization ◽  
Initial Conditions ◽  
Chaotic Synchronization ◽  
Chaotic Attractors ◽  
Coexisting Attractors

In this paper, we have studied the dynamics of two identical resistively coupled Chua's canonical circuits and have found that it is strongly affected by initial conditions, coupling strength and the presence of coexisting attractors. Depending on the coupling variable, chaotic synchronization has been observed both numerically and experimentally. Anti-phase synchronization has also been studied numerically clarifying some aspects of uncertainty in chaos synchronization.

PHASE SYNCHRONIZATION OF LINEARLY AND NONLINEARLY COUPLED OSCILLATORS WITH INTERNAL RESONANCE

2009 ◽  
Vol 23 (30) ◽  
pp. 5715-5726
Author(s):  
YONG LIU
Keyword(s):  
Coupling Strength ◽  
Coupled Oscillators ◽  
Internal Resonance ◽  
Natural Frequencies ◽  
Phase Synchronization ◽  
Coupled Systems ◽  
Nonlinear Coupling ◽  
Linear Coupling ◽  
Phase Dynamics ◽  
Diffuse Clouds

Phase synchronization between linearly and nonlinearly coupled systems with internal resonance is investigated in this paper. By introducing the conception of phase for a chaotic motion, it demonstrates that the detuning parameter σ between the two natural frequencies ω1and ω2affects phase dynamics, and with the increase in the linear coupling strength, the effect of phase synchronization between two sub-systems was enhanced, while increased firstly, and then decayed as nonlinear coupling strength increases. Further investigation reveals that the transition of phase states between the two oscillators are related to the critical changes of the Lyapunov exponents, which can also be explained by the diffuse clouds.

TRACKING PHASE SYNCHRONIZATION OF TWO COUPLED RÖSSLER OSCILLATORS WITH INTERNAL RESONANCE

2009 ◽  
Vol 23 (23) ◽  
pp. 4809-4816 ◽  
Author(s):  
YONG LIU
Keyword(s):  
Coupling Strength ◽  
Internal Resonance ◽  
Natural Frequencies ◽  
Phase Synchronization ◽  
Coupling Parameter ◽  
Primary Resonance ◽  
Coupled Systems ◽  
Nonlinear Coupling ◽  
Linear Coupling ◽  
Phase Dynamics

Phase synchronization between linearly and nonlinearly coupled systems with internal resonance is investigated in this paper. By introducing the conception of phase for a chaotic motion, we tune the linear coupling parameter to obtain the two Rössler oscillators in a synchronized regime and analyze the effect of a nonlinear coupling on the phase synchronized state. It demonstrates that the detuning parameter σ between the two natural frequencies ω1and ω2affects phase dynamics, and with the increase of the nonlinear coupling strength, for the primary resonance, the effect of phase synchronization between two sub-systems was decayed, while increasing with frequency ratio 1:2. Further investigation reveals that the transition of phase states between the two oscillators are related to the critical changes of the nonlinear coupling strength.

Chaotic Motion in a Flexible Rotating Beam and Synchronization

2017 ◽  
Vol 12 (4) ◽  
Author(s):  
B. Sandeep Reddy ◽  
Ashitava Ghosal
Keyword(s):  
Chaotic Motion ◽  
Chaos Synchronization ◽  
Initial Conditions ◽  
Lyapunov Stability Theory ◽  
Original System ◽  
Flexible Beam ◽  
Nonlinear Controller ◽  
Control Of Chaos ◽  
Sensitive Dependence ◽  
Parameter Values

A rotating flexible beam undergoing large deformation is known to exhibit chaotic motion for certain parameter values. This work deals with an approach for control of chaos known as chaos synchronization. A nonlinear controller based on the Lyapunov stability theory is developed, and it is shown that such a controller can avoid the sensitive dependence of initial conditions seen in all chaotic systems. The proposed controller ensures that the error between the controlled and the original system, for different initial conditions, asymptotically goes to zero. A numerical example using the parameters of a rotating power generating wind turbine blade is used to illustrate the theoretical approach.

Effects of initial conditions and coupling competition modes on behaviors of coupled non-identical fractional-order bistable oscillators

2016 ◽  
Vol 94 (11) ◽  
pp. 1158-1166
Author(s):  
Liming Wang
Keyword(s):  
Boundary Layer ◽  
Monte Carlo ◽  
Fractional Order ◽  
Coupled Oscillators ◽  
Chaos Synchronization ◽  
Phase Synchronization ◽  
Initial Conditions ◽  
Coupled System ◽  
Amplitude Death ◽  
Dynamic Behaviors

The effects of the initial conditions and the coupling competition modes on the dynamic behaviors of coupled non-identical fractional-order bistable oscillators are investigated intensively and the various phenomena are explored. The coupled system can be controlled to form chaos synchronization, chaos anti-phase synchronization, amplitude death, oscillation death, etc., by setting the initial conditions or selecting the coupling competition modes. Depending on whether the arbitrary initial conditions can let two coupled oscillators stop oscillating, the dynamic behaviors of the coupled system are further classified into three types, that is, both of oscillators stop oscillating, only one oscillator stops oscillating, and none of oscillators stop oscillating. Based on the principle of Monte Carlo method, the percentages of three types of dynamic behaviors are calculated for the different coupling competition modes and the dynamic behaviors of the coupled system are characterized from the perspective of statistics. Moreover, the mechanism behind the various phenomena is explained in detail by the concept of boundary layer and the optimum coupling competition modes are found.

scholarly journals On some kinds of dense orbits in general nonautonomous dynamical systems

2020 ◽  
Vol 7 (1) ◽  
pp. 163-175
Author(s):  
Mehdi Pourbarat
Keyword(s):  
Dynamical Systems ◽  
Initial Conditions ◽  
Point Of View ◽  
Topological Conjugacy ◽  
Topological Transitivity ◽  
Nonautonomous Systems ◽  
Nonautonomous Dynamical Systems ◽  
Sensitive Dependence ◽  
Dense Orbits

AbstractWe study the theory of universality for the nonautonomous dynamical systems from topological point of view related to hypercyclicity. The conditions are provided in a way that Birkhoff transitivity theorem can be extended. In the context of generalized linear nonautonomous systems, we show that either one of the topological transitivity or hypercyclicity give sensitive dependence on initial conditions. Meanwhile, some examples are presented for topological transitivity, hypercyclicity and topological conjugacy.

IS SENSITIVE DEPENDENCE ON INITIAL CONDITIONS NATURE’S SENSORY DEVICE?

1992 ◽  
Vol 02 (01) ◽  
pp. 193-199 ◽  
Author(s):  
RAY BROWN ◽  
LEON CHUA ◽  
BECKY POPP
Keyword(s):  
Initial Conditions ◽  
Sensory Perception ◽  
Sensory Features ◽  
Sensitive Dependence ◽  
Nonlinear Devices

In this letter we illustrate three methods of using nonlinear devices as sensors. We show that the sensory features of these devices is a result of sensitive dependence on parameters which we show is equivalent to sensitive dependence on initial conditions. As a result, we conjecture that sensitive dependence on initial conditions is nature’s sensory device in cases where remarkable feats of sensory perception are seen.

CELLULAR NEURAL NETWORKS CAN MIMIC SMALL-WORLD NETWORKS

1999 ◽  
Vol 09 (10) ◽  
pp. 2105-2126 ◽  
Author(s):  
TAO YANG ◽  
LEON O. CHUA
Keyword(s):  
Neural Networks ◽  
Coupling Strength ◽  
Initial Conditions ◽  
Computing Time ◽  
Small World ◽  
Cellular Neural Networks ◽  
Clustering Coefficient ◽  
High Survival ◽  
Game Of Life ◽  
Hole Detection

Small-world phenomenon can occur in coupled dynamical systems which are highly clustered at a local level and yet strongly coupled at the global level. We show that cellular neural networks (CNN's) can exhibit "small-world phenomenon". We generalize the "characteristic path length" from previous works on "small-world phenomenon" into a "characteristic coupling strength" for measuring the average coupling strength of the outputs of CNN's. We also provide a simplified algorithm for calculating the "characteristic coupling strength" with a reasonable amount of computing time. We define a "clustering coefficient" and show how it can be calculated by a horizontal "hole detection" CNN, followed by a vertical "hole detection" CNN. Evolutions of the game-of-life CNN with different initial conditions are used to illustrate the emergence of a "small-world phenomenon". Our results show that the well-known game-of-life CNN is not a small-world network. However, generalized CNN life games whose individuals have strong mobility and high survival rate can exhibit small-world phenomenon in a robust way. Our simulations confirm the conjecture that a population with a strong mobility is more likely to qualify as a small world. CNN games whose individuals have weak mobility can also exhibit a small-world phenomenon under a proper choice of initial conditions. However, the resulting small worlds depend strongly on the initial conditions, and are therefore not robust.

A PHYSICAL INTERPRETATION OF MELNIKOV’S METHOD

1992 ◽  
Vol 02 (01) ◽  
pp. 1-9 ◽  
Author(s):  
YOHANNES KETEMA
Keyword(s):  
Physical Interpretation ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Homoclinic Orbits ◽  
Poincare Map ◽  
Stable And Unstable Manifolds ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Sensitive Dependence ◽  
The Stability

This paper is concerned with analyzing Melnikov’s method in terms of the flow generated by a vector field in contrast to the approach based on the Poincare map and giving a physical interpretation of the method. It is shown that the direct implication of a transverse crossing between the stable and unstable manifolds to a saddle point of the Poincare map is the existence of two distinct preserved homoclinic orbits of the continuous time system. The stability of these orbits and their role in the phenomenon of sensitive dependence on initial conditions is discussed and a physical example is given.

TWOFOLD IMPACT OF DELAYED FEEDBACK ON COUPLED OSCILLATORS

2007 ◽  
Vol 17 (07) ◽  
pp. 2517-2530 ◽  
Author(s):  
OLEKSANDR V. POPOVYCH ◽  
VALERII KRACHKOVSKYI ◽  
PETER A. TASS
Keyword(s):  
Time Delay ◽  
Coupling Strength ◽  
Coupled Oscillators ◽  
Phase Synchronization ◽  
Threshold Value ◽  
Phase Oscillators ◽  
Intermediate Coupling ◽  
Rich Variety ◽  
Dynamical Regimes ◽  
The Impact

We present a detailed bifurcation analysis of desynchronization transitions in a system of two coupled phase oscillators with delay. The coupling between the oscillators combines a delayed self-feedback of each oscillator with an instantaneous mutual interaction. The delayed self-feedback leads to a rich variety of dynamical regimes, ranging from phase-locked and periodically modulated synchronized states to chaotic phase synchronization and desynchronization. We show that an increase of the coupling strength between oscillators may lead to a loss of synchronization. Intriguingly, the delay has a twofold influence on the oscillations: synchronizing for small and intermediate coupling strength and desynchronizing if the coupling strength exceeds a certain threshold value. We show that the desynchronization transition has the form of a crisis bifurcation of a chaotic attractor of chaotic phase synchronization. This study contributes to a better understanding of the impact of time delay on interacting oscillators.

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