DRY TURBULENCE FROM A TIME-DELAYED CHUA'S CIRCUIT

1993 ◽  
Vol 03 (02) ◽  
pp. 645-668 ◽  
Author(s):  
A. N. SHARKOVSKY ◽  
YU. MAISTRENKO ◽  
PH. DEREGEL ◽  
L. O. CHUA
Keyword(s):  
Difference Equation ◽  
Stability Region ◽  
Nonlinear Difference Equation ◽  
Chua’S Circuit ◽  
Stability Boundaries ◽  
Chua's Circuit ◽  
Infinite Dimensional ◽  
Chaotic Region ◽  
The Stability ◽  
Left Portion

In this paper, we consider an infinite-dimensional extension of Chua's circuit (Fig. 1) obtained by replacing the left portion of the circuit composed of the capacitance C2 and the inductance L by a lossless transmission line as shown in Fig. 2. As we shall see, if the remaining capacitance C1 is equal to zero, the dynamics of this so-called time-delayed Chua's circuit can be reduced to that of a scalar nonlinear difference equation. After deriving the corresponding 1-D map, it will be possible to determine without any approximation the analytical equation of the stability boundaries of cycles of every period n. Since the stability region is nonempty for each n, this proves rigorously that the time-delayed Chua's circuit exhibits the "period-adding" phenomenon where every two consecutive cycles are separated by a chaotic region.

DRY TURBULENCE AND PERIOD-ADDING PHENOMENA FROM A 1-D MAP WITH TIME DELAY

1995 ◽  
Vol 05 (05) ◽  
pp. 1283-1302 ◽  
Author(s):  
A. N. SHARKOVSKY ◽  
PH. DEREGEL ◽  
L. O. CHUA
Keyword(s):  
Stability Region ◽  
Exact Analytical Solution ◽  
Piecewise Linear ◽  
Interesting Property ◽  
Nonlinear Difference Equation ◽  
Chua’S Circuit ◽  
Chua's Circuit ◽  
Infinite Dimensional ◽  
Chaotic Region ◽  
The Stability

In this tutorial paper, we consider an infinite-dimensional extension of Chua's circuit, as shown in Fig. 1, where the transmission line is lossless. As we shall see, if the capacitance C1 is set to zero, the dynamics of this so-called time-delayed Chua's circuit can be reduced, without any approximation, to that of a continuous scalar nonlinear difference equation. This type of equation can lead to space-time chaos which, due to the absence of viscosity in our system, will be termed "dry turbulence". Another interesting property of this system occurs under certain conditions, when the corresponding 1-D map has two segments and is piecewise-linear. The extreme simplicity of this map will allow us to derive, without any approximation, the exact analytical solution of the stability boundaries of stable cycles of every period n. Since the stability region is non-empty for each n, this proves rigorously that the time-delayed Chua's circuit exhibits the "period-adding" phenomenon where every two consecutive cycles are separated by a chaotic region.

CHAOTIC BEHAVIOR OF ORBITS CLOSE TO A HETEROCLINIC CONTOUR

1996 ◽  
Vol 06 (01) ◽  
pp. 69-79 ◽  
Author(s):  
M. BLÁZQUEZ ◽  
E. TUMA
Keyword(s):  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
Dimensional Case ◽  
Chua’S Circuit ◽  
Closed Contour ◽  
Chua's Circuit ◽  
Infinite Dimensional ◽  
Focus Type

We study the behavior of the solutions in a neighborhood of a closed contour formed by two heteroclinic connections to two equilibrium points of saddle-focus type. We consider both the three-dimensional case, as in the well-known Chua's circuit, as well as the infinite-dimensional case.

STABILITY OF OBSERVER-BASED CHAOTIC COMMUNICATIONS FOR A CLASS OF LUR'E SYSTEMS

2002 ◽  
Vol 12 (07) ◽  
pp. 1605-1618 ◽  
Author(s):  
JOSE ALVAREZ-RAMIREZ ◽  
HECTOR PUEBLA ◽  
ILSE CERVANTES
Keyword(s):  
Numerical Simulations ◽  
Chaotic Oscillator ◽  
Chua’S Circuit ◽  
Lur’E Systems ◽  
Chua's Circuit ◽  
Information Signal ◽  
Synchronization Signal ◽  
Chaotic Communications ◽  
The Stability

In this paper, the stability of observer-based chaotic communications using Lur'e systems is presented. In this approach, the transmitter contains a chaotic oscillator with an input that is modulate by the information signal. The receiver is composed by a copy of the transmitter driven by a synchronization signal. Some effects of transmission noise on the demodulation procedure are discussed. Numerical simulations on Chua's circuit are provided to illustrate our findings.

scholarly journals Amplitude Modulation and Synchronization of Fractional-Order Memristor-Based Chua's Circuit

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
A. G. Radwan ◽  
K. Moaddy ◽  
I. Hashim
Keyword(s):  
Fractional Order ◽  
Amplitude Modulation ◽  
Lyapunov Stability Theory ◽  
Stability And Convergence ◽  
Chua’S Circuit ◽  
Nonlinear Controller ◽  
Chua's Circuit ◽  
Control Functions ◽  
Nonlinear Control Theory ◽  
The Stability

This paper presents a general synchronization technique and an amplitude modulation of chaotic generators. Conventional synchronization and antisynchronization are considered a very narrow subset from the proposed technique where the scale between the output response and the input response can be controlled via control functions and this scale may be either constant (positive, negative) or time dependent. The concept of the proposed technique is based on the nonlinear control theory and Lyapunov stability theory. The nonlinear controller is designed to ensure the stability and convergence of the proposed synchronization scheme. This technique is applied on the synchronization of two identical fractional-order Chua's circuit systems with memristor. Different examples are studied numerically with different system parameters, different orders, and with five alternative cases where the scaling functions are chosen to be positive/negative and constant/dynamic which covers all possible cases from conventional synchronization to the amplitude modulation cases to validate the proposed concept.

scholarly journals On a nonlinear difference equation with variable delay

2013 ◽  
Vol 46 (1) ◽  
Author(s):  
Dinh Cong Huong ◽  
Nguyen Van Mau
Keyword(s):  
Difference Equation ◽  
Periodic Solutions ◽  
Nonlinear Difference Equation ◽  
Variable Delay ◽  
Positive Periodic Solutions ◽  
Boundedness Of Solutions ◽  
The Stability

AbstractIn this paper, some results on the equi-boundedness of solutions, the stability of the zero and the existence of positive periodic solutions of nonlinear difference equation with variable delay

scholarly journals A nonlinear difference equation with two parameters

1985 ◽  
Vol 26 (4) ◽  
pp. 430-451 ◽  
Author(s):  
A. Brown
Keyword(s):  
Difference Equation ◽  
General Problem ◽  
Iteration Process ◽  
Nonlinear Difference Equation ◽  
Cubic Equation ◽  
The Difference ◽  
The Stability ◽  
Image Position ◽  
Two Parameters ◽  
Parameter Problem

AbstractThe paper is mainly concerned with the difference equationwhere k and m are parameters, with k > 0. This equation arises from a method proposed for solving a cubic equation by iteration and represents a standardised form of the general problem. In using the above equation it is essential to know when the iteration process converges and this is discussed by means of the usual stability criterion. Critical values are obtained for the occurrence of solutions with period two and period three and the stability of these solutions is also examined. This was done by considering the changes as k increases, for a give value of m, which makes it effectively a one-parameter problem, and it turns out that the change with k can differ strongly from the usual behaviour for a one-parameter difference equation. For m = 2, for example it appears that the usual picture of stable 2-cycle solutions giving way to stable 4-cycle solutions is valid for smaller values of k but the situation is recersed for larger values of k where stable 4-cycle solutions precede stable 2-cycle solutions. Similar anomalies arise for the 3-cycle solutions.

Bifurcation Analysis of a Class Fractional-Oder Nonlinear Chua’s Circuit System

2020 ◽  
Vol 24 (4) ◽  
pp. 549-556
Author(s):  
Zhe Zhang ◽  
Toshimitsu Ushio ◽  
Jing Zhang ◽  
Can Ding ◽  
Feng Liu ◽  
...  
Keyword(s):  
Fractional Order ◽  
Bifurcation Analysis ◽  
Hot Spot ◽  
Rapid Development ◽  
Academic Research ◽  
Practical Consideration ◽  
Chua’S Circuit ◽  
Chua's Circuit ◽  
Practical Engineering ◽  
The Stability

In recent years, with the rapid development of science and technology, dynamic characterization and control of the research circuit system has become not only theoretical but also practical consideration in academic research and practical engineering applications. Therefore, the complex behavior of a research circuit system has become a hot spot in the theoretical field. This thesis is aimed toward the stability criterion and bifurcation of the fractional-order Chua’s circuit system. Despite numerous studies relating to the Chua’s system, most of them focus on its sum of delays. Different from traditional bifurcation analysis of Chua’s circuit system, the parameters are chosen as the bifurcation parameters in this paper such that the stability and bifurcation of the fractional-order Chua’s system is analyzed from a new angle. Then, the conditions of the existence for Hopf bifurcations are achieved by analyzing its characteristic equation. Finally, the validity and rationality of the theory are verified by numerical simulation.

CHUA'S CIRCUIT WITH A DISCONTINUOUS NONLINEARITY

1993 ◽  
Vol 03 (01) ◽  
pp. 231-237 ◽  
Author(s):  
ADELHEID I. MAHLA ◽  
ÁLVARO G. BADAN PALHARES
Keyword(s):  
Poincaré Map ◽  
Chaotic Behavior ◽  
Piecewise Linear ◽  
Chua’S Circuit ◽  
Discontinuous Nonlinearity ◽  
Linear Element ◽  
Chua's Circuit ◽  
Stability Behavior ◽  
The Stability ◽  
Discrete Map

A discrete map can be obtained from Chua's circuit1 with a discontinuous nonlinearity. Chaotic behavior is observed in the simulation of Chua's circuit with a discontinuous piecewise-linear element. The Poincaré map can be used to study the stability behavior and opens the possibility for analyzing the chaotic behavior by determining the existence of a snap-back repeller.

scholarly journals Dynamics and Stability of Stepped Gun-Barrels with Moving Bullets

2008 ◽  
Vol 2008 ◽  
pp. 1-6 ◽  
Author(s):  
Mohammad Tawfik
Keyword(s):  
Finite Element ◽  
Finite Element Model ◽  
Stability Region ◽  
Element Model ◽  
Simple Eigenvalue ◽  
Bernoulli Beam ◽  
Stability Boundaries ◽  
The Stability ◽  
Euler Bernoulli Beam ◽  
Lowest Eigenvalue

The stability of an Euler-Bernoulli beam under the effect of a moving projectile will be reintroduced using simple eigenvalue analysis of a finite element model. The eigenvalues of the beam change with the mass, speed, and position of the projectile, thus, the eigenvalues are evaluated for the system with different speeds and masses at different positions until the lowest eigenvalue reaches zero indicating the instability occurrence. Then a map for the stability region may be obtained for different boundary conditions. Then the dynamics of the beam will be investigated using the Newmark algorithm at different values of speed and mass ratios. Finally, the effect of using stepped barrels on the stability and the dynamics is going to be investigated. It is concluded that the technique used to predict the stability boundaries is simple, accurate, and reliable, the mass of the barrel on the dynamics of the problem cannot be ignored, and that using the stepped barrels, with small increase in the diameter, enhances the stability and the dynamics of the barrel.

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