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.

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.

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.

CHUA’S CIRCUIT: AN OVERVIEW TEN YEARS LATER

1994 ◽  
Vol 04 (02) ◽  
pp. 117-159 ◽  
Author(s):  
LEON O. CHUA
Keyword(s):  
Secure Communication ◽  
Vector Fields ◽  
Piecewise Linear ◽  
International Workshop ◽  
Noise Spectrum ◽  
Chaotic Regime ◽  
Practical Significance ◽  
Chua’S Circuit ◽  
Chua's Circuit ◽  
New Phenomena

More than 200 papers, two special issues (Journal of Circuits, Systems, and Computers, March, June, 1993, and IEEE Trans. on Circuits and Systems, vol. 40, no. 10, October, 1993), an International Workshop on Chua’s Circuit: chaotic phenomena and applica tions at NOLTA’93, and a book (edited by R.N. Madan, World Scientific, 1993) on Chua’s circuit have been published since its inception a decade ago. This review paper attempts to present an overview of these timely publications, almost all within the last six months, and to identify four milestones of this very active research area. An important milestone is the recent fabrication of a monolithic Chua’s circuit. The robustness of this IC chip demonstrates that an array of Chua’s circuits can also be fabricated into a monolithic chip, thereby opening the floodgate to many unconventional applications in information technology, synergetics, and even music. The second milestone is the recent global unfolding of Chua’s circuit, obtained by adding a linear resistor in series with the inductor to obtain a canonical Chua’s circuit— now generally referred to as Chua’s oscillator. This circuit is most significant because it is structurally the simplest (it contains only 6 circuit elements) but dynamically the most complex among all nonlinear circuits and systems described by a 21-parameter family of continuous odd-symmetric piecewise-linear vector fields. The third milestone is the recent discovery of several important new phenomena in Chua’s circuits, e.g., stochastic resonance, chaos-chaos type intermittency, 1/f noise spectrum, etc. These new phenomena could have far-reaching theoretical and practical significance. The fourth milestone is the theoretical and experimental demonstration that Chua’s circuit can be easily controlled from a chaotic regime to a prescribed periodic or constant orbit, or it can be synchronized with 2 or more identical Chua’s circuits, operating in an oscillatory, or a chaotic regime. These recent breakthroughs have ushered in a new era where chaos is deliberately created and exploited for unconventional applications, e.g. secure communication.

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.

FLUCTUATIONS IN THE STATE OF CHAOS-CHAOS INTERMITTENCY OF CHUA’S CIRCUIT

1995 ◽  
Vol 05 (01) ◽  
pp. 271-273
Author(s):  
M. KOCH ◽  
R. TETZLAFF ◽  
D. WOLF
Keyword(s):  
Power Spectrum ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
The State ◽  
Circuit Parameter ◽  
Chua’S Circuit ◽  
External Excitation ◽  
Chua's Circuit ◽  
Parameter Values

We studied the power spectrum of the normalized voltage across the capacitor parallel to a piecewise-linear resistor of Chua’s circuit in the “chaos-chaos intermittency” state [Anishchenko et al., 1992]. The investigations included various initial conditions and circuit parameter values without and with external excitation. In all cases we found spectra showing a 1/ω2-decay over more than four decades.

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.

MAXIMUM DYNAMIC RANGE OF BIFURCATION OF CHUA'S CIRCUIT

1993 ◽  
Vol 03 (02) ◽  
pp. 471-481 ◽  
Author(s):  
A. A. A. NASSER ◽  
E. E. HOSNY ◽  
M. I. SOBHY
Keyword(s):  
Linear Function ◽  
Dynamic Range ◽  
Piecewise Linear ◽  
Piecewise Linear Function ◽  
Bifurcation Parameter ◽  
Chua’S Circuit ◽  
Chua's Circuit ◽  
Simulation Results

This paper includes a method for detecting the maximum possible range of bifurcations based upon the multilevel oscillation technique. An application of the method to Chua's circuit, and new simulation results using the slope of the piecewise-linear function as a bifurcation parameter are presented.

DIGITAL SIGNAL PROCESSOR-BASED INVESTIGATION OF CHUA'S CIRCUIT FAMILY

1993 ◽  
Vol 03 (02) ◽  
pp. 269-292 ◽  
Author(s):  
MICHAEL PETER KENNEDY ◽  
CHAI WAH WU ◽  
STANLEY PAU ◽  
JAMES TOW
Keyword(s):  
Digital Signal Processor ◽  
Piecewise Linear ◽  
Digital Signal ◽  
Chua’S Circuit ◽  
Single Chip ◽  
Integration Algorithm ◽  
Chua's Circuit ◽  
Multistep Integration ◽  
Circuit Family ◽  
Signal Processor

This paper is concerned with exploiting the architecture of a single-chip digital signal processor for integrating piecewise-linear ODEs. We show that DSPs can be usefully applied in the study of Chua's circuit family provided that one chooses a multistep integration algorithm which exploits their unique single-instruction multiply-and-accumulate feature.

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