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.

AN AUTONOMOUS CHAOTIC CELLULAR NEURAL NETWORK AND CHUA'S CIRCUIT

1993 ◽  
Vol 03 (02) ◽  
pp. 591-601 ◽  
Author(s):  
FAN ZOU ◽  
JOSEF A. NOSSEK
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Chaotic Behavior ◽  
Piecewise Linear ◽  
Cellular Neural Network ◽  
Piecewise Linear Function ◽  
Chua’S Circuit ◽  
Nonlinear Dynamical ◽  
Chua's Circuit ◽  
Parameter Values ◽  
Circuit Family

Cellular neural networks (CNN) are time-continuous nonlinear dynamical systems. Like in Chua's circuit, the nonlinearity of these networks is defined as a piecewise-linear function. For CNNs with at least three cells chaotic behavior may be possible in certain regions of parameter values. In this paper we report such a chaotic attractor with an autonomous three-cell CNN. It can be shown that the attractor has a structure very similar to the double-scroll Chua's attractor. Through some equivalent transformations this circuit, in three major subspaces of its state space, is shown to belong to Chua's circuit family, although originating from a completely different field.

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.

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.

scholarly journals ESTIMATING PROBABILITY DENSITY FUNCTIONS AND ENTROPIES OF CHUA'S CIRCUIT USING B-SPLINE FUNCTIONS

2012 ◽  
Vol 22 (05) ◽  
pp. 1250107 ◽  
Author(s):  
F. A. SAVACI ◽  
M. GÜNGÖR
Keyword(s):  
Probability Density ◽  
Probability Density Functions ◽  
The State ◽  
Spline Functions ◽  
Density Estimator ◽  
Density Functions ◽  
Bifurcation Parameter ◽  
Chua’S Circuit ◽  
B Spline ◽  
Chua's Circuit

In this paper, first the probability density functions (PDFs) of the states of Chua's circuit have been estimated using B-spline functions and then the state entropies of Chua's circuit with respect to the bifurcation parameter have been obtained. The results of the proposed B-spline density estimator have been compared with the results obtained from the Parzen density estimator.

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.

Monte Carlo Approach Towards Evaluating Random Number Generators Based on Mathematical Schemes Driven from Chua's Circuit

2020 ◽  
Author(s):  
Kasra Amini ◽  
Aidin Momtaz ◽  
Ehsan Qoreishi ◽  
Sarah Amini ◽  
Sanaz Haddadian
Keyword(s):  
Monte Carlo ◽  
Chaos Theory ◽  
Initial Conditions ◽  
Monte Carlo Integration ◽  
Mathematical Function ◽  
Distributed Data ◽  
Chua’S Circuit ◽  
Data Generation ◽  
Data Set ◽  
Chua's Circuit

The philosophical nature of randomly generated quantities is widely discussed in the realms of chaos theory. Although, the fundamental premise of the chaos theory does not assume any random behavior in the resulting series and considers them deterministic however highly dependent of the initial conditions of the system, one could address the problem of randomness, by using the output of a chaotic system, as the input of a mathematical function, aiming for the generation of randomly distributed values. For that matter, the voltages of the two capacitors in the classic configuration of a Chua's circuit have been measured. Having defined eight mathematical schemes for manipulating the inputted data set, the current manuscript focuses on the pragmatic and engineering criteria of the resulting data, in terms of randomness, and spectral distributions; hence proposing methods of random data generation. The ranking of schemes has been proceeded through a geometrical manifestation of the Monte Carlo Integration. And the suggested eight schemes are compared with the commercially common timer-based random generators. As the geometrical domain in the Monte Carlo Integration has defined in such a way that the most randomly distributed data set would result in a closer estimation of the number Pi, the suggested scheme working based on 'frequency indicator' is evaluated as the highest-ranked scheme in that regard, with  the estimated numerical value of 3.1424 for Pi.

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.

A BIPARAMETRIC BIFURCATION IN 3D CONTINUOUS PIECEWISE LINEAR SYSTEMS WITH TWO ZONES: APPLICATION TO CHUA'S CIRCUIT

2007 ◽  
Vol 17 (02) ◽  
pp. 445-457 ◽  
Author(s):  
E. FREIRE ◽  
E. PONCE ◽  
J. ROS
Keyword(s):  
Hopf Bifurcation ◽  
Linear Systems ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Continuous Systems ◽  
Chua’S Circuit ◽  
Chua's Circuit ◽  
Piecewise Linear Systems ◽  
Limit Cycle Bifurcation

In this paper, a possible degeneration of the focus-center-limit cycle bifurcation for piecewise smooth continuous systems is analyzed. The case of continuous piecewise linear systems with two zones is considered, and the coexistence of two limit cycles for certain values of parameters is justified. Finally, the Chua's circuit is shown to exhibit the analyzed bifurcation. The obtained bifurcation set in the parameter plane is similar to the degenerate Hopf bifurcation for differentiable systems.

ARNOL’D TONGUES, DEVIL’S STAIRCASE, AND SELF-SIMILARITY IN THE DRIVEN CHUA’S CIRCUIT

1994 ◽  
Vol 04 (06) ◽  
pp. 1743-1753 ◽  
Author(s):  
LADISLAV PIVKA ◽  
ALEXANDER L. ZHELEZNYAK ◽  
LEON O. CHUA
Keyword(s):  
Initial Conditions ◽  
Period Doubling ◽  
Chua’S Circuit ◽  
Self Similarity ◽  
Devil's Staircase ◽  
Algebraic Form ◽  
Chua's Circuit ◽  
Recurrent Relations ◽  
Devil’S Staircase ◽  
Self Similar

Empirical recurrent relations, governing the structure of the devil’s staircase in the driven Chua’s circuit are given, which reflect the self-similar structure in an algebraic form. In particular, it turns out that the same formulas hold for both winding and period numbers, but with different “initial conditions”. Some of the finer details such as period-doubling along with numerous coexistence phenomena within staircases of mode-locked states have been revealed by computing high-resolution bifurcation diagrams.

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