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.

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.

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.

Robust Synchronization of Fractional-Order Arneodo Chaotic Systems Using a Fractional Sliding Mode Control Strategy

2018 ◽  
pp. 1-22
Author(s):  
Samir Ladaci ◽  
Karima Rabah ◽  
Mohamed Lashab
Keyword(s):  
Sliding Mode Control ◽  
Fractional Order ◽  
Chaotic Systems ◽  
Sliding Mode ◽  
Chaotic Behavior ◽  
Control Design ◽  
Lyapunov Stability Theory ◽  
Mode Control ◽  
Control Configuration ◽  
The Stability

This chapter investigates a new control design methodology for the synchronization of fractional-order Arneodo chaotic systems using a fractional-order sliding mode control configuration. This class of nonlinear fractional-order systems shows a chaotic behavior for a set of model parameters. The stability analysis of the proposed fractional-order sliding mode control law is performed by means of the Lyapunov stability theory. Simulation examples on fractional-order Arneodo chaotic systems synchronization are provided in presence of disturbances and noises. These results illustrate the effectiveness and robustness of this control design approach.

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 An Adaptive Control Scheme for Nonholonomic Mobile Robot with Parametric Uncertainty

10.5772/5801 ◽  
2005 ◽  
Vol 2 (1) ◽  
pp. 8 ◽  
Author(s):  
F. Mnif ◽  
F. Touati
Keyword(s):  
Mobile Robot ◽  
Parametric Uncertainty ◽  
Lyapunov Stability Theory ◽  
Original System ◽  
Stability And Convergence ◽  
Nonholonomic Mobile Robot ◽  
Control Scheme ◽  
The Stability ◽  
Invariant Control ◽  
Time Invariant

This paper addresses the problem of stabilizing the dynamic model of a nonholonomic mobile robot. A discontinuous adaptive state feedback controller is derived to achieve global stability and convergence of the trajectories of the of the closed loop system in the presence of parameter modeling uncertainty. This task is achieved by a non smooth transformation in the original system followed by the derivation of a smooth time invariant control in the new coordinates. The stability and convergence analysis is built on Lyapunov stability theory.

scholarly journals A Numerical Study of Nonlinear Fractional Order Partial Integro-Differential Equation with a Weakly Singular Kernel

2021 ◽  
Vol 5 (3) ◽  
pp. 85
Author(s):  
Tayyaba Akram ◽  
Zeeshan Ali ◽  
Faranak Rabiei ◽  
Kamal Shah ◽  
Poom Kumam
Keyword(s):  
Differential Equation ◽  
Fractional Order ◽  
Numerical Study ◽  
Stability And Convergence ◽  
Weakly Singular Kernel ◽  
Integro Differential Equation ◽  
Suggested Technique ◽  
Collocation Approach ◽  
The Stability ◽  
Efficiency And Reliability

Fractional differential equations can present the physical pathways with the storage and inherited properties due to the memory factor of fractional order. The purpose of this work is to interpret the collocation approach for tackling the fractional partial integro-differential equation (FPIDE) by employing the extended cubic B-spline (ECBS). To determine the time approximation, we utilize the Caputo approach. The stability and convergence analysis have also been analyzed. The efficiency and reliability of the suggested technique are demonstrated by two numerical applications, which support the theoretical results and the effectiveness of the implemented algorithm.

scholarly journals An Efficient Hybrid Numerical Scheme for Nonlinear Multiterm Caputo Time and Riesz Space Fractional-Order Diffusion Equations with Delay

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
A. K. Omran ◽  
M. A. Zaky ◽  
A. S. Hendy ◽  
V. G. Pimenov
Keyword(s):  
Fractional Order ◽  
Reaction Diffusion ◽  
Riesz Space ◽  
Reaction Diffusion Equation ◽  
Stability And Convergence ◽  
Energy Estimates ◽  
Discrete Energy ◽  
Fully Discrete ◽  
Exponential Rate Of Convergence ◽  
The Stability

In this paper, we construct and analyze a linearized finite difference/Galerkin–Legendre spectral scheme for the nonlinear multiterm Caputo time fractional-order reaction-diffusion equation with time delay and Riesz space fractional derivatives. The temporal fractional orders in the considered model are taken as 0 < β 0 < β 1 < β 2 < ⋯ < β m < 1 . The problem is first approximated by the L 1 difference method on the temporal direction, and then, the Galerkin–Legendre spectral method is applied on the spatial discretization. Armed by an appropriate form of discrete fractional Grönwall inequalities, the stability and convergence of the fully discrete scheme are investigated by discrete energy estimates. We show that the proposed method is stable and has a convergent order of 2 − β m in time and an exponential rate of convergence in space. We finally provide some numerical experiments to show the efficacy of the theoretical results.

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.

ANTICIPATING SYNCHRONIZATION OF INTEGER ORDER AND FRACTIONAL ORDER HYPER-CHAOTIC CHEN SYSTEM

2012 ◽  
Vol 26 (32) ◽  
pp. 1250211 ◽  
Author(s):  
PENGZHEN DONG ◽  
GANG SHANG ◽  
JIE LIU
Keyword(s):  
Fractional Order ◽  
Chaotic Systems ◽  
Lyapunov Stability Theory ◽  
Chen System ◽  
Integer Order ◽  
Research Fields ◽  
Long Time ◽  
Anticipation Time ◽  
Error System ◽  
The Stability

Such a problem, how to resolve the problem of long-term unpredictability of chaotic systems, has puzzled researchers in nonlinear research fields for a long time during the last decades. Recently, Voss et al. had proposed a new scheme to research the anticipating synchronization of integral-order nonlinear systems for arbitrary initial values and anticipation time. Can this anticipating synchronization be achieved with hyper-chaotic systems? In this paper, we discussed the application of anticipating synchronization in hyper-chaotic systems. Setting integer order and commensurate fractional order hyper-chaotic Chen systems as our research objects, we carry out the research on anticipating synchronization of above two systems based on analyzing the stability of the error system with the Krasovskill–Lyapunov stability theory. Simulation experiments show anticipating synchronization can be achieved in both integer order and fractional order hyper-chaotic Chen system for arbitrary initial value and arbitrary anticipation time.

CHAOTIC DYNAMICS AND SYNCHRONIZATION OF FRACTIONAL-ORDER CHUA'S CIRCUITS WITH A PIECEWISE-LINEAR NONLINEARITY

2005 ◽  
Vol 19 (20) ◽  
pp. 3249-3259 ◽  
Author(s):  
JUN GUO LU
Keyword(s):  
Lyapunov Exponent ◽  
Fractional Order ◽  
Stability Theory ◽  
Chaos Synchronization ◽  
Chaotic Dynamics ◽  
Piecewise Linear ◽  
Chua’S Circuit ◽  
Chua's Circuit ◽  
Synchronization Method ◽  
Simulation Results

In this paper, we numerically investigate the chaotic behaviors of the fractional-order Chua's circuit with a piecewise-linear nonlinearity. We find that chaos exists in the fractional-order Chua's circuit with order less than 3. The lowest order we find to have chaos is 2.7 in the homogeneous fractional-order Chua's circuit and 2.8 in the unhomogeneous fractional-order Chua's circuit. Our results are validated by the existence of a positive Lyapunov exponent. A chaos synchronization method is also presented for synchronizing the homogeneous fractional-order chaotic Chua's systems. The approach, based on stability theory of fractional-order linear systems, is simple and theoretically rigorous. It does not require the computation of the conditional Lyapunov exponents. Simulation results are used to visualize and illustrate the effectiveness of the proposed synchronization method.

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