scholarly journals Exact Analysis and Physical Realization of the 6-Lobe Chua Corsage Memristor

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-21 ◽  
Author(s):  
Zubaer I. Mannan ◽  
Changju Yang ◽  
Shyam P. Adhikari ◽  
Hyongsuk Kim
Keyword(s):  
Initial Conditions ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Dynamical Analysis ◽  
Theoretic Approach ◽  
Physical Realization ◽  
Pulse Input ◽  
Nonlinear Memory ◽  
Highly Nonlinear ◽  
Circuit Components

A novel generic memristor, dubbed the 6-lobe Chua corsage memristor, is proposed with its nonlinear dynamical analysis and physical realization. The proposed corsage memristor contains four asymptotically stable equilibrium points on its complex and diversified dynamic routes which reveals a 4-state nonlinear memory device. The higher degree of versatility of its dynamic routes reveal that the proposed memristor has a variety of dynamic paths in response to different initial conditions and exhibits a highly nonlinear contiguous DC V-I curve. The DC V-I curve of the proposed memristor is endowed with an explicit analytical parametric representation. Moreover, the derived three formulas, exponential trajectories of state xnt, time period tfn, and minimum pulse amplitude VA, are required to analyze the movement of the state trajectories on the piecewise linear (PWL) dynamic route map (DRM) of the corsage memristor. These formulas are universal, that is, applicable to any PWL DRM curves for any DC or pulse input and with any number of segments. Nonlinear dynamics and circuit and system theoretic approach are employed to explain the asymptotic quad-stable behavior of the proposed corsage memristor and to design a novel real memristor emulator using off-the-shelf circuit components.

scholarly journals Nonlinear Dynamics, Switching Kinetics and Physical Realization of the Family of Chua Corsage Memristors

Electronics ◽  
2020 ◽  
Vol 9 (2) ◽  
pp. 369
Author(s):  
Zubaer Ibna Mannan ◽  
Hyongsuk Kim
Keyword(s):  
Memory Device ◽  
Negative Slope ◽  
Stable Limit ◽  
Theoretic Approach ◽  
Physical Realization ◽  
Stable States ◽  
Pulse Input ◽  
Asymptotic Stable ◽  
Highly Nonlinear ◽  
Circuit Components

This article reviews the nonlinear dynamical attributes, switching kinetics, bifurcation analysis, and physical realization of a family of generic memristors, namely, Chua corsage memristors (CCM). CCM family contains three 1-st order generic memristor dubbed as 2-lobe, 4-lobe, and 6-lobe Chua corsage memristors and can be distinguished in accordance with their asymptotic stable states. The 2-lobe CCM has two asymptotically stable equilibrium states and regarded as a binary memory device. In contrast, the versatile 4-lobe CCM and 6-lobe CCM are regarded as a multi-bit-per-cell memory device as they exhibit three and four asymptotic stable states, respectively, on their complex and diversified dynamic routes. Due to the diversified dynamic routes, the CC memristors exhibit a highly nonlinear DC V-I curve. Unlike most published highly-nonlinear DC V-I curves with several disconnected branches, the DC V-I curves of CCMs are contiguous along with a locally active negative slope region. Moreover, the DC V-I curves and parametric representations of the CCMs are explicitly analytical. Switching kinetics of the CCM family can be demonstrated with universal formulas of exponential state trajectories xn(t), time period tfn, and applied minimum pulse amplitude VA and width Δw. These formulas are regarded universal as they can be applied to any piecewise linear dynamic routes for any DC or pulse input and with any number of segments. When local activity, and bifurcation and chaos theorems are employed, CMMs exhibit unique stable limit cycles spawn from a supercritical Hopf bifurcation along with static attractors. In addition, the nonlinear circuit and system theoretic approach is applied to explain the asymptotic stability behavior of CCMs and to design real memristor emulators using off-the-shelf circuit components.

scholarly journals Three-Saddle-Foci Chaotic Behavior of a Modified Jerk Circuit with Chua’s Diode

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1803
Author(s):  
Pattrawut Chansangiam
Keyword(s):  
Equilibrium Point ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Initial Point ◽  
Electrical Circuit ◽  
Nonlinear Resistor ◽  
Voltage Current Characteristic ◽  
Time Waveform

This paper investigates the chaotic behavior of a modified jerk circuit with Chua’s diode. The Chua’s diode considered here is a nonlinear resistor having a symmetric piecewise linear voltage-current characteristic. To describe the system, we apply fundamental laws in electrical circuit theory to formulate a mathematical model in terms of a third-order (jerk) nonlinear differential equation, or equivalently, a system of three first-order differential equations. The analysis shows that this system has three collinear equilibrium points. The time waveform and the trajectories about each equilibrium point depend on its associated eigenvalues. We prove that all three equilibrium points are of type saddle focus, meaning that the trajectory of (x(t),y(t)) diverges in a spiral form but z(t) converges to the equilibrium point for any initial point (x(0),y(0),z(0)). Numerical simulation illustrates that the oscillations are dense, have no period, are highly sensitive to initial conditions, and have a chaotic hidden attractor.

HOPF BIFURCATION FROM LINES OF EQUILIBRIA WITHOUT PARAMETERS IN MEMRISTOR OSCILLATORS

2010 ◽  
Vol 20 (02) ◽  
pp. 437-450 ◽  
Author(s):  
MARCELO MESSIAS ◽  
CRISTIANE NESPOLI ◽  
VANESSA A. BOTTA
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbits ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
Fixed Set ◽  
Electronic Element ◽  
Stable Periodic ◽  
Parameter Values

The memristor is supposed to be the fourth fundamental electronic element in addition to the well-known resistor, inductor and capacitor. Named as a contraction for memory resistor, its theoretical existence was postulated in 1971 by L. O. Chua, based on symmetrical and logical properties observed in some electronic circuits. On the other hand its physical realization was announced only recently in a paper published on May 2008 issue of Nature by a research team from Hewlett–Packard Company. In this work, we present the bifurcation analysis of two memristor oscillators mathematical models, given by three-dimensional five-parameter piecewise-linear and cubic systems of ordinary differential equations. We show that depending on the parameter values, the systems may present the coexistence of both infinitely many stable periodic orbits and stable equilibrium points. The periodic orbits arise from the change in local stability of equilibrium points on a line of equilibria, for a fixed set of parameter values. This phenomenon is a kind of Hopf bifurcation without parameters. We have numerical evidences that such stable periodic orbits form an invariant surface, which is an attractor of the systems solutions. The results obtained imply that even for a fixed set of parameters the two systems studied may or may not present oscillations, depending on the initial condition considered in the phase space. Moreover, when they exist, the amplitude of the oscillations also depends on the initial conditions.

scholarly journals Dynamic Analysis and Circuit Implementation of a New 4D Lorenz-Type Hyperchaotic System

2019 ◽  
Vol 2019 ◽  
pp. 1-17 ◽  
Author(s):  
A. Al-khedhairi ◽  
A. Elsonbaty ◽  
A. H. Abdel Kader ◽  
A. A. Elsadany
Keyword(s):  
Initial Conditions ◽  
Equilibrium Points ◽  
Dynamical Analysis ◽  
Global Dynamics ◽  
Hyperchaotic System ◽  
Circuit Implementation ◽  
Local And Global Dynamics ◽  
Invariant Surfaces ◽  
The Rich ◽  
Global Invariant

This paper attempts to further extend the results of dynamical analysis carried out on a recent 4D Lorenz-type hyperchaotic system while exploring new analytical results concerns its local and global dynamics. In particular, the equilibrium points of the system along with solution’s continuous dependence on initial conditions are examined. Then, a detailed Z2 symmetrical Bogdanov-Takens bifurcation analysis of the hyperchaotic system is performed. Also, the possible first integrals and global invariant surfaces which exist in system’s phase space are analytically found. Theoretical results reveal the rich dynamics and the complexity of system behavior. Finally, numerical simulations and a proposed circuit implementation of the hyperchaotic system are provided to validate the present analytical study of the system.

scholarly journals Mathematical analysis of a measles transmission dynamics model in Bangladesh with double dose vaccination

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Md Abdul Kuddus ◽  
M. Mohiuddin ◽  
Azizur Rahman
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Transmission Rate ◽  
Equilibrium Points ◽  
Rank Correlation ◽  
Dynamical Analysis ◽  
Model Parameters ◽  
Progression Rate ◽  
Double Dose ◽  
The Basic Reproduction Number

AbstractAlthough the availability of the measles vaccine, it is still epidemic in many countries globally, including Bangladesh. Eradication of measles needs to keep the basic reproduction number less than one $$(\mathrm{i}.\mathrm{e}. \, \, {\mathrm{R}}_{0}<1)$$ ( i . e . R 0 < 1 ) . This paper investigates a modified (SVEIR) measles compartmental model with double dose vaccination in Bangladesh to simulate the measles prevalence. We perform a dynamical analysis of the resulting system and find that the model contains two equilibrium points: a disease-free equilibrium and an endemic equilibrium. The disease will be died out if the basic reproduction number is less than one $$(\mathrm{i}.\mathrm{e}. \, \, {\mathrm{ R}}_{0}<1)$$ ( i . e . R 0 < 1 ) , and if greater than one $$(\mathrm{i}.\mathrm{e}. \, \, {\mathrm{R}}_{0}>1)$$ ( i . e . R 0 > 1 ) epidemic occurs. While using the Routh-Hurwitz criteria, the equilibria are found to be locally asymptotically stable under the former condition on $${\mathrm{R}}_{0}$$ R 0 . The partial rank correlation coefficients (PRCCs), a global sensitivity analysis method is used to compute $${\mathrm{R}}_{0}$$ R 0 and measles prevalence $$\left({\mathrm{I}}^{*}\right)$$ I ∗ with respect to the estimated and fitted model parameters. We found that the transmission rate $$(\upbeta )$$ ( β ) had the most significant influence on measles prevalence. Numerical simulations were carried out to commissions our analytical outcomes. These findings show that how progression rate, transmission rate and double dose vaccination rate affect the dynamics of measles prevalence. The information that we generate from this study may help government and public health professionals in making strategies to deal with the omissions of a measles outbreak and thus control and prevent an epidemic in Bangladesh.

3D Printing — The Basins of Tristability in the Lorenz System

2017 ◽  
Vol 27 (08) ◽  
pp. 1750128 ◽  
Author(s):  
Anda Xiong ◽  
Julien C. Sprott ◽  
Jingxuan Lyu ◽  
Xilu Wang
Keyword(s):  
Lorenz System ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basins Of Attraction ◽  
Symmetric Pair ◽  
State Space Approach ◽  
3D Printers ◽  
The Lorenz System ◽  
Almost All ◽  
Space Approach

The famous Lorenz system is studied and analyzed for a particular set of parameters originally proposed by Lorenz. With those parameters, the system has a single globally attracting strange attractor, meaning that almost all initial conditions in its 3D state space approach the attractor as time advances. However, with a slight change in one of the parameters, the chaotic attractor coexists with a symmetric pair of stable equilibrium points, and the resulting tri-stable system has three intertwined basins of attraction. The advent of 3D printers now makes it possible to visualize the topology of such basins of attraction as the results presented here illustrate.

Nonlinear Measure for Nonlinear Dynamic Processes Using Convergence Area: Typical Case Studies

2021 ◽  
Author(s):  
Joanofarc Xavier ◽  
S.K. Patnayak ◽  
Rames Panda
Keyword(s):  
Phase Portrait ◽  
Case Studies ◽  
Initial Conditions ◽  
Operating Point ◽  
Linear Approximations ◽  
Promising Tool ◽  
Highly Nonlinear ◽  
Phase Portrait Analysis ◽  
Nonlinear Dynamic Processes ◽  
Nonlinear Measure

Abstract Several industrial chemical processes exhibit severe nonlinearity. This paper addresses the computational and nonlinear issues occurring in many typical industrial problems in aspects of its stability, strength of nonlinearity and input output dynamics. In this article, initially, a prospective investigation is conducted on various nonlinear processes through phase portrait analysis to understand their stability status at different initial conditions about the vicinity of the operating point of the process. To estimate the degree of nonlinearity, for input perturbations from its nominal value, a novel nonlinear measure is put forward, that anticipates on the converging area between the nonlinear and their linearized responses. The nonlinearity strength is fixed between 0 and 1 to classify processes to be mild, medium or highly nonlinear. The most suitable operating point, for which the system remains asymptotically stable is clearly identified from the phase portrait. The metric can be contemplated as a promising tool to measure the nonlinearity of Industrial case studies at different linear approximations. Numerical simulations are executed in Matlab to compute , which conveys that the nonlinear dynamics of each Industrial example is very sensitive to input perturbations at different linear approximations. In addition to the identified metric, nonlinear lemmas are framed to select appropriate control schemes for the processes based on its numerical value of nonlinearity..

A new hyperchaotic particle motion system and its control

2019 ◽  
Vol 30 (12) ◽  
pp. 2050004
Author(s):  
Ning Cui ◽  
Junhong Li
Keyword(s):  
Particle Motion ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Multiple Equilibrium ◽  
Hopf Bifurcations ◽  
Hyperchaotic System ◽  
Motion System ◽  
The Rich ◽  
Theoretical Analyses ◽  
Multiple Equilibrium Points

This paper formulates a new hyperchaotic system for particle motion. The continuous dependence on initial conditions of the system’s solution and the equilibrium stability, bifurcation, energy function of the system are analyzed. The hyperchaotic behaviors in the motion of the particle on a horizontal smooth plane are also investigated. It shows that the rich dynamic behaviors of the system, including the degenerate Hopf bifurcations and nondegenerate Hopf bifurcations at multiple equilibrium points, the irregular variation of Hamiltonian energy, and the hyperchaotic attractors. These results generalize and improve some known results about the particle motion system. Furthermore, the constraint of hyperchaos control is obtained by applying Lagrange’s method and the constraint change the system from a hyperchaotic state to asymptotically state. The numerical simulations are carried out to verify theoretical analyses and to exhibit the rich hyperchaotic behaviors.

scholarly journals Crossing Limit Cycles of Planar Piecewise Linear Hamiltonian Systems without Equilibrium Points

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 755
Author(s):  
Rebiha Benterki ◽  
Jaume LLibre
Keyword(s):  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Upper Bounds ◽  
Linear Hamiltonian Systems ◽  
Straight Lines

In this paper, we study the existence of limit cycles of planar piecewise linear Hamiltonian systems without equilibrium points. Firstly, we prove that if these systems are separated by a parabola, they can have at most two crossing limit cycles, and if they are separated by a hyperbola or an ellipse, they can have at most three crossing limit cycles. Additionally, we prove that these upper bounds are reached. Secondly, we show that there is an example of two crossing limit cycles when these systems have four zones separated by three straight lines.

scholarly journals A Fractional-Order Discrete Noninvertible Map of Cubic Type: Dynamics, Control, and Synchronization

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-21
Author(s):  
Xiaojun Liu ◽  
Ling Hong ◽  
Lixin Yang ◽  
Dafeng Tang
Keyword(s):  
Fractional Order ◽  
Initial Conditions ◽  
Dimensional Space ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
State Variables ◽  
Discrete Maps ◽  
The Stability ◽  
Control And Synchronization ◽  
Simultaneous Variation

In this paper, a new fractional-order discrete noninvertible map of cubic type is presented. Firstly, the stability of the equilibrium points for the map is examined. Secondly, the dynamics of the map with two different initial conditions is studied by numerical simulation when a parameter or a derivative order is varied. A series of attractors are displayed in various forms of periodic and chaotic ones. Furthermore, bifurcations with the simultaneous variation of both a parameter and the order are also analyzed in the three-dimensional space. Interior crises are found in the map as a parameter or an order varies. Thirdly, based on the stability theory of fractional-order discrete maps, a stabilization controller is proposed to control the chaos of the map and the asymptotic convergence of the state variables is determined. Finally, the synchronization between the proposed map and a fractional-order discrete Loren map is investigated. Numerical simulations are used to verify the effectiveness of the designed synchronization controllers.

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