Analysis of the Dynamics of Piecewise Linear Memristors

2016 ◽  
Vol 26 (13) ◽  
pp. 1650217 ◽  
Author(s):  
Fangfang Jiang ◽  
Zhicheng Ji ◽  
Qing-Guo Wang ◽  
Jitao Sun
Keyword(s):  
Piecewise Linear ◽  
Equilibrium Points ◽  
Characteristic Curve ◽  
Piecewise Linear Function ◽  
Excited Oscillation ◽  
Exciting Source ◽  
Memristive Circuits ◽  
The Stability ◽  
Straight Lines ◽  
The Mathematical Model

In this paper, we consider a class of flux controlled memristive circuits with a piecewise linear memristor (i.e. the characteristic curve of the memristor is given by a piecewise linear function). The mathematical model is described by a discontinuous planar piecewise smooth differential system, which is defined on three zones separated by two parallel straight lines [Formula: see text] (called as discontinuity lines in discontinuous differential systems). We first investigate the stability of equilibrium points and the existence and uniqueness of a crossing limit cycle for the memristor-based circuit under self-excited oscillation. We then analyze the existence of periodic orbits of forced nonlinear oscillation for the memristive circuit with an external exciting source. Finally, we give numerical simulations to show good matches between our theoretical and simulation results.

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.

Characteristics of Memristor Series-Parallel Connection Circuit with Applications

2013 ◽  
Vol 284-287 ◽  
pp. 2485-2489
Author(s):  
Wei Hong Yin ◽  
Li Dan Wang ◽  
Shu Kai Duan
Keyword(s):  
Large Scale ◽  
Piecewise Linear ◽  
Piecewise Linear Function ◽  
Parallel Connection ◽  
Low Energy Consumption ◽  
Circuit Component ◽  
Parallel Network ◽  
The Mathematical Model ◽  
Nonlinear Resistance ◽  
Resistance Value

Memristor is a nonlinear resistance with the ability of memory and its resistance depends on the amount of charge or magnetic flux. As a new circuit component, the memristor is not only expected to promote the evolution of electronic industry but also has huge potential application in many fields because of its small size, low energy consumption and be large scale integrated easily. In this paper according to the mathematical model of memristor based on a piecewise-linear function, the characteristics of memristor series-parallel connection circuit are analyzed by MATLAB and SPICE simulations. A series of simulation results show that the memristive series-parallel network is still a nonlinear element, the resistance value of memristor depends on the initial condition greatly and the electronic characteristics are more complex. It is also applied to the ladder signal generator which can be used in the controller circuit.

scholarly journals Stability Analysis of Mathematical Models of the Dynamics of Spread of Meningitis with the Effects of Vaccination, Campaigns and Treatment

2020 ◽  
Vol 17 (1) ◽  
pp. 71-81
Author(s):  
Sulma Sulma ◽  
Syamsuddin Toaha ◽  
Kasbawati Kasbawati
Keyword(s):  
Infectious Disease ◽  
Equilibrium Point ◽  
Equilibrium Points ◽  
Vaccination Campaign ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Infected People ◽  
The Stability ◽  
Vaccination Campaigns ◽  
The Mathematical Model

Meningitis is an infectious disease caused by bacteria, viruses, and protosoa and has the potential to cause an outbreak. Vaccination and campaign are carried out as an effort to prevent the spread of meningitis, treatment reduces the number of deaths caused by the disease and the number of infected people. This study aims to analyze and determine the stability of equilibrium point of the mathematical model of the spread of meningitis using five compartments namely susceptibles, carriers, infected without symptoms, infected with symptoms, and recovered with the effect of vaccination, campaign, and treatment. The results obtained from the analysis of the model that there are two equilibrium points, namely non endemic and endemic equilibrium points. If a certain condition is met then the non endemic equilibrium point will be asymptotically stable. Numerical simulations show that the spread of disease decreases with the influence of vaccination, campaign, and treatment.

scholarly journals A Novel Method for Constructing Grid Multi-Wing Butterfly Chaotic Attractors via Nonlinear Coupling Control

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Yun Huang
Keyword(s):  
Piecewise Linear ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
Piecewise Linear Function ◽  
Chaotic Attractors ◽  
Nonlinear Coupling ◽  
Linear Coupling ◽  
First Order ◽  
Novel Method ◽  
Chaotic Characteristic

A new method is presented to construct grid multi-wing butterfly chaotic attractors. Based on the three-dimensional Lorenz system, two first-order differential equations are added along with one linear coupling controller, respectively. And a piecewise linear function, which is taken into the linear coupling controller, is designed to form a nonlinear coupling controller; thus a five-dimensional chaotic system is produced, which is able to generate gird multi-wing butterfly chaotic attractors. Through the analysis of the equilibrium points, Lyapunov exponent spectrums, bifurcation diagrams, and Poincaré mapping in this system, the chaotic characteristic of the system is verified. Apart from the research above, an electronic circuit is designed to implement the system. The circuit experimental results are in accordance with the results of numerical simulation, which verify the availability and feasibility of this method.

The Optimal Design of Saving Water on Kaifeng Agriculture Irrigated Area

2013 ◽  
Vol 726-731 ◽  
pp. 3100-3104
Author(s):  
Chang Hu ◽  
Hui Li Yang ◽  
Dang Sheng Li
Keyword(s):  
Programming Problem ◽  
Optimal Allocation ◽  
Yellow River ◽  
Piecewise Linear ◽  
Underground Water ◽  
Piecewise Linear Function ◽  
Decision Variables ◽  
Kaifeng City ◽  
Set Up ◽  
The Mathematical Model

The agricultural irrigation water of Kaifeng city is mainly supplied by the Yellow River and the local underground water. According to conditions of the irrigation area water supply and groundwater reduction for many years, the paper puts forward the optimal allocation and the supplying water programs on using combination of surface water and groundwater. By using variable one-dimensional search technology, it will express nonlinearity function is approximated by piecewise linear function so as to set up the mathematical model of controlled object, so the nonlinear programming problem will be turned into linear programming problem. Thus choose different decision variables and different parameters to use computing in order to get to approach the optimal allocation of water resources.

scholarly journals Some Properties of Membership Functions Composed of Triangle Functions and Piecewise Linear FunctionsThis work has been partially supported in 2019-2020 by the domestic research grant of University of Marketing and Distribution Sciences in Kobe (Japan).

2021 ◽  
Vol 29 (2) ◽  
pp. 103-115
Author(s):  
Takashi Mitsuishi
Keyword(s):  
Fuzzy Sets ◽  
Linear Function ◽  
Piecewise Linear ◽  
Gaussian Function ◽  
Piecewise Linear Function ◽  
Approximate Reasoning ◽  
Evaluation Function ◽  
Membership Functions ◽  
Composite Function ◽  
Straight Lines

Summary. IF-THEN rules in fuzzy inference is composed of multiple fuzzy sets (membership functions). IF-THEN rules can therefore be considered as a pair of membership functions [7]. The evaluation function of fuzzy control is composite function with fuzzy approximate reasoning and is functional on the set of membership functions. We obtained continuity of the evaluation function and compactness of the set of membership functions [12]. Therefore, we proved the existence of pair of membership functions, which maximizes (minimizes) evaluation function and is considered IF-THEN rules, in the set of membership functions by using extreme value theorem. The set of membership functions (fuzzy sets) is defined in this article to verifier our proofs before by Mizar [9], [10], [4]. Membership functions composed of triangle function, piecewise linear function and Gaussian function used in practice are formalized using existing functions. On the other hand, not only curve membership functions mentioned above but also membership functions composed of straight lines (piecewise linear function) like triangular and trapezoidal functions are formalized. Moreover, different from the definition in [3] formalizations of triangular and trapezoidal function composed of two straight lines, minimum function and maximum functions are proposed. We prove, using the Mizar [2], [1] formalism, some properties of membership functions such as continuity and periodicity [13], [8].

scholarly journals Stability Analysis of Divorce Dynamics Models

2020 ◽  
Vol 17 (2) ◽  
pp. 267-279
Author(s):  
Syamsir Muaraf ◽  
Syamsuddin Toaha ◽  
Kasbawati Kasbawati
Keyword(s):  
Stability Analysis ◽  
Local Stability ◽  
Reproduction Number ◽  
Transmission Rate ◽  
Equilibrium Points ◽  
Existence And Stability ◽  
Next Generation Matrix ◽  
The Stability ◽  
Dynamics Models ◽  
The Mathematical Model

This article examines the mathematical model of divorce. This model consists of four population classes, namely the Married class (M), the population class who experiences separation of separated beds (S), the population class who is divorced by Divorce (D), and the population class who experiences depression or stress due to divorce Hardship (H). This study focuses on the stability analysis of divorce-free and endemic equilibrium points. Local stability was analyzed using linearization and eigenvalues ​​methods. In addition, the basic reproduction number  is provided via the next generation matrix method. The existence and stability of the equilibrium point are determined from . The results showed that the rate of interaction between population M and populations other than H is very influential on efforts to minimize divorce. Divorce can be minimized when the transmission rate is reduced to . Reducing the transmission rate and increasing the rate of transfer from split bed class to married class can turn divorce endemic cases into non-endemic cases. A numerical simulation is given to confirm the analysis results.

Memristor Based van der Pol Oscillation Circuit

2014 ◽  
Vol 24 (12) ◽  
pp. 1450154 ◽  
Author(s):  
Yimin Lu ◽  
Xianfeng Huang ◽  
Shaobin He ◽  
Dongdong Wang ◽  
Bo Zhang
Keyword(s):  
Analog Circuit ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Circuit Element ◽  
Nonlinear Properties ◽  
Oscillation Circuit ◽  
Excited Oscillation ◽  
Exciting Source ◽  
The Stability ◽  
Passive Circuit

The memristor is referred to as the fourth fundamental passive circuit element of which inherent nonlinear properties offer to construct the chaos circuits. In this paper, a flux-controlled memristor circuit is developed, and then a van der Pol oscillator is implemented based on this new memristor circuit. The stability of the circuit, the occurring conditions of Hopf bifurcation and limit circle of the self-excited oscillation are analyzed; meanwhile, under the condition of the circuit with an external exciting source, the circuit exhibits a complicated nonlinear dynamic behavior, and chaos occurs within a certain parameter set. The memristor based van der Pol oscillator, furthermore, has been created by an analog circuit utilizing active elements, and there is a good agreement between the circuit responses and numerical simulations of the van der Pol equation. In the consequence, a new approach has been proposed to generate chaos within a nonautonomous circuit system.

BIFURCATION IN A PIECEWISE LINEAR CIRCUIT WITH SWITCHING BOUNDARIES

2012 ◽  
Vol 22 (02) ◽  
pp. 1250034 ◽  
Author(s):  
ZHENGDI ZHANG ◽  
QINSHENG BI
Keyword(s):  
Periodic Orbits ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Period Doubling ◽  
Dynamical Evolution ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Generalized Jacobian ◽  
Linear Circuit ◽  
The Stability

By introducing time-dependent power source, a periodically excited piecewise linear circuit with double-scroll is established. In the absence of the excitation, all possible equilibrium points as well as the stability conditions are presented. Analyzing the corresponding characteristic equations with perturbation method, Hopf bifurcation conditions associated with the equilibria are derived, which can be demonstrated by the numerical simulations. The Hopf bifurcations of the two symmetric equilibrium points may cause two symmetric periodic orbits, which lead to single-scroll chaotic attractors via sequences of period-doubling bifurcations with the variation of the parameters. The two chaotic attractors expand to interact with each other to form an enlarged chaotic attractor with double-scroll. The behaviors on the switching boundaries are investigated by the generalized Jacobian matrix. When periodic excitation is applied to work on the circuit, three periodic orbits with the frequency of the excitation may exist, which can be called generalized equilibrium points (GEPs) with the same characteristic polynomials as those of the corresponding equilibrium points for the autonomous case. It is shown that when the trajectories do not pass across the switching boundaries, the solutions are the same as the GEPs. However, when the trajectories pass across the switching boundaries, complicated behaviors will take place. Three forms of chaotic attractors via different bifurcations can be observed and the influence of the switching boundaries on the phase portraits is discussed to explore the mechanism of the dynamical evolution.

scholarly journals Model Matematika Untuk Kontrol Campak Menggunakan Vaksinasi

2013 ◽  
Vol 2 (2) ◽  
pp. 81
Author(s):  
Maesaroh Ulfa ◽  
Sugiyanto Sugiyanto
Keyword(s):  
Vaccination Coverage ◽  
Measles Virus ◽  
Equilibrium Points ◽  
Herd Immunity ◽  
Effective Strategy ◽  
Literature Study ◽  
Measles Vaccination ◽  
Deadly Disease ◽  
The Stability ◽  
The Mathematical Model

Measles (also known as Rubeola, measles 9 day) is a highly contagious virus infection, characterized by fever, cough, conjunctiva (inflammation of the tissue lining of the eye) and skin rash. The disease is caused by infection of measles virus paramyxovirus cluster. It is a deadly disease. Vaccination is the most effective strategy to prevent the disease. It is generally given to children. This research aims to establish a model of the effect of measles vaccination, forming the point of equilibrium and analyze the stability, create a simulation model and interpret them, and to know the design to optimize the vaccination coverage required, so it can reduce the spread of this disease. This research was conducted by the method of literature study. It is expected to provide an overview of the mathematical model used to control measles vaccination with division of classes SEIR. The steps taken is identifying the problem, formulating assumptions to simplifying the model, making the transfer diagram, defining parameters, determining the equilibrium points and analyzing the stability, simulating the model, and forming the design to optimize the vaccination. Then from this research can be obtained free balance point of endemic and diseases and their stability. Based on the results obtained, the simulation is done by taking the data in Yogyakarta, and obtained vaccination coverage with two doses that can increase the herd immunity with lower vaccination coverage.

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