scholarly journals Synchronization of Periodic Self-Oscillators Interacting via Memristor-Based Coupling

2020 ◽  
Vol 30 (07) ◽  
pp. 2050096 ◽  
Author(s):  
Ivan A. Korneev ◽  
Vladimir V. Semenov ◽  
Tatiana E. Vadivasova
Keyword(s):  
Numerical Simulation ◽  
Phase Space ◽  
Initial Conditions ◽  
Phase Locking ◽  
Equilibrium Points ◽  
State Equation ◽  
Harmonic Reduction ◽  
Synchronization Region ◽  
Parameter Values ◽  
Line Of Equilibria

A model of two self-sustained oscillators interacting through memristive coupling is studied. The memristive coupling is realized by using a cubic memristor model. Numerical simulation is combined with theoretical analysis by means of quasi-harmonic reduction. It is shown that the specifics of the memristor nonlinearity results in the appearance of infinitely many equilibrium points which form a line of equilibria in the phase space of the system under study. It is established that the possibility to observe the effect of phase locking in the considered system depends on both parameter values and initial conditions. Consequently, the boundaries of the synchronization region are determined by the initial conditions. It is demonstrated that introducing or adding a small term into the memristor state equation gives rise to the disappearance of the line of equilibria and eliminates the dependence of synchronization on the initial conditions.

Bifurcations Leading to Nonlinear Oscillations in a 3D Piecewise Linear Memristor Oscillator

2014 ◽  
Vol 24 (01) ◽  
pp. 1430001 ◽  
Author(s):  
Marluce da Cruz Scarabello ◽  
Marcelo Messias
Keyword(s):  
Phase Space ◽  
Nonlinear Oscillations ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Central Zone ◽  
Electric Circuit ◽  
Invariant Set ◽  
Piecewise Linear System ◽  
Electronic Elements ◽  
Parameter Values

In this paper, we make a bifurcation analysis of a mathematical model for an electric circuit formed by the four fundamental electronic elements: one memristor, one capacitor, one inductor and one resistor. The considered model is given by a discontinuous piecewise linear system of ordinary differential equations, defined on three zones in ℝ3, determined by |z| < 1 (called the central zone) and |z| > 1 (the external zones). We show that the z-axis is filled by equilibrium points of the system, and analyze the linear stability of the equilibria in each zone. Due to the existence of this line of equilibria, the phase space ℝ3 is foliated by invariant planes transversal to the z-axis and parallel to each other, in each zone. In this way, each solution is contained in a three-piece invariant set formed by part of a plane contained in the central zone, which is extended by two half planes in the external zones. We also show that the system may present nonlinear oscillations, given by the existence of infinitely many periodic orbits, each one belonging to one such invariant set and passing by two of the three zones or passing by the three zones. These orbits arise due to homoclinic and heteroclinic bifurcations, obtained varying one parameter in the studied model, and may also exist for some fixed sets of parameter values. This intricate phase space may bring some light to the understanding of these memristor properties. The analytical and numerical results obtained extend the analysis presented in [Itoh & Chua, 2009; Messias et al., 2010].

CHUA’S EQUATION WITH CUBIC NONLINEARITY

1996 ◽  
Vol 06 (12a) ◽  
pp. 2175-2222 ◽  
Author(s):  
ANSHAN HUANG ◽  
LADISLAV PIVKA ◽  
CHAI WAH WU ◽  
MARTIN FRANZ
Keyword(s):  
Computer Program ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Tabular Form ◽  
Cubic Nonlinearity ◽  
Bifurcation Phenomena ◽  
Hands On ◽  
Autonomous Differential Equations ◽  
Parameter Values

In this tutorial paper we present one of the simplest autonomous differential equations capable of generating chaotic behavior. Some of the fundamental routes to chaos and bifurcation phenomena are demonstrated with examples. A brief discussion of equilibrium points and their stability is given. For the convenience of the reader, a short computer program written in QuickBASIC is included to give the reader a possibility of quick hands-on experience with the generation of chaotic phenomena without using sophisticated numerical simulators. All the necessary parameter values and initial conditions are provided in a tabular form. Eigenvalue diagrams showing regions with particular eigenvalue patterns are given.

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 Stability Analysis of the Chaotic Lorenz System with a State-Feedback Controller

2021 ◽  
Author(s):  
Dan Jones
Keyword(s):  
Strange Attractor ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Geometric Approach ◽  
Stable Equilibrium Point ◽  
The Lorenz System ◽  
The Stability ◽  
Parameter Values

The Lorenz model is considered a benchmark system in chaotic dynamics in that it displays extraordinary sensitivity to initial conditions and the strange attractor phenomenon. Even though the system tends to amplify perturbations, it is indeed possible to convert a strange attractor to a non-chaotic one using various control schemes. In this work it is shown that the chaotic behavior of the Lorenz system can be suppressed through the use of a feedback loop driven by a quotient controller. The stability of the controlled Lorenz system is evaluated near its equilibrium points using Routh-Hurwitz testing, and the global stability of the controlled system is established using a geometric approach. It is shown that the controlled Lorenz system has only one globally stable equilibrium point for the set of parameter values under consideration.

Numerical Simulation on Temperature Field in Multi-Layers Inside-Beam Powder Feeding when Laser Cladding

2012 ◽  
Vol 499 ◽  
pp. 114-119 ◽  
Author(s):  
Ming Di Wang ◽  
Shi Hong Shi ◽  
X.B. Liu ◽  
Cheng Fa Song ◽  
Li Ning Sun
Keyword(s):  
Numerical Simulation ◽  
Heat Source ◽  
Laser Beam ◽  
Cooling Rate ◽  
Light Source ◽  
Laser Cladding ◽  
Laser Scanning ◽  
Initial Conditions ◽  
Cladding Layer ◽  
Powder Feeding

Numerical simulation of laser cladding is the main research topics for many universities and academes, but all researchers used the Gaussian laser light source. Due to using inside-beam powder feeding for laser cladding, the laser is dispersed by the cone-shaped mirror, and then be focused by the annular mirror, the laser can be assumed as the light source of uniform intensity.In this paper,the temperature of powder during landing selected as the initial conditions, and adopting the life-and-death unit method, the moving point heat source and the uniform heat source are realized. In the thickness direction, using the small melt layer stacking method, a finite element model has been established, and layer unit is acted layer by layer, then a virtual reality laser cladding manu-facturing process is simulated. Calculated results show that the surface temperature of the cladding layer depends on the laser scanning speed, powder feed rate, defocus distance. As cladding layers increases, due to the heat conduction into the base too late, bath temperature will gradually increase. The highest temperature is not at the laser beam, but at the later point of the laser beam. In the clad-ding process, the temperature cooling rate of the cladding layer in high temperature section is great, and in the low-temperature, cooling rate is relatively small. These conclusions are also similar with the normal laser cladding. Finally, some experiments validate the simulation results. The trends of simulating temperature are fit to the actual temperature, and the temperature gradient can also ex-plain the actual shape of cross-section.

Numerical Simulation Study on High-Temperature Ventilated Cavitating Flow Considering the Compressibility of Gases

2012 ◽  
Vol 569 ◽  
pp. 395-399
Author(s):  
Jing Zhao ◽  
Guo Yu Wang ◽  
Yan Zhao ◽  
Yue Ju Liu
Keyword(s):  
Numerical Simulation ◽  
State Equation ◽  
Field Structure ◽  
Cavitating Flow ◽  
Simulation Approach ◽  
Gas Velocity ◽  
Simulation Data ◽  
Ventilated Cavity ◽  
Supercavitating Flow ◽  
Fluctuation Range

A numerical simulation approach of ventilated cavity considering the compressibility of gases is established in this paper, introducing the gas state equation into the calculation of ventilated supercavitating flow. Based on the comparison of computing results and experimental data, we analyzes the differences between ventilated cavitating flow fields with and without considered the compressibility of gases. The effect of ventilation on the ventilated supercavitating flow field structure is discussed considering the compressibility of gases. The results show that the simulation data of cavity form and resistance, which takes the compressibility of gases into account, accord well with the experimental ones. With the raising of ventilation temperature, the gas fraction in the front cavity and the gas velocity in the cavity increase, and the cavity becomes flat. The resistance becomes lower at high ventilation temperature, but its fluctuation range becomes larger than that at low temperature.

scholarly journals MATHEMATICAL MODEL OF THREE SPECIES FOOD CHAIN INTERACTION WITH MIXED FUNCTIONAL RESPONSE

2012 ◽  
Vol 09 ◽  
pp. 334-340 ◽  
Author(s):  
MADA SANJAYA WS ◽  
ISMAIL BIN MOHD ◽  
MUSTAFA MAMAT ◽  
ZABIDIN SALLEH
Keyword(s):  
Mathematical Model ◽  
Functional Response ◽  
Food Chain ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Bifurcation Points ◽  
Dynamical Behaviors ◽  
Practical Life ◽  
Parameter Values ◽  
Chain Interaction

In this paper, we study mathematical model of ecology with a tritrophic food chain composed of a classical Lotka-Volterra functional response for prey and predator, and a Holling type-III functional response for predator and super predator. There are two equilibrium points of the system. In the parameter space, there are passages from instability to stability, which are called Hopf bifurcation points. For the first equilibrium point, it is possible to find bifurcation points analytically and to prove that the system has periodic solutions around these points. Furthermore the dynamical behaviors of this model are investigated. Models for biologically reasonable parameter values, exhibits stable, unstable periodic and limit cycles. The dynamical behavior is found to be very sensitive to parameter values as well as the parameters of the practical life. Computer simulations are carried out to explain the analytical findings.

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.

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