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.

PIECEWISE-LINEAR CHAOTIC SYSTEMS WITH A SINGLE EQUILIBRIUM POINT

2000 ◽  
Vol 10 (09) ◽  
pp. 2015-2060 ◽  
Author(s):  
TAO YANG ◽  
LEON O. CHUA
Keyword(s):  
Linear Function ◽  
Autonomous System ◽  
Chaotic Systems ◽  
Random Search ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
Piecewise Linear Function ◽  
Chaotic Attractors ◽  
Bifurcation Diagrams

As a unique paradigm for chaos, the various versions of Chua's circuits and equations consists of a three-dimensional autonomous system with a three-segment piecewise-linear function which gives rise to three equilibrium points. This paper considers the possibility of simplifying the system configurations of piecewise-linear chaotic systems based on the structures of Chua's systems. We study a new class of piecewise-linear three-dimensional autonomous system with a three-segment piecewise-linear function. However, unlike Chua's systems, the systems we study in this paper have only single equilibrium points. To find chaotic attractors from this class of systems, we use a systematic random-search process to search the parameter space. The searching process consists of three stages. For the first stage, we simply count the number of points on a Poincaré section and find candidates for chaotic attractors. At the second stage, Lyapunov exponents are calculated for selecting chaotic attractors from the candidates. Finally, bifurcation diagrams constructed around the located chaotic attractors are used to find different types of chaotic attractors. Many qualitatively different chaotic attractors of this class of systems had been found and presented in this paper. Another method to simplify the configurations of a piecewise-linear chaotic system is to reduce the number of segments of the piecewise-linear function. We have developed some chaotic systems with a two-segment piecewise-linear function and which gives rise to two equilibrium points. Many color illustrations of chaotic attractors and bifurcation diagrams are presented.

ON CONSTRUCTING COMPLEX GRID MULTIWING CHAOTIC SYSTEM BY SWITCHING CONTROL AND MIRROR SYMMETRY CONVERSION

2013 ◽  
Vol 23 (07) ◽  
pp. 1350115 ◽  
Author(s):  
CHAOXIA ZHANG ◽  
SIMIN YU ◽  
QIANHUA HE ◽  
JINXIN RUAN
Keyword(s):  
Chaotic System ◽  
Mirror Symmetry ◽  
Chaotic Systems ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
Quadratic Function ◽  
Chaotic Attractors ◽  
Novel Method ◽  
Switching Controller ◽  
Index 2

This paper further investigates a novel method, namely the mirror and double-mirror symmetry conversion both in z direction and in y direction, for generating complex grid multiwing chaotic attractors from a three-dimensional quadratic chaotic system. First, by designing a switching controller with even-symmetry multisegment quadratic function to extend the number of saddle-focus equilibrium points with index 2 in x direction, multiwing chaotic attractors are obtained. Based on this approach, then, different from the methods proposed in the previous literature, by the mirror and double-mirror symmetry conversion with respect to x-axis and y-axis respectively, various intended grid n × m-wing chaotic systems can be obtained. The principle and method for generating grid multiwing are also given. Numerical simulations and circuit realizations have demonstrated the feasibility and effectiveness of the proposed approaches.

scholarly journals DESIGN OF TIME DELAYED CHAOTIC CIRCUIT WITH THRESHOLD CONTROLLER

2011 ◽  
Vol 21 (03) ◽  
pp. 725-735 ◽  
Author(s):  
K. SRINIVASAN ◽  
I. RAJA MOHAMED ◽  
K. MURALI ◽  
M. LAKSHMANAN ◽  
SUDESHNA SINHA
Keyword(s):  
Numerical Simulations ◽  
Linear Function ◽  
Piecewise Linear ◽  
Piecewise Linear Function ◽  
Operational Amplifiers ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Chaotic Oscillator ◽  
Threshold Values ◽  
Chaotic Circuit

A novel time delayed chaotic oscillator exhibiting mono- and double scroll complex chaotic attractors is designed. This circuit consists of only a few operational amplifiers and diodes and employs a threshold controller for flexibility. It efficiently implements a piecewise linear function. The control of piecewise linear function facilitates controlling the shape of the attractors. This is demonstrated by constructing the phase portraits of the attractors through numerical simulations and hardware experiments. Based on these studies, we find that this circuit can produce multi-scroll chaotic attractors by just introducing more number of threshold values.

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.

A NOVEL METHOD TO SIMULATE FORCE-CHEST DISPLACEMENT RELATIONSHIP DURING CARDIOPULMONARY RESUSCITATION

2011 ◽  
Vol 23 (06) ◽  
pp. 519-526
Author(s):  
Xinwu Xie ◽  
Gaofeng Wei ◽  
Rui Liu ◽  
Qiuming Sun ◽  
Aijuan Ni ◽  
...  
Keyword(s):  
Cardiopulmonary Resuscitation ◽  
Mechanical System ◽  
Piecewise Linear ◽  
Variable Stiffness ◽  
Simulation Method ◽  
Piecewise Linear Function ◽  
Hysteresis Curve ◽  
System A ◽  
Novel Method ◽  
The Relationship

Survival from cardiac arrest is dependent on timely cardiopulmonary resuscitation (CPR). Experimental and modeling work had shown that the relationship between compression force and sternal displacement had a tendency of hysteresis curve, which in manikins was rather lineal. A novel method was introduced to improve the mechanical characteristics of the manikins using a variable-stiffness springs group-damper structure, in which the spring's group and the damper simulate the elastic and damping of human chest respectively. To do the simulation, the model of the human chest's mechanical during CPR based on Gruben's pre-result was modified and the elastic part was fitted by piecewise linear function to get the springs' stiffness. The variable-stiffness springs group system was designed accordingly to simulate the human chest' stiffness during CPR, and a damper was designed to simulate the damping of chest. The damper and the variable-stiffness springs group were paralleling combined, forming a mechanical system. A sample system was realized and the test results showed that there were nonlinear elasticity and adequate viscosity in this system, whose coefficients were adjacent to Gruben's 'typical' human. According to the method, the mechanical system could be changed to accurately simulate different models, and its stiffness could be easily adjusted by varying the interval between the nest springs' top. The simulation method could be used directly in manikin, whose mechanical character would be improved and then more adequate training would be provided.

scholarly journals Global Dynamics of the Hastings-Powell System

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Luis N. Coria
Keyword(s):  
Numerical Simulations ◽  
Equilibrium Points ◽  
Heteroclinic Orbits ◽  
Invariant Sets ◽  
Chaotic Attractors ◽  
Global Dynamics ◽  
State Variables ◽  
System Parameters ◽  
First Order ◽  
Extremum Conditions

This paper studies the problem of bounding a domain that contains all compact invariant sets of the Hastings-Powell system. The results were obtained using the first-order extremum conditions and the iterative theorem to a biologically meaningful model. As a result, we calculate the bounds given by a tetrahedron with excisions, described by several inequalities of the state variables and system parameters. Therefore, a region is identified where all the system dynamics are located, that is, its compact invariant sets: equilibrium points, periodic-homoclinic-heteroclinic orbits, and chaotic attractors. It was also possible to formulate a nonexistence condition of the compact invariant sets. Additionally, numerical simulations provide examples of the calculated boundaries for the chaotic attractors or periodic orbits. The results provide insights regarding the global dynamics of the system.

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.

Generation of chaotic attractors without equilibria via piecewise linear systems

2017 ◽  
Vol 28 (01) ◽  
pp. 1750008 ◽  
Author(s):  
R. J. Escalante-González ◽  
E. Campos-Cantón
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Chaotic Attractor ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Chaotic Attractors ◽  
Necessary And Sufficient ◽  
Switched Dynamical System ◽  
Theoretical Results

In this paper, we present a mechanism of generation of a class of switched dynamical system without equilibrium points that generates a chaotic attractor. The switched dynamical systems are based on piecewise linear (PWL) systems. The theoretical results are formally given through a theorem and corollary which give necessary and sufficient conditions to guarantee that a linear affine dynamical system has no equilibria. Numerical results are in accordance with the theory.

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.

DYNAMICS OF A PARTICULAR LORENZ TYPE SYSTEM

2010 ◽  
Vol 21 (07) ◽  
pp. 973-982 ◽  
Author(s):  
GIORGIO E. TESTONI ◽  
PAULO C. RECH
Keyword(s):  
Equilibrium Points ◽  
Three Dimensional ◽  
Type System ◽  
Equation System ◽  
Largest Lyapunov Exponent ◽  
Differential Equation System ◽  
Order Ordinary Differential Equation ◽  
First Order ◽  
The Stability ◽  
The Right

In this paper we analytically and numerically investigate the dynamics of a nonlinear three-dimensional autonomous first-order ordinary differential equation system, obtained from paradigmatic Lorenz system by suppressing the y variable in the right-hand side of the second equation. The Routh–Hurwitz criterion is used to decide on the stability of the nontrivial equilibrium points of the system, as a function of the parameters. The dynamics of the system is numerically characterized by using diagrams that associate colors to largest Lyapunov exponent values in the parameter-space. Additionally, phase-space plots and bifurcation diagrams are used to characterize periodic and chaotic attractors.

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