2 × 2-SCROLL ATTRACTORS GENERATED IN A THREE-DIMENSIONAL SMOOTH AUTONOMOUS SYSTEM

2007 ◽  
Vol 17 (11) ◽  
pp. 4153-4157 ◽  
Author(s):  
WENBO LIU ◽  
WALLACE K. S. TANG ◽  
GUANRONG CHEN
Keyword(s):  
Numerical Simulations ◽  
Continuous Time ◽  
Autonomous System ◽  
Electronic Circuit ◽  
Three Dimensional ◽  
Chaotic Attractors ◽  
First Time

In this letter, a three-dimensional continuous-time smooth autonomous system with quadratic nonlinear terms is proposed for generating multiscroll chaotic attractors. Observation of 2 × 2-scroll attractors generated from this kind of system is reported for the first time. The result is confirmed by both numerical simulations and electronic circuit experiments.

A Novel Cubic–Equilibrium Chaotic System with Coexisting Hidden Attractors: Analysis, and Circuit Implementation

2017 ◽  
Vol 27 (04) ◽  
pp. 1850066 ◽  
Author(s):  
Viet-Thanh Pham ◽  
Christos Volos ◽  
Sajad Jafari ◽  
Tomasz Kapitaniak
Keyword(s):  
Chaotic System ◽  
Autonomous System ◽  
Chaotic Systems ◽  
Electronic Circuit ◽  
Complex Dynamics ◽  
Three Dimensional ◽  
Chaotic Attractors ◽  
Circuit Implementation ◽  
Hidden Attractors ◽  
Dynamical Properties

Chaotic systems with a curve of equilibria have attracted considerable interest in theoretical researches and engineering applications because they are categorized as systems with hidden attractors. In this paper, we introduce a new three-dimensional autonomous system with cubic equilibrium. Fundamental dynamical properties and complex dynamics of the system have been investigated. Of particular interest is the coexistence of chaotic attractors in the proposed system. Furthermore, we have designed and implemented an electronic circuit to verify the feasibility of such a system with cubic equilibrium.

A 3D Autonomous System with Infinitely Many Chaotic Attractors

2019 ◽  
Vol 29 (12) ◽  
pp. 1950166 ◽  
Author(s):  
Ting Yang ◽  
Qigui Yang
Keyword(s):  
Dynamical System ◽  
Differential Equations ◽  
Autonomous System ◽  
Three Dimensional ◽  
Chaotic Attractors ◽  
Double Zero ◽  
Finite Dimensional ◽  
Periodic Attractors ◽  
The Stability ◽  
Autonomous Dynamical System

Intuitively, a finite-dimensional autonomous system of ordinary differential equations can only generate finitely many chaotic attractors. Amazingly, however, this paper finds a three-dimensional autonomous dynamical system that can generate infinitely many chaotic attractors. Specifically, this system can generate infinitely many coexisting chaotic attractors and infinitely many coexisting periodic attractors in the following three cases: (i) no equilibria, (ii) only infinitely many nonhyperbolic double-zero equilibria, and (iii) both infinitely many hyperbolic saddles and nonhyperbolic pure-imaginary equilibria. By analyzing the stability of double-zero and pure-imaginary equilibria, it is shown that the classic Shil’nikov criteria fail in verifying the existence of chaos in the above three cases.

Ratchet mechanism of drops climbing a vibrated oblique plate

2017 ◽  
Vol 835 ◽  
Author(s):  
Hang Ding ◽  
Xi Zhu ◽  
Peng Gao ◽  
Xi-Yun Lu
Keyword(s):  
Contact Angle ◽  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Contact Angle Hysteresis ◽  
Three Dimensional ◽  
Direct Numerical Simulations ◽  
Ratchet Mechanism ◽  
Wetted Area ◽  
First Time ◽  
Good Agreement

In this paper, we investigate the ratchet mechanism of drops climbing a vibrated oblique plate based on three-dimensional direct numerical simulations, which for the first time reproduce the existing experiment (Brunet et al., Phys. Rev. Lett., vol. 99, 2007, 144501). With the help of numerical simulations, we identify an interesting and important wetting behaviour of the climbing drop; that is, the breaking of symmetry due to the inclination of the plate with respect to the acceleration leads to a hysteresis of the wetted area in one period of harmonic vibration. In particular, the average wetted area in the downhill stage is larger than that in the uphill stage, which is found to be responsible for the uphill net motion of the drop. A new hydrodynamic model is proposed to interpret the ratchet mechanism, taking account of the effects of the acceleration and contact angle hysteresis. The predictions of the theoretical analysis are in good agreement with the numerical results.

scholarly journals Integer and Fractional GeneralT-System and Its Application to Control Chaos and Synchronization

2015 ◽  
Vol 2015 ◽  
pp. 1-14
Author(s):  
Mihaela Neamţu ◽  
Anamaria Liţoiu ◽  
Petru C. Strain
Keyword(s):  
Numerical Simulations ◽  
Fractional Order ◽  
Electronic Circuit ◽  
Three Dimensional ◽  
Hopf Bifurcations ◽  
Secure Communications ◽  
Bifurcation Parameter ◽  
Potential Applications ◽  
Chaos Attractor ◽  
Control And Synchronization

We propose a three-dimensional autonomous nonlinear system, called the generalTsystem, which has potential applications in secure communications and the electronic circuit. For the generalTsystem with delayed feedback, regarding the delay as bifurcation parameter, we investigate the effect of the time delay on its dynamics. We determine conditions for the existence of the Hopf bifurcations and analyze their direction and stability. Also, the fractional order generalT-system is proposed and analyzed. We provide some numerical simulations, where the chaos attractor and the dynamics of the Lyapunov coefficients are taken into consideration. The effectiveness of the chaotic control and synchronization on schemes for the new fractional order chaotic system are verified by numerical simulations.

Generating Multiple Spherical Attractors via Parameter Switching Control

2021 ◽  
Vol 31 (16) ◽  
Author(s):  
Changchun Sun ◽  
Qicheng Xu
Keyword(s):  
Continuous Time ◽  
Three Dimensional ◽  
Switching Control ◽  
Control Technique ◽  
Chaotic Attractors ◽  
Time System ◽  
Offset Boosting ◽  
State Dependent ◽  
Parameter Switching ◽  
Continuous Time System

A three-dimensional smooth continuous-time system with a parameter and two quadratic terms is constructed and a spherical attractor is generated. There exist multiple coexisting spherical attractors based on offset boosting. Two classes of switching signals that depend on the time and the state are designed respectively. By employing a parameter switching control technique, multiple spherical attractors can be generated. Simultaneously, complex chaotic attractors can also be generated by designing a state-dependent switching signal. Numerical examples and corresponding simulations show the effectiveness of the switching control technique.

CAN A THREE-DIMENSIONAL SMOOTH AUTONOMOUS QUADRATIC CHAOTIC SYSTEM GENERATE A SINGLE FOUR-SCROLL ATTRACTOR?

2004 ◽  
Vol 14 (04) ◽  
pp. 1395-1403 ◽  
Author(s):  
WENBO LIU ◽  
GUANRONG CHEN
Keyword(s):  
Phase Space ◽  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Chaotic System ◽  
Chaotic Attractor ◽  
Three Dimensional ◽  
Chaotic Attractors ◽  
Circuit Implementation ◽  
System Parameters ◽  
New Chaotic System

Recently, we have investigated a new chaotic system of three-dimensional autonomous quadratic ordinary differential equations, and found that the system visually displays a four-scroll chaotic attractor confirmed by both numerical simulations and circuit implementation. In this paper, we further study the following question: Is it really true that this system can generate a four-scroll chaotic attractor, or is it only a numerical artifact? By a more careful theoretical analysis along with some further numerical simulations, we conclude that the four-scroll chaotic attractor of this system, which we observed on both computer and oscilloscope, cannot actually exist in theory. The fact is that this system has two co-existing two-scroll chaotic attractors that are arbitrarily close in the phase space for some system parameters, therefore extremely tiny numerical round-off errors or signal fluctuations will nudge the system orbit to switch from one attractor to another, thereby forming the seemingly single four-scroll chaotic attractor on screen display.

scholarly journals Analysis and electronic circuit implementation of an integer-and fractional-order Shimizu-Morioka system

2021 ◽  
Vol 67 (6 Nov-Dec) ◽  
Author(s):  
François Kapche Tagne ◽  
Guillaume Honoré KOM ◽  
Marceline Motchongom Tingue ◽  
Pierre Kisito Talla ◽  
V. Kamdoum Tamba
Keyword(s):  
Numerical Simulations ◽  
Fractional Order ◽  
Close Agreement ◽  
Electronic Circuit ◽  
Dynamical Behavior ◽  
Chaotic Attractors ◽  
Transient Chaos ◽  
Circuit Implementation ◽  
Bifurcation Structures ◽  
Electronic Circuit Implementation

The dynamics of an integer-order and fractional-order Lorenz like system called Shimizu-Morioka system is investigated in this paper. It is shown thatinteger-order Shimizu-Morioka system displays bistable chaotic attractors, monostable chaotic attractors and coexistence between bistable and monostable chaotic attractors. For suitable choose of parameters, the fractional-order Shimizu-Morioka system exhibits bistable chaotic attractors, monostable chaotic attractors, metastable chaos (i.e. transient chaos) and spiking oscillations. The bifurcation structures reveal that the fractional-order derivative affects considerably the dynamics of Shimizu-Morioka system. The chain fractance circuit is used to designand implement the integer- and fractional-order Shimizu-Morioka system in Pspice. A close agreement is observed between PSpice based circuit simulations and numerical simulations analysis. The results obtained in this work were not reported previously in the interger as well as in fractional-order Shimizu-Morioka system and thus represent an important contribution which may help us in better understanding of the dynamical behavior of this class of systems.

A NEW HYPERCHAOTIC SYSTEM AND ITS CIRCUIT IMPLEMENTATION

2010 ◽  
Vol 20 (04) ◽  
pp. 1201-1208 ◽  
Author(s):  
MINGHUA LIU ◽  
JIUCHAO FENG ◽  
CHI K. TSE
Keyword(s):  
Nonlinear Control ◽  
Bifurcation Diagram ◽  
Continuous Time ◽  
Control Parameter ◽  
Electronic Circuit ◽  
Three Dimensional ◽  
Hyperchaotic System ◽  
Circuit Implementation ◽  
Time Autonomous ◽  
New Hyperchaotic System

A four-dimensional continuous-time autonomous hyperchaotic system is proposed in this letter. This system is constructed by incorporating a nonlinear control to a three-dimensional continuous-time autonomous chaotic system. The hyperchaotic system is analyzed by studying the spectrum of Lyapunov exponents and the corresponding bifurcation diagram. The system exhibits chaotic, periodic, hyperchaotic behaviors for different values of a selected control parameter. Also, a simple electronic circuit is designed and implemented. Simulations and experimental observations verify the analytical results.

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.

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