scholarly journals Multiple Coexisting Attractors and Hysteresis in the Generalized Ueda Oscillator

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Kehui Sun ◽  
A Di-li Duo Li-kun ◽  
Yanqing Dong ◽  
Huihai Wang ◽  
Ke Zhong
Keyword(s):  
Nonlinear Function ◽  
Power Dynamics ◽  
Largest Lyapunov Exponent ◽  
Hysteresis Phenomenon ◽  
Coexisting Attractors ◽  
Restoring Force ◽  
Dynamical Behaviors ◽  
Promising Application ◽  
Multiple Coexisting Attractors ◽  
Forced Nonlinear Oscillator

A periodically forced nonlinear oscillator called the generalized Ueda oscillator is proposed. The restoring force term of this equation consists of a nonlinear functionsgn(x)and an absolute function with a variant power. Dynamics is investigated by detailed numerical analysis as well as dynamic simulation, including the largest Lyapunov exponent, phase diagrams, and bifurcation diagrams. Multiple coexisting attractors and complex hysteresis phenomenon are observed. The results show that this system has rich dynamical behaviors, and it has a promising application in the fields of science and engineering.

Various Types of Coexisting Attractors in a New 4D Autonomous Chaotic System

2017 ◽  
Vol 27 (09) ◽  
pp. 1750142 ◽  
Author(s):  
Qiang Lai ◽  
Akif Akgul ◽  
Xiao-Wen Zhao ◽  
Huiqin Pei
Keyword(s):  
Numerical Simulation ◽  
Dynamic Behavior ◽  
Chaotic System ◽  
Limit Cycles ◽  
Electronic Circuit ◽  
Strange Attractors ◽  
Bifurcation Diagrams ◽  
Coexisting Attractors ◽  
Signum Function ◽  
Multiple Coexisting Attractors

An unique 4D autonomous chaotic system with signum function term is proposed in this paper. The system has four unstable equilibria and various types of coexisting attractors appear. Four-wing and four-scroll strange attractors are observed in the system and they will be broken into two coexisting butterfly attractors and two coexisting double-scroll attractors with the variation of the parameters. Numerical simulation shows that the system has various types of multiple coexisting attractors including two butterfly attractors with four limit cycles, two double-scroll attractors with a limit cycle, four single-scroll strange attractors, four limit cycles with regard to different parameters and initial values. The coexistence of the attractors is determined by the bifurcation diagrams. The chaotic and hyperchaotic properties of the attractors are verified by the Lyapunov exponents. Moreover, we present an electronic circuit to experimentally realize the dynamic behavior of the system.

Various Attractors, Coexisting Attractors and Antimonotonicity in a Simple Fourth-Order Memristive Twin-T Oscillator

2018 ◽  
Vol 28 (04) ◽  
pp. 1850050 ◽  
Author(s):  
Ling Zhou ◽  
Chunhua Wang ◽  
Xin Zhang ◽  
Wei Yao
Keyword(s):  
Fourth Order ◽  
Chaotic Systems ◽  
Basin Of Attraction ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Coexisting Attractors ◽  
Dynamical Behaviors ◽  
Random Bit Generator ◽  
In Series ◽  
Dynamical Features

By replacing the resistor in a Twin-T network with a generalized flux-controlled memristor, this paper proposes a simple fourth-order memristive Twin-T oscillator. Rich dynamical behaviors can be observed in the dynamical system. The most striking feature is that this system has various periodic orbits and various chaotic attractors generated by adjusting parameter [Formula: see text]. At the same time, coexisting attractors and antimonotonicity are also detected (especially, two full Feigenbaum remerging trees in series are observed in such autonomous chaotic systems). Their dynamical features are analyzed by phase portraits, Lyapunov exponents, bifurcation diagrams and basin of attraction. Moreover, hardware experiments on a breadboard are carried out. Experimental measurements are in accordance with the simulation results. Finally, a multi-channel random bit generator is designed for encryption applications. Numerical results illustrate the usefulness of the random bit generator.

A Novel Simple Hyperchaotic System with Two Coexisting Attractors

2019 ◽  
Vol 29 (14) ◽  
pp. 1950203 ◽  
Author(s):  
Jiaopeng Yang ◽  
Zhengrong Liu
Keyword(s):  
Complex Dynamics ◽  
Chaotic Dynamic ◽  
Cross Product ◽  
Hyperchaotic System ◽  
Compound Structure ◽  
Coexisting Attractors ◽  
Dynamical Behaviors ◽  
Periodic Attractors ◽  
New Hyperchaotic System ◽  
New System

This article introduces a new hyperchaotic system of four-dimensional autonomous ordinary differential equations, with only cubic cross-product nonlinearities, which can respectively display two hyperchaotic attractors with only nonhyperbolic equilibria line. Several issues such as basic dynamical behaviors, routes to chaos, bifurcations, periodic windows, and the compound structure of the new hyperchaotic and chaotic system are investigated, either theoretically or numerically. Of particular interest is the fact that the two coexisting attractors of the new hyperchaotic system are symmetrical, and this hyperchaotic system can generate plenty of complex dynamics including two coexisting chaotic or periodic attractors. Moreover, some chaotic features of the attractor are justified numerically. Finally, 0-1 test is used to analyze and describe the complex chaotic dynamic behavior of the new system.

scholarly journals Multistability in a Fractional-Order Centrifugal Flywheel Governor System and Its Adaptive Control

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Bo Yan ◽  
Shaobo He ◽  
Shaojie Wang
Keyword(s):  
Fractional Order ◽  
Lyapunov Stability Theory ◽  
Adaptive Controller ◽  
Basins Of Attraction ◽  
Entropy Measure ◽  
Coexisting Attractors ◽  
Minimum Order ◽  
System Parameters ◽  
Multiple Coexisting Attractors ◽  
Derivative Order

In this paper, a 4D fractional-order centrifugal flywheel governor system is proposed. Dynamics including the multistability of the system with the variation of system parameters and the derivative order are investigated by Lyapunov exponents (LEs), bifurcation diagram, phase portrait, entropy measure, and basins of attraction, numerically. It shows that the minimum order for chaos of the fractional-order centrifugal flywheel governor system is q = 0.97, and the system has rich dynamics and produces multiple coexisting attractors. Moreover, the system is controlled by introducing the adaptive controller which is proved by the Lyapunov stability theory. Numerical analysis results verify the effectiveness of the proposed method.

Responses and energy transmissibility of a viscoelastic isolation system with a power-form restoring force under delayed feedback control

2015 ◽  
Vol 23 (14) ◽  
pp. 2291-2306 ◽  
Author(s):  
Dongmei Huang ◽  
Wei Xu ◽  
Wenxian Xie ◽  
Wei Liu
Keyword(s):  
Feedback Control ◽  
Frequency Response ◽  
Vibration Isolation ◽  
Multiple Scales ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Restoring Force ◽  
Isolation System ◽  
Feedback Parameter ◽  
Dynamical Behaviors

In this paper, combination of cubic nonlinearity and time delay is designed to improve the performance of a viscoelastic isolation system with a power-form restoring force. By the method of multiple scales, the amplitude-frequency response, stability, backbone curve and energy transmissibility are considered. More specifically, three nonlinear cubic delayed feedback control methodologies are examined: position, velocity and acceleration delayed feedback. It is found that the viscoelastic damping coefficient can induce multi-valued response, especially frequency island phenomenon. In this regard, the isolation system indicates the softening behavior for under-linear restoring force and hardening behavior for over-linear restoring force. And equivalent damping and jump avoidance condition are first proposed to interpret the effect of feedback control loop on dynamical behaviors. Furthermore, with the purpose of improving the stability and reducing the vibration, suitable feedback parameter pairs are determined by the frequency response together with stability conditions. Finally, the vibration isolation property is predicted based on energy transmissibility in different cases. Results show that the strategy proposed in this paper is practicable and feedback control parameters are significant factors to alter dynamical behaviors, and more importantly, to improve the isolation effectiveness for the viscoelastic isolation system.

Dynamic Response of Nonlinear Oscillators With Hysteresis

2015 ◽  
Author(s):  
Biagio Carboni ◽  
Walter Lacarbonara
Keyword(s):  
Hysteresis Loops ◽  
Base Excitation ◽  
Response Curves ◽  
Restoring Force ◽  
Single Degree Of Freedom ◽  
Wire Ropes ◽  
Dynamical Behaviors ◽  
Nonlinear Features ◽  
Restoring Forces ◽  
Force Displacement

The nonlinear features of the steady-state periodic response of hysteretic oscillators are investigated. Frequency-response curves of base-excited single-degree-of-freedom (SDOF) systems possessing different hysteretic restoring forces are numerically obtained employing a continuation procedure based on the Jacobian of the Poincaré map. The memory-dependent restoring forces are expressed as a direct summation of linear and cubic elastic components and a hysteretic part described by a modified version of the Bouc-Wen law. The resulting force-displacement curves feature a pinching around the origin. Depending on the hysteresis material parameters (which regulate the shapes of the hysteresis loops), the oscillator exhibits hardening, softening and softening-hardening behaviors in which the switching from softening to hardening takes place above certain base excitation amplitudes. A comprehensive analysis in the parameters space is performed to identify the thresholds of these different behaviors. The restoring force features here considered have been experimentally obtained by means of an original rheological device comprising assemblies of steel and shape memory wire ropes. This study is carried out also with the aim of designing the restoring forces which give rise to dynamical behaviors useful for a variety of applications.

scholarly journals IMAGE ENCRYPTION BASED ON TWO-DIMENSIONAL FRACTIONAL QUADRIC POLYNOMIAL MAP

Fractals ◽  
2021 ◽  
pp. 2140041
Author(s):  
ZE-YU LIU ◽  
TIE-CHENG XIA ◽  
HUA-RONG FENG ◽  
CHANG-YOU MA
Keyword(s):  
Image Encryption ◽  
Color Image ◽  
Chaotic Map ◽  
Encryption Algorithm ◽  
Phase Portraits ◽  
Largest Lyapunov Exponent ◽  
Two Dimensional ◽  
Polynomial Map ◽  
Dynamical Behaviors ◽  
Image Encryption Algorithm

A new fractional two-dimensional quadric polynomial discrete chaotic map (2D-QPDM) with the discrete fractional difference is proposed. Afterwards, the new dynamical behaviors are observed, so that the bifurcation diagrams, the largest Lyapunov exponent plot and the phase portraits of the proposed map are given, respectively. The new discrete fractional map is exploited into color image encryption algorithm and it is illustrated with several examples. The proposed image encryption algorithm is analyzed in six aspects which indicates that the proposed algorithm is superior to other known algorithms as a conclusion.

scholarly journals A locally Active Discrete Memristor Model and Its Application in a Hyperchaotic Map

2021 ◽  
Author(s):  
Minglin Ma ◽  
Yang Yang ◽  
Zhicheng Qiu ◽  
Yuexi Peng ◽  
Yichuang Sun ◽  
...  
Keyword(s):  
Simulation Analysis ◽  
Engineering Application ◽  
Two Dimensional ◽  
Coexisting Attractors ◽  
Distribution Diagram ◽  
Dynamical Behaviors ◽  
Memristive System ◽  
Definition Of ◽  
First Time ◽  
Lyapunov Exponent Spectrum

Abstract The continuous memristor is a popular topic of research in recent years, however, there is rare discussion about the discrete memristor model, especially the locally active discrete memristor model. This paper proposes a locally active discrete memristor model for the first time and proves the three fingerprints characteristics of this model according to the definition of generalized memristor. A novel hyperchaotic map is constructed by coupling the discrete memristor with a two-dimensional generalized square map. The dynamical behaviors are analyzed with attractor phase diagram, bifurcation diagram, Lyapunov exponent spectrum, and dynamic behavior distribution diagram. Numerical simulation analysis shows that there is significant improvement in the hyperchaotic area, the quasi-periodic area and the chaotic complexity of the two-dimensional map when applying the locally active discrete memristor. In addition, antimonotonicity and transient chaos behaviors of system are reported. In particular, the coexisting attractors can be observed in this discrete memristive system, resulting from the different initial values of the memristor. Results of theoretical analysis are well verified with hardware experimental measurements. This paper lays a great foundation for future analysis and engineering application of the discrete memristor and relevant the study of other hyperchaotic maps.

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