scholarly journals Symmetry Breaking, Coexisting Bubbles, Multistability, and Its Control for a Simple Jerk System with Hyperbolic Tangent Nonlinearity

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-24 ◽  
Author(s):  
Léandre Kamdjeu Kengne ◽  
Jacques Kengne ◽  
Justin Roger Mboupda Pone ◽  
Hervé Thierry Kamdem Tagne
Keyword(s):  
Coupling Parameter ◽  
Constant Term ◽  
Basins Of Attraction ◽  
Phase Portraits ◽  
Largest Lyapunov Exponent ◽  
Nonlinear Phenomena ◽  
Hyperbolic Tangent ◽  
Critical Transitions ◽  
Jerk System ◽  
Theoretical Results

Symmetry is an important property found in a large number of nonlinear systems. The study of chaotic systems with symmetry is well documented. However, the literature is unfortunately very poor concerning the dynamics of such systems when their symmetry is altered or broken. In this paper, we investigate the dynamics of a simple jerk system with hyperbolic tangent nonlinearity (Kengne et al., Chaos Solitons, and Fractals, 2017) whose symmetry is broken by adding a constant term modeling an external excitation force. We demonstrate that the modified system experiences several unusual and striking nonlinear phenomena including coexisting bifurcation branches, hysteretic dynamics, coexisting asymmetric bubbles, critical transitions, and multiple (i.e., up to six) coexisting asymmetric attractors for some suitable ranges of system parameters. These features are highlighted by exploiting common nonlinear analysis tools such as graphs of largest Lyapunov exponent, bifurcation diagrams, phase portraits, and basins of attraction. The control of multistability is investigated by using the method of linear augmentation. We demonstrate that the multistable system can be converted to a monostable state by smoothly adjusting the coupling parameter. The theoretical results are confirmed by performing a series of PSpice simulations based on an electronic analogue of the system.

An Autonomous Simple Chaotic Jerk System with Stable and Unstable Equilibria Using Reverse Sine Hyperbolic Functions

2020 ◽  
Vol 30 (05) ◽  
pp. 2050070 ◽  
Author(s):  
Manoj Joshi ◽  
Ashish Ranjan
Keyword(s):  
Stable Equilibrium ◽  
Chaotic Behavior ◽  
Basins Of Attraction ◽  
Electrical Circuit ◽  
Phase Portraits ◽  
Hidden Attractors ◽  
Hyperbolic Functions ◽  
Simple Implementation ◽  
Route To Chaos ◽  
Jerk System

This article introduces a new simple jerk system with sine hyperbolic nonlinearity which gives the hidden attractor. An autonomous simple implementation of jerk system experiences an important and striking feature of hidden attractors with both stable equilibrium and unstable equilibrium using a reverse nonlinearity function with parametrically controlled approach. Some basic properties of the system are well studied and analyzed in terms of route to chaos, basins of attraction, Lyapunov exponent (LE), bifurcation sequences, coexistence of attractor and phase portraits. The chaotic behavior of the new system is investigated through numerical simulation and their equivalent electrical circuit implementation using single amplifier with few passive elements. The justification of theoretical observation of the proposed chaotic system is perfectly observed in PSPICE simulation and laboratory experiment.

Control of Multistability in a Self-Excited Memristive Hyperchaotic Oscillator

2019 ◽  
Vol 29 (09) ◽  
pp. 1950119 ◽  
Author(s):  
T. Fonzin Fozin ◽  
R. Kengne ◽  
J. Kengne ◽  
K. Srinivasan ◽  
M. Souffo Tagueu ◽  
...  
Keyword(s):  
Coupling Parameter ◽  
Basins Of Attraction ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Weak Values ◽  
Controlled System ◽  
Nonlinear Dynamical ◽  
System Parameters ◽  
Control Scheme ◽  
Lyapunov Exponent Spectrum

This paper investigates the control of multistability in a self-excited memristive hyperchaotic oscillator using linear augmentation method. Such a method is advantageous in the case of system parameters that are inaccessible. The effectiveness of the applied control scheme is revealed numerically through the nonlinear dynamical tools including bifurcation diagrams, Lyapunov exponent spectrum, phase portraits, basins of attraction and relative basin sizes. Results of such numerical methods reveal that the asymmetric pair of chaotic attractors which were coexisting with the symmetric periodic one in the system, are progressively annihilated as the coupling parameter is increasing. The main transitions observed in the control system are the coexistence of three distinct attractors for weak values of the coupling strength. Above a certain critical value of the coupling parameter, only two attractors are now coexisting within the system. Finally, for higher values of the control strength, the controlled system becomes regular and monostable.

Dynamical Effects of Offset Terms on a Modified Chua’s Oscillator and Its Circuit Implementation

2021 ◽  
Vol 31 (16) ◽  
Author(s):  
Léandre Kamdjeu Kengne ◽  
Karthikeyan Rajagopal ◽  
Nestor Tsafack ◽  
Paul Didier Kamdem Kuate ◽  
Balamurali Ramakrishnan ◽  
...  
Keyword(s):  
Initial Conditions ◽  
Basins Of Attraction ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Nonlinear Phenomena ◽  
Hidden Attractor ◽  
Chua’S Oscillator ◽  
Kirchhoff’S Laws ◽  
Attraction Basins ◽  
The Mathematical Model

This paper addresses the effects of offset terms on the dynamics of a modified Chua’s oscillator. The mathematical model is derived using Kirchhoff’s laws. The model is analyzed with the help of the maximal Lyapunov exponent, bifurcation diagrams, phase portraits, and basins of attraction. The investigations show that the offset terms break the symmetry of the system, generating more complex nonlinear phenomena like coexisting asymmetric bifurcations, coexisting asymmetric attractors, asymmetric double-scroll chaotic attractors and asymmetric attraction basins. Also, a hidden attractor (period-1 limit cycle) is found when varying the initial conditions. More interestingly, this latter attractor coexists with all other self-excited ones. A microcontroller-based implementation of the circuit is carried out to verify the numerical investigations.

Nonlinear Dynamics of an Imperfect Microbeam Under an Axial Load and Electric Excitation

2011 ◽  
Author(s):  
Laura Ruzziconi ◽  
Mohammad I. Younis ◽  
Stefano Lenci
Keyword(s):  
Nonlinear Dynamics ◽  
Microelectromechanical Systems ◽  
Complex Dynamics ◽  
Nonlinear Behavior ◽  
Single Mode ◽  
Phase Portraits ◽  
Nonlinear Phenomena ◽  
Theoretical Point ◽  
Initial Shape ◽  
Electric Excitation

This study is motivated by the growing attention, both from a practical and a theoretical point of view, toward the nonlinear behavior of microelectromechanical systems (MEMS). We analyze the nonlinear dynamics of an imperfect microbeam under an axial force and electric excitation. The imperfection of the microbeam, typically due to microfabrication processes, is simulated assuming the microbeam to be of a shallow arched initial shape. The device has a bistable static behavior. The aim is that of illustrating the nonlinear phenomena, which arise due to the coupling of mechanical and electrical nonlinearities, and discussing their usefulness for the engineering design of the microstructure. We derive a single-mode-reduced-order model by combining the classical Galerkin technique and the Pade´ approximation. Despite its apparent simplicity, this model is able to capture the main features of the complex dynamics of the device. Extensive numerical simulations are performed using frequency response diagrams, attractor-basins phase portraits, and frequency-dynamic voltage behavior charts. We investigate the overall scenario, up to the inevitable escape, obtaining the theoretical boundaries of appearance and disappearance of the main attractors. The main features of the nonlinear dynamics are discussed, stressing their existence and their practical relevance. We focus on the coexistence of robust attractors, which leads to a considerable versatility of behavior. This is a very attractive feature in MEMS applications. The ranges of coexistence are analyzed in detail, remarkably at high values of the dynamic excitation, where the penetration of the escape (dynamic pull-in) inside the double well may prevent the safe jump between the attractors.

scholarly journals An optimal fourth-order family of modified Cauchy methods for finding solutions of nonlinear equations and their dynamical behavior

2019 ◽  
Vol 17 (1) ◽  
pp. 1567-1598
Author(s):  
Tianbao Liu ◽  
Xiwen Qin ◽  
Qiuyue Li
Keyword(s):  
Nonlinear Equations ◽  
Fourth Order ◽  
Dynamical Behavior ◽  
Basins Of Attraction ◽  
Second Derivative ◽  
Third Order ◽  
Numerical Tests ◽  
The Family ◽  
Theoretical Results ◽  
High Computational Efficiency

Abstract In this paper, we derive and analyze a new one-parameter family of modified Cauchy method free from second derivative for obtaining simple roots of nonlinear equations by using Padé approximant. The convergence analysis of the family is also considered, and the methods have convergence order three. Based on the family of third-order method, in order to increase the order of the convergence, a new optimal fourth-order family of modified Cauchy methods is obtained by using weight function. We also perform some numerical tests and the comparison with existing optimal fourth-order methods to show the high computational efficiency of the proposed scheme, which confirm our theoretical results. The basins of attraction of this optimal fourth-order family and existing fourth-order methods are presented and compared to illustrate some elements of the proposed family have equal or better stable behavior in many aspects. Furthermore, from the fractal graphics, with the increase of the value m of the series in iterative methods, the chaotic behaviors of the methods become more and more complex, which also reflected in some existing fourth-order methods.

Multistability Control of Hysteresis and Parallel Bifurcation Branches through a Linear Augmentation Scheme

2019 ◽  
Vol 75 (1) ◽  
pp. 11-21 ◽  
Author(s):  
T. Fonzin Fozin ◽  
G. D. Leutcho ◽  
A. Tchagna Kouanou ◽  
G. B. Tanekou ◽  
R. Kengne ◽  
...  
Keyword(s):  
Coupling Parameter ◽  
Phase Portraits ◽  
Coexisting Attractors ◽  
Nonlinear Dynamical ◽  
Numerical Investigations ◽  
System Parameters ◽  
Chua’S Oscillator ◽  
Control Scheme ◽  
Periodic Attractors ◽  
Basin Of Attractions

AbstractMultistability analysis has received intensive attention in recently, however, its control in systems with more than two coexisting attractors are still to be discovered. This paper reports numerically the multistability control of five disconnected attractors in a self-excited simplified hyperchaotic canonical Chua’s oscillator (hereafter referred to as SHCCO) using a linear augmentation scheme. Such a method is appropriate in the case where system parameters are inaccessible. The five distinct attractors are uncovered through the combination of hysteresis and parallel bifurcation techniques. The effectiveness of the applied control scheme is revealed through the nonlinear dynamical tools including bifurcation diagrams, Lyapunov’s exponent spectrum, phase portraits and a cross section basin of attractions. The results of such numerical investigations revealed that the asymmetric pair of chaotic and periodic attractors which were coexisting with the symmetric periodic one in the SHCCO are progressively annihilated as the coupling parameter is increasing. Monostability is achieved in the system through three main crises. First, the two asymmetric periodic attractors are annihilated through an interior crisis after which only three attractors survive in the system. Then, comes a boundary crisis which leads to the disappearance of the symmetric attractor in the system. Finally, through a symmetry restoring crisis, a unique symmetric attractor is obtained for higher values of the control parameter and the system is now monostable.

The Effects of a Constant Excitation Force on the Dynamics of an Infinite-Equilibrium Chaotic System Without Linear Terms: Analysis, Control and Circuit Simulation

2020 ◽  
Vol 30 (15) ◽  
pp. 2050234
Author(s):  
L. Kamdjeu Kengne ◽  
Z. Tabekoueng Njitacke ◽  
J. R. Mboupda Pone ◽  
H. T. Kamdem Tagne
Keyword(s):  
Chaotic System ◽  
Analog Circuit ◽  
Circuit Simulation ◽  
Equilibrium Points ◽  
Nonlinear Behavior ◽  
Basins Of Attraction ◽  
Phase Portraits ◽  
Unique Contribution ◽  
Chaotic Signals ◽  
Excitation Force

In this paper, the effects of a bias term modeling a constant excitation force on the dynamics of an infinite-equilibrium chaotic system without linear terms are investigated. As a result, it is found that the bias term reduces the number of equilibrium points (transition from infinite-equilibria to only two equilibria) and breaks the symmetry of the model. The nonlinear behavior of the system is highlighted in terms of bifurcation diagrams, maximal Lyapunov exponent plots, phase portraits, and basins of attraction. Some interesting phenomena are found including, for instance, hysteretic dynamics, multistability, and coexisting bifurcation branches when monitoring the system parameters and the bias term. Also, we demonstrate that it is possible to control the offset and amplitude of the chaotic signals generated. Compared to some few cases previously reported on systems without linear terms, the plethora of behaviors found in this work represents a unique contribution in comparison with such type of systems. A suitable analog circuit is designed and used to support the theoretical analysis via a series of Pspice simulations.

scholarly journals Complex Dynamics of an Impulsive Chemostat Model

2019 ◽  
Vol 29 (08) ◽  
pp. 1950101 ◽  
Author(s):  
Jin Yang ◽  
Yuanshun Tan ◽  
Robert A. Cheke
Keyword(s):  
Periodic Solution ◽  
Global Stability ◽  
Substrate Concentration ◽  
Complex Dynamics ◽  
Control Strategies ◽  
Phase Portraits ◽  
Global Dynamics ◽  
Chemostat Model ◽  
Existence And Stability ◽  
Theoretical Results

We propose a novel impulsive chemostat model with the substrate concentration as the basis for the implementation of control strategies, and then investigate the model’s global dynamics. The exact domains of the impulsive and phase sets are discussed in the light of phase portraits of the model, and then we define the Poincaré map and study its complex properties. Furthermore, the existence and stability of the microorganism eradication periodic solution are addressed, and the analysis of a transcritical bifurcation reveals that an order-1 periodic solution is generated. We also provide the conditions for the global stability of an order-1 periodic solution and show the existence of order-[Formula: see text] [Formula: see text] periodic solutions. Moreover, the PRCC results and bifurcation analyses not only substantiate our results, but also indicate that the proposed system exists with complex dynamics. Finally, biological implications related to the theoretical results are discussed.

scholarly journals Image Encryption Technology Based on Fractional Two-Dimensional Triangle Function Combination Discrete Chaotic Map Coupled with Menezes-Vanstone Elliptic Curve Cryptosystem

2018 ◽  
Vol 2018 ◽  
pp. 1-24 ◽  
Author(s):  
Zeyu Liu ◽  
Tiecheng Xia ◽  
Jinbo Wang
Keyword(s):  
Elliptic Curve ◽  
Image Encryption ◽  
Chaotic Map ◽  
Phase Portraits ◽  
Largest Lyapunov Exponent ◽  
Elliptic Curve Cryptosystem ◽  
Two Dimensional ◽  
Triangle Function ◽  
Function Combination ◽  
Secret Keys

A new fractional two-dimensional triangle function combination discrete chaotic map (2D-TFCDM) with the discrete fractional difference is proposed. We observe the bifurcation behaviors and draw the bifurcation diagrams, the largest Lyapunov exponent plot, and the phase portraits of the proposed map, respectively. On the application side, we apply the proposed discrete fractional map into image encryption with the secret keys ciphered by Menezes-Vanstone Elliptic Curve Cryptosystem (MVECC). Finally, the image encryption algorithm is analysed in four main aspects that indicate the proposed algorithm is better than others.

scholarly journals Complexity of a Duopoly Game in the Electricity Market with Delayed Bounded Rationality

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Junhai Ma ◽  
Hongliang Tu
Keyword(s):  
Nash Equilibrium ◽  
Bounded Rationality ◽  
Electricity Market ◽  
Complex Dynamics ◽  
Fractal Dimensions ◽  
Practical Significance ◽  
Stable Region ◽  
Phase Portraits ◽  
Largest Lyapunov Exponent ◽  
Game Model

According to a triopoly game model in the electricity market with bounded rational players, a new Cournot duopoly game model with delayed bounded rationality is established. The model is closer to the reality of the electricity market and worth spreading in oligopoly. By using the theory of bifurcations of dynamical systems, local stable region of Nash equilibrium point is obtained. Its complex dynamics is demonstrated by means of the largest Lyapunov exponent, bifurcation diagrams, phase portraits, and fractal dimensions. Since the output adjustment speed parameters are varied, the stability of Nash equilibrium gives rise to complex dynamics such as cycles of higher order and chaos. Furthermore, by using the straight-line stabilization method, the chaos can be eliminated. This paper has an important theoretical and practical significance to the electricity market under the background of developing new energy.

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