Multistability Control of Space Magnetization in Hyperjerk Oscillator: A Case Study

2020 ◽  
Vol 15 (5) ◽  
Author(s):  
Gervais Dolvis Leutcho ◽  
Jacques Kengne ◽  
Theophile Fonzin Fozin ◽  
K. Srinivasan ◽  
Z. Njitacke Tabekoueng ◽  
...  
Keyword(s):  
Coupling Strength ◽  
Chaotic Attractor ◽  
Critical Value ◽  
Basins Of Attraction ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Stable States ◽  
Multiple Attractors ◽  
Fixed Set ◽  
Hyperjerk System

Abstract In this paper, multistability control of a 5D autonomous hyperjerk oscillator through linear augmentation scheme is investigated. The space magnetization is characterized by the coexistence of five different stable states including an asymmetric pair of chaotic attractors, an asymmetric pair of period-3 cycle, and a symmetric chaotic attractor for a given/fixed set of parameters. The linear augmentation method is applied here to control, for the first time, five coexisting attractors. Standard Lyapunov exponents, bifurcation diagrams, basins of attraction, and 3D phase portraits are presented as methods to conduct the efficaciousness of the control scheme. The results of the applied methods reveal that the monostable chaotic attractor is obtained through three important crises when varying the coupling strength. In particular, below the first critical value of the coupling strength, five distinct attractors are coexisting. Above that critical value, three and then two chaotic attractors are now coexisting, respectively. While for higher values of the coupling strength, only the symmetric chaotic attractor is viewed in the controlled system. The process of annihilation of coexisting multiple attractors to monostable one is confirmed experimentally. The important results of the controlled hyperjerk system with its unique survived chaotic attractor are suited in applications like secure communications.

Antimonotonicity, Chaos and Multiple Attractors in a Novel Autonomous Jerk Circuit

2017 ◽  
Vol 27 (07) ◽  
pp. 1750100 ◽  
Author(s):  
J. Kengne ◽  
A. Nguomkam Negou ◽  
Z. T. Njitacke
Keyword(s):  
Three Dimensional ◽  
Period Doubling ◽  
Basins Of Attraction ◽  
Electrical Circuit ◽  
Semiconductor Diode ◽  
Chaotic Attractors ◽  
Coexisting Attractors ◽  
Multiple Attractors ◽  
Systematic Analysis ◽  
The Stability

We perform a systematic analysis of a system consisting of a novel jerk circuit obtained by replacing the single semiconductor diode of the original jerk circuit described in [Sprott, 2011a] with a pair of semiconductor diodes connected in antiparallel. The model is described by a continuous time three-dimensional autonomous system with hyperbolic sine nonlinearity, and may be viewed as a control system with nonlinear velocity feedback. The stability of the (unique) fixed point, the local bifurcations, and the discrete symmetries of the model equations are discussed. The complex behavior of the system is categorized in terms of its parameters by using bifurcation diagrams, Lyapunov exponents, time series, Poincaré sections, and basins of attraction. Antimonotonicity, period doubling bifurcation, symmetry restoring crises, chaos, and coexisting bifurcations are reported. More interestingly, one of the key contributions of this work is the finding of various regions in the parameters’ space in which the proposed (“elegant”) jerk circuit experiences the unusual phenomenon of multiple competing attractors (i.e. coexistence of four disconnected periodic and chaotic attractors). The basins of attraction of various coexisting attractors display complexity (i.e. fractal basins boundaries), thus suggesting possible jumps between coexisting attractors in experiment. Results of theoretical analyses are perfectly traced by laboratory experimental measurements. To the best of the authors’ knowledge, the jerk circuit/system introduced in this work represents the simplest electrical circuit (only a quadruple op amplifier chip without any analog multiplier chip) reported to date capable of four disconnected periodic and chaotic attractors for the same parameters setting.

scholarly journals Sequence of Routes to Chaos in a Lorenz-Type System

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Fangyan Yang ◽  
Yongming Cao ◽  
Lijuan Chen ◽  
Qingdu Li
Keyword(s):  
Limit Cycles ◽  
Chaotic Attractor ◽  
Type System ◽  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Basins Of Attraction ◽  
Chaotic Attractors ◽  
Main Route ◽  
Route To Chaos ◽  
Period Doubling Bifurcations

This paper reports a new bifurcation pattern observed in a Lorenz-type system. The pattern is composed of a main bifurcation route to chaos (n=1) and a sequence of sub-bifurcation routes with n=3,4,5,…,14 isolated sub-branches to chaos. When n is odd, the n isolated sub-branches are from a period-n limit cycle, followed by twin period-n limit cycles via a pitchfork bifurcation, twin chaotic attractors via period-doubling bifurcations, and a symmetric chaotic attractor via boundary crisis. When n is even, the n isolated sub-branches are from twin period-n/2 limit cycles, which become twin chaotic attractors via period-doubling bifurcations. The paper also shows that the main route and the sub-routes can coexist peacefully by studying basins of attraction.

Multistability and Bubbling Route to Chaos in a Deterministic Model for Geomagnetic Field Reversals

2019 ◽  
Vol 29 (12) ◽  
pp. 1930034
Author(s):  
Paulo C. Rech ◽  
Sudarshan Dhua ◽  
N. C. Pati
Keyword(s):  
Fixed Point ◽  
Geomagnetic Field ◽  
Deterministic Model ◽  
Initial Conditions ◽  
Period Doubling ◽  
Basins Of Attraction ◽  
Phase Portraits ◽  
Multiple Attractors ◽  
System A ◽  
Two Parameters

We report coexisting multiple attractors and birth of chaos via period-bubbling cascades in a model of geomagnetic field reversals. The model system comprises a set of three coupled first-order quadratic nonlinear equations with three control parameters. Up to seven kinds of multistable attractors, viz. fixed point-periodic, fixed point-chaotic, periodic–periodic, periodic-chaotic, chaotic–chaotic, fixed point-periodic–periodic, fixed point-periodic-chaotic are obtained depending on the initial conditions for critical parameter sets. Antimonotonicity is a striking characteristic feature of nonlinear systems through which a full Feigenbaum tree corresponding to creation and annihilation of period-doubling cascades is developed. By analyzing the two-parameters dependent dynamics of the system, a critical biparameter zone is identified, where antimonotonicity comes into existence. The complex dynamical behaviors of the system are explored using phase portraits, bifurcation diagrams, Lyapunov exponents, isoperiodic diagram, and basins of attraction.

Generating Four-Wing Hyperchaotic Attractor and Two-Wing, Three-Wing, and Four-Wing Chaotic Attractors in 4D Memristive System

2017 ◽  
Vol 27 (02) ◽  
pp. 1750027 ◽  
Author(s):  
Ling Zhou ◽  
Chunhua Wang ◽  
Lili Zhou
Keyword(s):  
Chaotic System ◽  
Three Dimensional ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Hyperchaotic System ◽  
Multiple Attractors ◽  
System A ◽  
Dynamical Behaviors ◽  
Memristive System ◽  
Line Of Equilibria

By adding only one smooth flux-controlled memristor into a three-dimensional (3D) pseudo four-wing chaotic system, a new real four-wing hyperchaotic system is constructed in this paper. It is interesting to see that this new memristive chaotic system can generate a four-wing hyperchaotic attractor with a line of equilibria. Moreover, it can generate two-, three- and four-wing chaotic attractors with the variation of a single parameter which denotes the strength of the memristor. At the same time, various coexisting multiple attractors (e.g. three-wing attractors, four-wing attractors and attractors with state transition under the same system parameters) are observed in this system, which means that extreme multistability arises. The complex dynamical behaviors of the proposed system are analyzed by Lyapunov exponents (LEs), phase portraits, Poincaré maps, and time series. An electronic circuit is finally designed to implement the hyperchaotic memristive system.

Transformations of Closed Invariant Curves and Closed-Invariant-Curve-Like Chaotic Attractors in Piecewise Smooth Systems

2021 ◽  
Vol 31 (03) ◽  
pp. 2130009
Author(s):  
Zhanybai T. Zhusubaliyev ◽  
Viktor Avrutin ◽  
Frank Bastian
Keyword(s):  
Chaotic Attractor ◽  
Small Amplitude ◽  
Homoclinic Bifurcation ◽  
Piecewise Smooth ◽  
Invariant Curve ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Invariant Curves ◽  
Piecewise Smooth Systems ◽  
Smooth Map

The paper describes some aspects of sudden transformations of closed invariant curves in a 2D piecewise smooth map. In particular, using detailed numerically calculated phase portraits, we discuss transitions from smooth to piecewise smooth closed invariant curves. We show that such transitions may occur not only when a closed invariant curve collides with a border but also via a homoclinic bifurcation. Furthermore, we describe an unusual transformation from a closed invariant curve to a large amplitude chaotic attractor and demonstrate that this transition occurs in two steps, involving a small amplitude closed-invariant-curve-like chaotic attractor.

Interacting convection modes in a saturated porous medium of nearly square planform: four modes

2017 ◽  
Vol 82 (3) ◽  
pp. 526-547 ◽  
Author(s):  
Brendan J. Florio ◽  
Andrew P. Bassom ◽  
Konstantinos Sakellariou ◽  
Thomas Stemler
Keyword(s):  
Porous Medium ◽  
Rayleigh Number ◽  
Saturated Porous Medium ◽  
Critical Rayleigh Number ◽  
Critical Value ◽  
Basins Of Attraction ◽  
Stable States ◽  
Pitchfork Bifurcations ◽  
Multiple Stable States ◽  
Box Dimensions

Abstract Convection can occur in a confined saturated porous box when the associated Rayleigh number exceeds a threshold critical value: the identity of the preferred onset convection mode depends sensitively on the geometry of the box. Here we discuss examples for which the box dimensions are such that four modes share a common critical Rayleigh number. Perturbation theory is used to derive a system of coupled ordinary differential equations that governs the nonlinear interaction of the four modes and an analysis of this set is undertaken. In particular, it is demonstrated that as the Rayleigh number is increased beyond critical so a series of pitchfork bifurcations occur and multiple stable states are identified that correspond to the survival of just one of the four modes. The basins of attraction for each mode in the 4D phase space are described by a reduction to a suitable 3D counterpart.

A novel 4D chaotic system with multistability: Dynamical analysis, circuit implementation, control design

2019 ◽  
Vol 33 (21) ◽  
pp. 1950240 ◽  
Author(s):  
Jian-Jun He ◽  
Bang-Cheng Lai
Keyword(s):  
Chaotic System ◽  
Chaotic Attractor ◽  
Electronic Circuit ◽  
Period Doubling ◽  
Control Design ◽  
Critical Value ◽  
Dynamical Analysis ◽  
Chaotic Attractors ◽  
Hopf Bifurcations ◽  
The Novel

The purpose of this work is to introduce a novel 4D chaotic system and investigate its multistability. The novel system has an unstable origin and two stable symmetrical hyperbolic equilibria. When the parameter increases across a critical value, the equilibria lose their stability and double Hopf bifurcations occur with the appearance of limit cycles. A pair of point, periodic, chaotic attractors are observed in the system from different initial values for given parameters. The chaos of the system is yielded via period-doubling bifurcation. A double-scroll chaotic attractor is numerically observed as well. By using the electronic circuit, the chaotic attractor of the system is realized. The control problem of the system is reported. An effective controller is designed to stabilize the system.

scholarly journals Coexistence of Multiple Points, Limit Cycles, and Strange Attractors in a Simple Autonomous Hyperjerk Circuit with Hyperbolic Sine Function

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-24
Author(s):  
M. Fouodji Tsotsop ◽  
J. Kengne ◽  
G. Kenne ◽  
Z. Tabekoueng Njitacke
Keyword(s):  
Dynamic Properties ◽  
Period Doubling ◽  
Phase Portraits ◽  
Stable States ◽  
Multiple Points ◽  
Offset Boosting ◽  
Hyperbolic Sine ◽  
Theoretical Predictions ◽  
Hyperjerk System ◽  
Bifurcation Chaos

In this contribution, a new elegant hyperjerk system with three equilibria and hyperbolic sine nonlinearity is investigated. In contrast to other models of hyperjerk systems where either hidden or self-excited attractors are obtained, the case reported in this work represents a unique one which displays the coexistence of self-excited chaotic attractors and stable fixed points. The dynamic properties of the new system are explored in terms of equilibrium point analyses, symmetry and dissipation, and existence of attractors as well. Common analysis tools (i.e., bifurcation diagram, Lyapunov exponents, and phase portraits) are used to highlight some important phenomena such as period-doubling bifurcation, chaos, periodic windows, and symmetric restoring crises. More interestingly, the system under consideration shows the coexistence of several types of stable states, including the coexistence of two, three, four, six, eight, and ten coexisting attractors. In addition, the system is shown to display antimonotonicity and offset boosting. Laboratory experimental measurements show a very good coherence with the theoretical predictions.

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.

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