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

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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Colpitts Chaotic Oscillator Coupling with a Generalized Memristor

Mathematical Problems in Engineering ◽

10.1155/2015/249102 ◽

2015 ◽

Vol 2015 ◽

pp. 1-9 ◽

Cited By ~ 7

Author(s):

Ling Lu ◽

Changdi Li ◽

Zicheng Zhao ◽

Bocheng Bao ◽

Quan Xu

Keyword(s):

Mathematical Model ◽

Equilibrium Point ◽

Fourth Order ◽

Chaotic Attractors ◽

Chaotic Oscillator ◽

Nonlinear Phenomena ◽

Unique Equilibrium ◽

First Order ◽

The Mathematical Model ◽

Order Parallel

By introducing a generalized memristor into a fourth-order Colpitts chaotic oscillator, a new memristive Colpitts chaotic oscillator is proposed in this paper. The generalized memristor is equivalent to a diode bridge cascaded with a first-order parallel RC filter. Chaotic attractors of the oscillator are numerically revealed from the mathematical model and experimentally captured from the physical circuit. The dynamics of the memristive Colpitts chaotic oscillator is investigated both theoretically and numerically, from which it can be found that the oscillator has a unique equilibrium point and displays complex nonlinear phenomena.

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Localization of Hidden Attractors in Chua’s System With Absolute Nonlinearity and Its FPGA Implementation

Frontiers in Physics ◽

10.3389/fphy.2021.788329 ◽

2021 ◽

Vol 9 ◽

Author(s):

Xianming Wu ◽

Huihai Wang ◽

Shaobo He

Keyword(s):

Basin Of Attraction ◽

Digital Circuit ◽

Fpga Implementation ◽

Chaotic Attractors ◽

Chua’S Circuit ◽

Hidden Attractor ◽

Hidden Attractors ◽

Chua's Circuit ◽

The Mathematical Model ◽

The Basin Of Attraction

Investigation of the classical self-excited and hidden attractors in the modified Chua’s circuit is a hot and interesting topic. In this article, a novel Chua’s circuit system with an absolute item is investigated. According to the mathematical model, dynamic characteristics are analyzed, including symmetry, equilibrium stability analysis, Hopf bifurcation analysis, Lyapunov exponents, bifurcation diagram, and the basin of attraction. The hidden attractors are located theoretically. Then, the coexistence of the hidden limit cycle and self-excited chaotic attractors are observed numerically and experimentally. The numerical simulation results are consistent with the FPGA implementation results. It shows that the hidden attractor can be localized in the digital circuit.

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

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127419300349 ◽

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.

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Hidden Strange Nonchaotic Attractors

Mathematics ◽

10.3390/math9060652 ◽

2021 ◽

Vol 9 (6) ◽

pp. 652

Author(s):

Marius-F. Danca ◽

Nikolay Kuznetsov

Keyword(s):

Finite Time ◽

Initial Conditions ◽

Power Spectra ◽

Recurrence Plot ◽

Chaotic Attractors ◽

Largest Lyapunov Exponent ◽

Hidden Attractor ◽

Test Power ◽

Finite Time Lyapunov Exponents ◽

Strange Nonchaotic Attractors

In this paper, it is found numerically that the previously found hidden chaotic attractors of the Rabinovich–Fabrikant system actually present the characteristics of strange nonchaotic attractors. For a range of the bifurcation parameter, the hidden attractor is manifestly fractal with aperiodic dynamics, and even the finite-time largest Lyapunov exponent, a measure of trajectory separation with nearby initial conditions, is negative. To verify these characteristics numerically, the finite-time Lyapunov exponents, ‘0-1’ test, power spectra density, and recurrence plot are used. Beside the considered hidden strange nonchaotic attractor, a self-excited chaotic attractor and a quasiperiodic attractor of the Rabinovich–Fabrikant system are comparatively analyzed.

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Hidden Attractor and Multistability in a Novel Memristor-Based System Without Symmetry

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127421501686 ◽

2021 ◽

Vol 31 (11) ◽

pp. 2150168

Author(s):

Musha Ji’e ◽

Dengwei Yan ◽

Lidan Wang ◽

Shukai Duan

Keyword(s):

Lyapunov Exponents ◽

Chaotic System ◽

Chaotic Systems ◽

Initial Conditions ◽

Equilibrium Points ◽

Control Unit ◽

Basins Of Attraction ◽

Phase Portraits ◽

Dynamic Behaviors ◽

Potential Applications

Memristor, as a typical nonlinear element, is able to produce chaotic signals in chaotic systems easily. Chaotic systems have potential applications in secure communications, information encryption, and other fields. Therefore, it is of importance to generate abundant dynamic behaviors in a single chaotic system. In this paper, a novel memristor-based chaotic system without equilibrium points is proposed. One of the essential features is the absence of symmetry in this system, which increases the complexity of the new system. Then, the nonlinear dynamic behaviors of the system are analyzed in terms of chaos diagrams, bifurcation diagrams, Poincaré maps, Lyapunov exponent spectra, the sum of Lyapunov exponents, phase portraits, 0–1 test, recurrence analysis and instantaneous phase. The results of the sum of Lyapunov exponents show that the given system is a quasi-Hamiltonian system with certain initial conditions (IC) and parameters. Next, other critical phenomena, such as hidden multi-scroll attractors, abundant coexistence characteristics, are found characterized through basins of attraction and others. Especially, it reveals some rare phenomena in other systems that multiple hidden hyperchaotic attractors coexist. Finally, the circuit implementation based on Micro Control Unit (MCU) confirms theoretical analysis and the numerical simulation.

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BIFURCATION AND CHAOS IN COUPLED BVP OSCILLATORS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127404009983 ◽

2004 ◽

Vol 14 (04) ◽

pp. 1305-1324 ◽

Cited By ~ 19

Author(s):

TETSUSHI UETA ◽

HISAYO MIYAZAKI ◽

TAKUJI KOUSAKA ◽

HIROSHI KAWAKAMI

Keyword(s):

Autonomous System ◽

Period Doubling ◽

Chaotic Attractors ◽

Nonlinear Phenomena ◽

Van Der Pol ◽

Bifurcation And Chaos ◽

Experimental Laboratory ◽

Chaotic Parameter ◽

Parameter Values ◽

The Mathematical Model

Bonhöffer–van der Pol(BVP) oscillator is a classic model exhibiting typical nonlinear phenomena in the planar autonomous system. This paper gives an analysis of equilibria, periodic solutions, strange attractors of two BVP oscillators coupled by a resister. When an oscillator is fixed its parameter values in nonoscillatory region and the others in oscillatory region, create the double scroll attractor due to the coupling. Bifurcation diagrams are obtained numerically from the mathematical model and chaotic parameter regions are clarified. We also confirm the existence of period-doubling cascades and chaotic attractors in the experimental laboratory.

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

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4046639 ◽

2020 ◽

Vol 15 (5) ◽

Cited By ~ 2

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.

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BIFURCATION STRUCTURE AT 1/3-SUBHARMONIC RESONANCE IN AN ASYMMETRIC NONLINEAR ELASTIC OSCILLATOR

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127496000904 ◽

1996 ◽

Vol 06 (08) ◽

pp. 1529-1546 ◽

Cited By ~ 13

Author(s):

G. REGA ◽

A. SALVATORI

Keyword(s):

Invariant Manifolds ◽

Initial Conditions ◽

Basins Of Attraction ◽

Subharmonic Resonance ◽

Phase Portraits ◽

Global Dynamics ◽

Bifurcation Structure ◽

Response Curves ◽

Nonlinear Elastic ◽

Insight Into

The attractor-basin bifurcation structure in an asymmetric nonlinear oscillator representative of the planar finite forced dynamics of elastic structural systems with initial curvature is studied at the 1/3-subharmonic resonance regime. Local and global analyses are made by means of different computational tools to obtain frequency-response curves of coexisting regular solutions, bifurcation diagrams ensuing from different sets of initial conditions, manifolds structure of direct and inverse saddles corresponding to unstable periodic solutions, basins of attraction at different values of the control parameter. Deep insight into the global dynamics of the system and its evolution is achieved through the analysis of synthetic attractor-basin-manifold phase portraits. The topological mechanisms which entail onset and disappearance of various attractors, and the main and secondary evolutions to chaos, are identified. Special attention is devoted to the analysis of sudden bifurcational events characterizing the system global dynamics, associated with the topological behavior of the invariant manifolds of several direct and inverse saddles. Features of basin metamorphosis, attractor-basin accessibility, and window occurrence are examined. The approach followed, consisting in combined bifurcation analysis of the attractor-basin structure and of the manifold structure, is thought to be useful for a variety of dynamical systems.

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On the genetic divergence of two adjacent populations living in a homogeneous habitat

Izvestiya VUZ Applied Nonlinear Dynamics ◽

10.18500/0869-6632-2021-29-5-706-726 ◽

2021 ◽

Vol 29 (5) ◽

pp. 706-726

Author(s):

Efim Frisman ◽

◽

Matvej Kulakov ◽

Keyword(s):

Growth Rates ◽

Genetic Divergence ◽

Initial Conditions ◽

Pitchfork Bifurcation ◽

Basins Of Attraction ◽

Phase Portraits ◽

First Case ◽

Fundamental Possibility ◽

Mean Fitness ◽

Population Ratio

The purpose is to study the mechanisms leading to the genetic divergence, i.e. stable genetic differences between two adjacent populations coupled by migration of individuals. We considered the case when the fitness of individuals is strictly determined genetically by a single diallelic locus with alleles A and a, the population is panmictic and Mendel's laws of inheritance hold. The dynamic model contains three phase variables: concentration of allele A in each population and fraction (weight) of the first population in the total population size. We assume that the numbers of coupled populations change independently or strictly synchronously. In the first case, the growth rates are determined by fitness of homo- and heterozygotes, the mean fitness of the each population and the initial concentrations of alleles. In the second case, the growth rates are the same. Methods. To study the model, we used the qualitative theory of differential equations studies, including the construction of parametric and phase portraits, basins of attraction and bifurcation diagrams. We studied local bifurcations that provide the fundamental possibility of genetic divergence. Results. If heterozygote fitness is higher than homozygotes, then both populations are polymorphic with the same concentration of homologous alleles. If the heterozygotes fitness is reduced, then over time the populations will have the same monomorphism in one allele, regardless of the type of population changes. In this case, the dynamics is bistable. We showed that the divergence in the model is a result of subcritical pitchfork bifurcation of an unstable polymorphic state. As a result, the genetic divergent state is unstable and exists as part of the transient process to one of monomorphic state. Conclusion. Divergence is stable only for populations that maintain a population ratio in a certain way. In this case, it is preceded by a saddle-node bifurcation and dynamics is quad-stable, i.e. depending on the initial conditions, two types of stable monomorphism and divergence are possible simultaneously.

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

Complexity ◽

10.1155/2020/2340934 ◽

2020 ◽

Vol 2020 ◽

pp. 1-24 ◽

Cited By ~ 1

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.

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