Numerical Study of Multiple Attractors in the Parallel Inductor–Capacitor–Memristor Circuit

2017 ◽  
Vol 27 (11) ◽  
pp. 1730036 ◽  
Author(s):  
Zbigniew Galias
Keyword(s):  
Numerical Study ◽  
Basins Of Attraction ◽  
Coexisting Attractors ◽  
Multiple Attractors ◽  
Return Map ◽  
Interval Newton Method ◽  
Coexistence Of Attractors ◽  
Periodic Attractors ◽  
Stable Periodic ◽  
Trapping Regions

Dynamical phenomena in the parallel inductor–capacitor–memristor circuit are studied numerically. A systematic search for coexisting attractors is carried out. The existence of multiple attractors is observed and bifurcation diagrams are constructed. Basins of attraction are computed. The coexistence of attractors is proved using interval analysis tools. The existence of periodic attractors is confirmed by applying the interval Newton method to prove the existence of stable periodic orbits of an associated return map. For numerically observed chaotic attractors the existence of attractors is proved by constructing trapping regions enclosing chaotic trajectories of the return map. The existence of topological chaos is proved using the method of covering relations.

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.

NUMERICAL STUDY OF COEXISTING ATTRACTORS FOR THE HÉNON MAP

2013 ◽  
Vol 23 (07) ◽  
pp. 1330025 ◽  
Author(s):  
ZBIGNIEW GALIAS ◽  
WARWICK TUCKER
Keyword(s):  
Periodic Orbits ◽  
Parameter Space ◽  
Interval Analysis ◽  
Numerical Study ◽  
Continuation Method ◽  
Basins Of Attraction ◽  
Hénon Map ◽  
Coexisting Attractors ◽  
Henon Map ◽  
Parameter Values

The question of coexisting attractors for the Hénon map is studied numerically by performing an exhaustive search in the parameter space. As a result, several parameter values for which more than two attractors coexist are found. Using tools from interval analysis, we show rigorously that the attractors exist. In the case of periodic orbits, we verify that they are stable, and thus proper sinks. Regions of existence in parameter space of the found sinks are located using a continuation method; the basins of attraction are found numerically.

THE NATURE OF ATTRACTOR BASINS IN MULTISTABLE SYSTEMS

2008 ◽  
Vol 18 (06) ◽  
pp. 1675-1688 ◽  
Author(s):  
MANISH DEV SHRIMALI ◽  
AWADHESH PRASAD ◽  
RAM RAMASWAMY ◽  
ULRIKE FEUDEL
Keyword(s):  
Power Law ◽  
Basins Of Attraction ◽  
Bifurcation Parameter ◽  
Multiple Attractors ◽  
Saddle Node Bifurcation ◽  
Basin Boundary ◽  
Abrupt Changes ◽  
Periodic Attractors ◽  
Weak Dissipation ◽  
Multistable Systems

In systems that exhibit multistability, namely those that have more than one coexisting attractor, the basins of attraction evolve in specific ways with the creation of each new attractor. These multiple attractors can be created via different mechanisms. When an attractor is formed via a saddle-node bifurcation, the size of its basin increases as a power-law in the bifurcation parameter. In systems with weak dissipation, the basins of low-order periodic attractors increase linearly, while those of high-order periodic attractors decay exponentially as the dissipation is increased. These general features are illustrated for autonomous as well as driven mappings. In addition, the boundaries of the basins can also change from being smooth to fractal when a new attractor appears. Transitions in the basin boundary morphology are reflected in abrupt changes in the dependence of the uncertainty exponent on the bifurcation parameter.

Sequences of Global Bifurcations and the Related Outcomes after Crisis of the Resonant Attractor in a Nonlinear Oscillator

1997 ◽  
Vol 07 (11) ◽  
pp. 2437-2457 ◽  
Author(s):  
W. Szemplińska-Stupnicka ◽  
E. Tyrkiel
Keyword(s):  
Nonlinear Oscillator ◽  
Duffing Oscillator ◽  
Nonlinear Resonance ◽  
Basins Of Attraction ◽  
Global Bifurcations ◽  
System Behavior ◽  
Coexisting Attractors ◽  
System Parameters

The problem of the system behavior after annihilation of the resonant attractor in the region of the nonlinear resonance hysteresis is considered. The sequences of global bifurcations, in connection with the associated metamorphoses of basins of attraction of coexisting attractors, are examined. The study allows one to reveal the mechanism that governs the phenomenon of the post crisis ensuing transient trajectory to settle onto one or another remote attractor. The problem is studied in detail for the twin-well potential Duffing oscillator. The boundary which splits the considered region of system parameters into two subdomains, where the outcome is unique or the two outcomes are possible, is defined.

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.

scholarly journals Stability and Multistability of a Bounded Rational Mixed Duopoly Model with Homogeneous Product

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-16 ◽  
Author(s):  
Wei Zhou ◽  
Na Zhao ◽  
Tong Chu ◽  
Ying-Xiang Chang
Keyword(s):  
Joint Venture ◽  
Fractal Structure ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basins Of Attraction ◽  
Mixed Duopoly ◽  
Formation Mechanisms ◽  
Multiple Attractors ◽  
Homogeneous Product ◽  
The Stability

In this paper, a mixed duopoly dynamic model with bounded rationality is built, where a public-private joint venture and a private enterprise produce homogeneous products and compete in the same market. The purpose of this research is to study the stability and the multistability of the established model. The local stability of all the equilibrium points is discussed by using Jury condition, and the stability region of the Nash equilibrium point has been given. A special fractal structure called “hub of periodicity” has been found in the two-parameter space by numerical simulation. In addition, the phenomena of multistability (also called coexistence of multiple attractors) are also studied using basins of attraction and 1-D bifurcation diagrams with adiabatic initial conditions. We find that there are two different coexistences of multiple attractors. And, the fractal structure of the attracting basin is also analyzed, and the formation mechanisms of “holes” and “contact” bifurcation have been revealed. At last, the long-term profits of the enterprises are studied. We find that some enterprises can even make more profits under a chaotic circumstance.

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.

THE FEASIBLE DOMAINS AND THEIR BIFURCATIONS IN AN EXTENDED LOGISTIC MODEL WITH AN EXTERNAL INTERFERENCE

2007 ◽  
Vol 17 (03) ◽  
pp. 877-889 ◽  
Author(s):  
EN-GUO GU
Keyword(s):  
Discrete Model ◽  
Logistic Model ◽  
Focal Point ◽  
Basins Of Attraction ◽  
Two Dimensional ◽  
External Interference ◽  
Feasible Set ◽  
Coexistence Of Attractors

This paper is devoted to the study of the properties of basins of attraction and the domains of feasible trajectories (discrete trajectories having an ecological sense) generated by a family of two-dimensional map T related to a discrete model of populations generation. The inverse of T has vanishing denominator giving rise to nonclassical singularities: a nondefinition line, a focal point and a prefocal line. Furthermore, the differences and relations between the feasible set, the feasible domains and the basins of attraction are presented. A phenomena of coexistence of attractors is shown. The structure of a chaotic repellor is interpreted by use of the singularities.

scholarly journals Robustness of Supercavitating Vehicles Based on Multistability Analysis

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Yipin Lv ◽  
Tianhong Xiong ◽  
Wenjun Yi ◽  
Jun Guan
Keyword(s):  
Equilibrium Points ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Small Range ◽  
Practical Application ◽  
Stable States ◽  
Supercavitating Vehicles ◽  
Control Gain ◽  
Stable Periodic ◽  
Setting Parameters

Supercavity can increase speed of underwater vehicles greatly. However, external interferences always lead to instability of vehicles. This paper focuses on robustness of supercavitating vehicles. Based on a 4-dimensional dynamic model, the existence of multistability is verified in supercavitating system through simulation, and the robustness of vehicles varying with parameters is analyzed by basins of attraction. Results of the research disclose that the supercavitating system has three stable states in some regions of parameters space, namely, stable, periodic, and chaotic states, while in other regions it has various multistability, such as coexistence of two types of stable equilibrium points, coexistence of a limit cycle with a chaotic attractor, and coexistence of 1-periodic cycle with 2-periodic cycle. Provided that cavitation number varies within a small range, with increase of the feedback control gain of fin deflection angle, size of basin of attraction becomes smaller and robustness of the system becomes weaker. In practical application, robustness of supercavitating vehicles can be improved by setting parameters of system or adjusting initial launching conditions.

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