A new class of two-dimensional rational maps with self-excited and hidden attractors

2021 ◽  
Author(s):  
Li-Ping Zhang ◽  
Yang Liu ◽  
Zhou-Chao Wei ◽  
Hai-Bo Jiang ◽  
Qin-Sheng Bi
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Complex Dynamics ◽  
Basins Of Attraction ◽  
Phase Portraits ◽  
Rational Maps ◽  
Two Dimensional ◽  
Hidden Attractors ◽  
New Class ◽  
Rational Term

Abstract This paper studies a new class of two-dimensional rational maps exhibiting self-excited and hidden attractors. The mathematical model of these maps is firstly formulated by introducing a rational term. The analysis of existence and stabilities of the fixed points in these maps suggests that there are four types of fixed points, i.e., no fixed point, one single fixed point, two fixed points and a line of fixed points. To investigate the complex dynamics of these rational maps with different types of fixed points, numerical analysis tools, such as time histories, phase portraits, basins of attraction, Lyapunov exponent spectrum, Lyapunov (Kaplan-Yorke) dimension and bifurcation diagrams, are employed. Our extensive numerical simulations identify both self-excited and hidden attractors, which were rarely reported in the literature. Therefore, the multi-stability of these maps, especially the hidden one, is further explored in the present work.

Controlling Hidden Dynamics and Multistability of a Class of Two-Dimensional Maps via Linear Augmentation

2021 ◽  
Vol 31 (03) ◽  
pp. 2150047
Author(s):  
Liping Zhang ◽  
Haibo Jiang ◽  
Yang Liu ◽  
Zhouchao Wei ◽  
Qinsheng Bi
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Numerical Simulations ◽  
Coupling Strength ◽  
Complex Dynamics ◽  
Discrete Dynamical Systems ◽  
Two Dimensional ◽  
Stable Fixed Point ◽  
Hidden Attractor ◽  
Hidden Attractors

This paper reports the complex dynamics of a class of two-dimensional maps containing hidden attractors via linear augmentation. Firstly, the method of linear augmentation for continuous dynamical systems is generalized to discrete dynamical systems. Then three cases of a class of two-dimensional maps that exhibit hidden dynamics, the maps with no fixed point and the maps with one stable fixed point, are studied. Our numerical simulations show the effectiveness of the linear augmentation method. As the coupling strength of the controller increases or decreases, hidden attractor can be annihilated or altered to be self-excited, and multistability of the map can be controlled to being bistable or monostable.

A New Class of Three-Dimensional Maps with Hidden Chaotic Dynamics

2016 ◽  
Vol 26 (12) ◽  
pp. 1650206 ◽  
Author(s):  
Haibo Jiang ◽  
Yang Liu ◽  
Zhouchao Wei ◽  
Liping Zhang
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Chaotic Dynamics ◽  
Three Dimensional ◽  
Computational Procedure ◽  
Basins Of Attraction ◽  
Chaotic Attractors ◽  
Computer Search ◽  
New Class ◽  
Special Concern

This paper studies a new class of three-dimensional maps in a Jerk-like structure with a special concern of their hidden chaotic dynamics. Our investigation focuses on the hidden chaotic attractors in three typical scenarios of fixed points, namely no fixed point, single fixed point, and two fixed points. A systematic computer search is performed to explore possible hidden chaotic attractors, and a number of examples of the proposed maps are used for demonstration. Numerical results show that the routes to hidden chaotic attractors are complex, and the basins of attraction for the hidden chaotic attractors could be tiny, so that using the standard computational procedure for localization is impossible.

scholarly journals Common Fixed Points for Pairs of Mappings with Variable Contractive Parameters

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
J. R. Morales ◽  
E. M. Rojas ◽  
Ravindra K. Bisht
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Common Fixed Point ◽  
Common Fixed Points ◽  
Contraction Mappings ◽  
New Class

We establish some common fixed point results for a new class of pair of contraction mappings having functions as contractive parameters, and satisfying minimal noncommutative operators property.

scholarly journals Convergence theorems for generalized contractions and applications

Filomat ◽  
2020 ◽  
Vol 34 (3) ◽  
pp. 945-964
Author(s):  
Mudasir Younis ◽  
Deepak Singh ◽  
Stojan Radenovic ◽  
Mohammad Imdad
Keyword(s):  
Fixed Point ◽  
Graph Theory ◽  
Fixed Points ◽  
Open Problem ◽  
Fixed Point Theorems ◽  
Balance Model ◽  
Population Balance Model ◽  
New Techniques ◽  
New Class ◽  
Article Deal

The principal results in this article deal with the existence of fixed points of a new class of generalized F-contraction. In our approach, by visualizing some non-trivial examples we will obtain better geometrical interpretation. Our main results substantially improve the theory of F-contraction mappings and the related fixed point theorems. In section-4, application to graph theory is entrusted and proved results are endorsed by an example through graph. The presented new techniques give the possibility to justify the existence problems of the solutions for some class of integral equations. For the future aspects of our study, an open problem is suggested regarding discretized population balance model, whose solution may be derived from the established techniques.

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.

A New Class of Two-Dimensional Chaotic Maps with Closed Curve Fixed Points

2019 ◽  
Vol 29 (07) ◽  
pp. 1950094 ◽  
Author(s):  
Haibo Jiang ◽  
Yang Liu ◽  
Zhouchao Wei ◽  
Liping Zhang
Keyword(s):  
Fixed Points ◽  
Nonlinear Function ◽  
Closed Curve ◽  
Computer Search ◽  
Two Dimensional ◽  
Closed Curves ◽  
New Class ◽  
Dynamical Behaviors ◽  
The Mathematical Model ◽  
Search Program

This paper constructs a new class of two-dimensional maps with closed curve fixed points. Firstly, the mathematical model of these maps is formulated by introducing a nonlinear function. Different types of fixed points which form a closed curve are shown by choosing proper parameters of the nonlinear function. The stabilities of these fixed points are studied to show that these fixed points are all nonhyperbolic. Then a computer search program is employed to explore the chaotic attractors in these maps, and several simple maps whose fixed points form different shapes of closed curves are presented. Complex dynamical behaviors of these maps are investigated by using the phase-basin portrait, Lyapunov exponents, and bifurcation diagrams.

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.

scholarly journals Fixed point results on θ-metric spaces via simulation functions

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3365-3375 ◽  
Author(s):  
Ankush Chanda ◽  
Bosko Damjanovic ◽  
Lakshmi Dey
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Existence And Uniqueness ◽  
Recent Article ◽  
Metric Spaces ◽  
New Class ◽  
Simulation Functions

In a recent article, Khojasteh et al. introduced a new class of simulation functions, Z-contractions, with blending over known contractive conditions in the literature. Subsequently, in this paper, we extend and generalize the results in ?-metric context and we discuss some fixed point results in connection with existing ones. Also, we originate the notion of modified Z-contractions and explore the existence and uniqueness of fixed points of such functions on the said spaces. Finally we include examples to instantiate our main results.

scholarly journals Strong convergence of the Ishikawa iteration for Lipschitz α-hemicontractive mappings

2015 ◽  
Vol 53 (1) ◽  
pp. 151-161
Author(s):  
Micah Okwuchukwu Osilike ◽  
Anthony Chibuike Onah
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Fixed Points ◽  
Iteration Scheme ◽  
Iteration Algorithm ◽  
Mann Iteration ◽  
Ishikawa Iteration ◽  
New Class ◽  
Hemicontractive Mappings ◽  
Demicontractive Mappings

Abstract A new class of α-hemicontractive maps T for which the strong convergence of the Ishikawa iteration algorithm to a fixed point of T is assured is introduced and studied. The study is a continuation of a recent study of a new class of α-demicontractive mappings T by L. Mărușter and Ș. Mărușter, Mathematical and Computer Modeling 54 (2011) 2486-2492 in which they proved strong convergence of the Mann iteration scheme to a fixed point of T. Our class of α-hemicontractive maps is more general than the class of α-demicontractive maps. No compactness assumption is imposed on the operator or it’s domain, and no additional requirement is imposed on the set of fixed points.

scholarly journals On the approximation of fixed points for a new class of generalized Berinde mappings

2016 ◽  
Vol 32 (3) ◽  
pp. 363-374
Author(s):  
BESSEM SAMET ◽  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fixed Points ◽  
Point Theorem ◽  
New Class ◽  
Contractive Type

In this paper, we introduce a new class of operators, for which a fixed point theorem is proven. This class of mappings is very large and unifies several classes of contractive type operators from the literature, including Berinde mappings. Such fact is proven via a comparison with various metrical contractive type mappings.

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