A Chaotic Quadratic Bistable Hyperjerk System with Hidden Attractors and a Wide Range of Sample Entropy: Impulsive Stabilization

2021 ◽  
Vol 31 (16) ◽  
Author(s):  
M. D. Vijayakumar ◽  
Alireza Bahramian ◽  
Hayder Natiq ◽  
Karthikeyan Rajagopal ◽  
Iqtadar Hussain
Keyword(s):  
Impulsive Control ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basin Of Attraction ◽  
Sample Entropy ◽  
Hidden Attractors ◽  
System Equilibrium ◽  
Wide Range ◽  
Hyperjerk System ◽  
The Basin Of Attraction

Hidden attractors generated by the interactions of dynamical variables may have no equilibrium point in their basin of attraction. They have grabbed the attention of mathematicians who investigate strange attractors. Besides, quadratic hyperjerk systems are under the magnifying glass of these mathematicians because of their elegant structures. In this paper, a quadratic hyperjerk system is introduced that can generate chaotic attractors. The dynamical behaviors of the oscillator are investigated by plotting their Lyapunov exponents and bifurcation diagrams. The multistability of the hyperjerk system is investigated using the basin of attraction. It is revealed that the system is bistable when one of its attractors is hidden. Besides, the complexity of the systems’ attractors is investigated using sample entropy as the complexity feature. It is revealed how changing the parameters can affect the complexity of the systems’ time series. In addition, one of the hyperjerk system equilibrium points is stabilized using impulsive control. All real initial conditions become the equilibrium points of the basin of attraction using the stabilizing method.

scholarly journals Chaotic Dynamics of a Mixed Rayleigh–Liénard Oscillator Driven by Parametric Periodic Damping and External Excitations

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Yélomè Judicaël Fernando Kpomahou ◽  
Laurent Amoussou Hinvi ◽  
Joseph Adébiyi Adéchinan ◽  
Clément Hodévèwan Miwadinou
Keyword(s):  
Chaotic Dynamics ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basin Of Attraction ◽  
Melnikov Method ◽  
Constant Term ◽  
Geometrical Shape ◽  
Analytical Prediction ◽  
Coexistence Of Attractors ◽  
The Basin Of Attraction

In this paper, chaotic dynamics of a mixed Rayleigh–Liénard oscillator driven by parametric periodic damping and external excitations is investigated analytically and numerically. The equilibrium points and their stability evolutions are analytically analyzed, and the transitions of dynamical behaviors are explored in detail. Furthermore, from the Melnikov method, the analytical criterion for the appearance of the homoclinic chaos is derived. Analytical prediction is tested against numerical simulations based on the basin of attraction of initial conditions. As a result, it is found that for ω = ν , the chaotic region decreases and disappears when the amplitude of the parametric periodic damping excitation increases. Moreover, increasing of F 1 and F 0 provokes an erosion of the basin of attraction and a modification of the geometrical shape of the chaotic attractors. For ω ≠ ν and η = 0.8 , the fractality of the basin of attraction increases as the amplitude of the external periodic excitation and constant term increase. Bifurcation structures of our system are performed through the fourth-order Runge–Kutta ode 45 algorithm. It is found that the system displays a remarkable route to chaos. It is also found that the system exhibits monostable and bistable oscillations as well as the phenomenon of coexistence of attractors.

Boundary of the Basin of Attraction for Weakly Damped Primary Resonance

1995 ◽  
Vol 62 (4) ◽  
pp. 941-946 ◽  
Author(s):  
R. Haberman ◽  
E. K. Ho
Keyword(s):  
Homoclinic Orbit ◽  
Initial Conditions ◽  
Basin Of Attraction ◽  
Primary Resonance ◽  
Analytic Formula ◽  
Melnikov Integral ◽  
Small Dissipation ◽  
Stable Periodic ◽  
The Basin Of Attraction ◽  
Selection Of

The dissipatively perturbed Hamiltonian system corresponding to primary resonance is analyzed in the case in which two competing stable periodic responses exist. The method of averaging fails as the trajectory approaches the unperturbed homoclinic orbit (separatrix). By using the small dissipation of the Hamiltonian (the Melnikov integral) near the homoclinic orbit, the boundaries of the basin of attraction are determined analytically in an asymptotically accurate way. The selection of the two competing periodic responses is influenced by small changes in the initial conditions. The analytic formula is shown to agree well with numerical computations.

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

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.

Analysis of the Orbits of Electrostatic MEMS Resonators

2008 ◽  
Author(s):  
F. Najar ◽  
E. M. Abdel-Rahman ◽  
A. H. Nayfeh ◽  
S. Choura
Keyword(s):  
Initial Conditions ◽  
Order Approximation ◽  
Differential Quadrature Method ◽  
Basin Of Attraction ◽  
Mems Resonator ◽  
Mems Resonators ◽  
Electrostatic Mems ◽  
Differential Integral Equation ◽  
First Order Approximation ◽  
The Basin Of Attraction

We study the dynamic behavior of an electrostatic MEMS resonator using a model that accounts for the system nonlinearities due to mid-plane stretching and electrostatic forcing. The partial-differential-integral equation and associated boundary conditions representing the system dynamics are discretized using the Differential Quadrature Method (DQM) and the Finite Difference Method (FDM) for the space and time derivatives, respectively. The resulting model is analyzed to determine the periodic orbits of the resonator and their stability. Simultaneous resonances are identified for large orbits. Finally, we develop a first-order approximation of the microbeam dynamic response, which reveals an erosion of the basin of attraction of the stable orbits that depends heavily on the amplitude and frequency of the AC excitation. Simulations show that the smoothness of the boundary of the basin of attraction can be lost to be replaced by fractal tongues, which increase the sensitivity of the microbeam response to initial conditions. As a result, the locations of the stable and unstable fixed points are likely to be disturbed.

Finding Method and Analysis of Hidden Chaotic Attractors for Plasma Chaotic System From Physical and Mechanistic Perspectives

2020 ◽  
Vol 30 (05) ◽  
pp. 2050072 ◽  
Author(s):  
Yingjuan Yang ◽  
Guoyuan Qi ◽  
Jianbing Hu ◽  
Philippe Faradja
Keyword(s):  
Chaotic Attractor ◽  
Equilibrium Points ◽  
Basin Of Attraction ◽  
The Self ◽  
Numerical Continuation ◽  
Chaotic Attractors ◽  
Phase Portrait Analysis ◽  
The Stability ◽  
Two Parameters ◽  
The Basin Of Attraction

A method for finding hidden chaotic attractors in the plasma system is presented. Using the Routh–Hurwitz criterion, the stability distribution associated with two parameters is identified to find the region around the equilibrium points of the stable nodes, stable focus-nodes, saddles and saddle-foci for the purpose of investigating hidden chaos. A physical interpretation is provided of the stability distribution for each type of equilibrium point. The basin of attraction and parameter region of hidden chaos are identified by excluding the self-excited chaotic attractors of all equilibrium points. Homotopy and numerical continuation are also employed to check whether the basin of chaotic attraction intersects with the neighborhood of a saddle equilibrium. Bifurcation analysis, phase portrait analysis, and basins of different dynamical attraction are used as tools to distinguish visually the self-excited chaotic attractor and hidden chaotic attractor. The Casimir power reflects the error power between the dissipative energy and the energy supplied by the whistler field. It explains physically, analytically, and numerically the conditions that generate the different dynamics, such as sinks, periodic orbits, and chaos.

Infinite Number of Hidden Attractors in Memristor-Based Autonomous Duffing Oscillator

2018 ◽  
Vol 28 (01) ◽  
pp. 1850013 ◽  
Author(s):  
Vaibhav Varshney ◽  
S. Sabarathinam ◽  
Awadhesh Prasad ◽  
K. Thamilmaran
Keyword(s):  
Analog Circuit ◽  
Piecewise Linear ◽  
Duffing Oscillator ◽  
Basin Of Attraction ◽  
Nonlinear Circuits ◽  
Hidden Attractors ◽  
Statistical Measures ◽  
Wide Range ◽  
Hidden Oscillations ◽  
First Time

In this paper, the recent and emerging phenomenon of hidden oscillations is observed in a newly implemented memristor-based autonomous Duffing oscillator for the first time. The hidden oscillations are presented and quantified by various statistical measures. The system shows a large number of hidden attractors for a wide range of the system parameters. This study indicates that hidden oscillations can exist not only in piecewise-linear but also in smooth nonlinear circuits and systems. The distribution of Lyapunov exponents and the basin of attraction are explored to understand the nature of the hidden oscillations. We have also discussed the new phenomenon of periodic line invariant. An experimental demonstration is also presented using real time analog circuit.

scholarly journals Family of Bistable Attractors Contained in an Unstable Dissipative Switching System Associated to a SNLF

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
J. L. Echenausía-Monroy ◽  
J. H. García-López ◽  
R. Jaimes-Reátegui ◽  
D. López-Mancilla ◽  
G. Huerta-Cuellar
Keyword(s):  
Global Attractor ◽  
Control Parameter ◽  
Initial Conditions ◽  
Basin Of Attraction ◽  
Nonlinear Function ◽  
Bifurcation Parameter ◽  
Switching System ◽  
Generator System ◽  
The Basin Of Attraction

This work presents a multiscroll generator system, which addresses the issue by the implementation of 9-level saturated nonlinear function, SNLF, being modified with a new control parameter that acts as a bifurcation parameter. By means of the modification of the newly introduced parameter, it is possible to control the number of scrolls to generate. The proposed system has richer dynamics than the original, not only presenting the generation of a global attractor; it is capable of generating monostable and bistable multiscrolls. The study of the basin of attraction for the natural attractor generation (9-scroll SNLF) shows the restrictions in the initial conditions space where the system is capable of presenting dynamical responses, limiting its possible electronic implementations.

Hidden Attractors and Dynamical Behaviors in an Extended Rikitake System

2015 ◽  
Vol 25 (02) ◽  
pp. 1550028 ◽  
Author(s):  
Zhouchao Wei ◽  
Wei Zhang ◽  
Zhen Wang ◽  
Minghui Yao
Keyword(s):  
Hopf Bifurcation ◽  
Initial Conditions ◽  
Dynamics Analysis ◽  
Hidden Attractor ◽  
Hidden Attractors ◽  
System Parameters ◽  
Dynamical Behaviors ◽  
Rikitake System ◽  
Wide Range ◽  
Stable Equilibria

In this paper, an extended Rikitake system is studied. Several issues, such as Hopf bifurcation, coexistence of stable equilibria and hidden attractor, and dynamics analysis at infinity are investigated either analytically or numerically. Especially, by a simple linear transformation, the wide range of hidden attractors is noticed, and the Lyapunov exponents diagram is given. The obtained results show that the unstable periodic solution generated by Hopf bifurcation leads to the hidden attractor. The existence of hidden attractors that may render the system's behavior unpredictable not only depends on the value of system parameters but also on the value of initial conditions. The phenomena are important and potentially problematic in engineering applications.

Effect of Delay on the Boundary of the Basin of Attraction in a Self-Excited Single Graded-Response Neuron

1997 ◽  
Vol 9 (2) ◽  
pp. 319-336 ◽  
Author(s):  
K. Pakdaman ◽  
C. P. Malta ◽  
C. Grotta-Ragazzo ◽  
J.-F. Vibert
Keyword(s):  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Transmission Delays ◽  
The Past ◽  
Graded Response ◽  
The Stability ◽  
Delay Dependent ◽  
The Basin Of Attraction

Little attention has been paid in the past to the effects of interunit transmission delays (representing a xonal and synaptic delays) ontheboundary of the basin of attraction of stable equilibrium points in neural networks. As a first step toward a better understanding of the influence of delay, we study the dynamics of a single graded-response neuron with a delayed excitatory self-connection. The behavior of this system is representative of that of a family of networks composed of graded-response neurons in which most trajectories converge to stable equilibrium points for any delay value. It is shown that changing the delay modifies the “location” of the boundary of the basin of attraction of the stable equilibrium points without affecting the stability of the equilibria. The dynamics of trajectories on the boundary are also delay dependent and influence the transient regime of trajectories within the adjacent basins. Our results suggest that when dealing with networks with delay, it is important to study not only the effect of the delay on the asymptotic convergence of the system but also on the boundary of the basins of attraction of the equilibria.

1-D MAPS, CHAOS AND NEURAL NETWORKS FOR INFORMATION PROCESSING

1996 ◽  
Vol 06 (04) ◽  
pp. 627-646 ◽  
Author(s):  
YU.V. ANDREYEV ◽  
A.S. DMITRIEV ◽  
D.A. KUMINOV ◽  
L.O. CHUA ◽  
C.W. WU
Keyword(s):  
Neural Networks ◽  
Information Processing ◽  
Associative Memory ◽  
Complex Dynamics ◽  
Partial Information ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Basin Of Attraction ◽  
The State ◽  
The Basin Of Attraction

An application of complex dynamics and chaos in neural networks to information processing is studied. Mathematical models based on piecewise-linear maps implementing basic functions of information processing via complex dynamics and chaos are discussed. Realizations of these models by neural networks are presented. In contrast to other methods of using neural networks and associative memory to store information, the information is stored in dynamical attractors such as limit cycles, rather than equilibrium points. Retrieval of information corresponds to getting the state into the basin of attraction of the attractor. We show that noise-corrupted information or partial information are sufficient to drive the state into the basin of attraction of the attractor, thus these systems exhibit the property of associative memory.

Sign in / Sign up

Export Citation Format

Share Document