scholarly journals A New Circumscribed Self-Excited Spherical Strange Attractor

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Ramesh Ramamoorthy ◽  
Sajjad Shaukat Jamal ◽  
Iqtadar Hussain ◽  
Mahtab Mehrabbeik ◽  
Sajad Jafari ◽  
...  
Keyword(s):  
Strange Attractor ◽  
Unique Feature ◽  
Initial Conditions ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Spherical Coordinates ◽  
Chaotic Flows ◽  
Pass Through ◽  
Third Dimension ◽  
The Basin Of Attraction

Studying new chaotic flows with specific characteristics has been an open-ended field of exploring nonlinear dynamics. Investigation of chaotic flows is an area of research that has been taken into consideration for many years; thus, it helps in a better understanding of the chaotic systems. In this paper, an original chaotic 3D system, which has not been investigated yet, is presented in spherical coordinates. A unique feature of the proposed system is that its velocity becomes zero for a specific value of the radius variable. Hence, the system’s attractor is expected to be stuck on one side of a plane in spherical coordinates and inside or outside a sphere in the corresponding Cartesian coordinates. It means that the attractor cannot pass through the sphere or even touch it. The introduced system owns two unstable equilibria and a self-excited strange attractor. The 1D and 2D system’s bifurcation diagrams concerning the alteration of two bifurcation parameters are plotted to investigate the system’s dynamical properties. Moreover, the system’s Lyapunov exponents in the corresponding period of bifurcation parameters are calculated. Then, two 2D basins of attraction for two different third dimension values are explored. Based on the basin of attraction, it can be found that the sphere has attraction itself, partially, and some initial conditions are led to the sphere, not to the strange attractor. Ultimately, the connecting curves of the proposed system are explored to find an informative 1D set in addition to the system’s equilibria.

THE BASIN OF ATTRACTION OF LOZI MAPPINGS

2009 ◽  
Vol 19 (03) ◽  
pp. 1043-1049 ◽  
Author(s):  
DIOGO BAPTISTA ◽  
RICARDO SEVERINO ◽  
SANDRA VINAGRE
Keyword(s):  
Strange Attractor ◽  
Basin Of Attraction ◽  
Parameter Values ◽  
The Basin Of Attraction

For parameter values which assure its existence, we characterize the basin of attraction for the Lozi map strange attractor.

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.

On-Off Intermittency and Riddled Basins of Attraction in a Coupled Map System

1998 ◽  
Vol 01 (02n03) ◽  
pp. 161-180 ◽  
Author(s):  
J. Laugesen ◽  
E. Mosekilde ◽  
Yu. L. Maistrenko ◽  
V. L. Maistrenko
Keyword(s):  
Periodic Orbits ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Chaotic State ◽  
One Dimensional ◽  
Type Iii ◽  
Different Types ◽  
One Dimensional Maps ◽  
The Basin Of Attraction

The paper examines the appearance of on-off intermittency and riddled basins of attraction in a system of two coupled one-dimensional maps, each displaying type-III intermittency. The bifurcation curves for the transverse destablilization of low periodic orbits embeded in the synchronized chaotic state are obtained. Different types of riddling bifurcation are discussed, and we show how the existence of an absorbing area inside the basin of attraction can account for the distinction between local and global riddling as well as for the distinction between hysteric and non-hysteric blowout. We also discuss the role of the so-called mixed absorbing area that exists immediately after a soft riddling bifurcation. Finally, we study the on-off intermittency that is observed after a non-hysteric blowout bifurcaton.

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.

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.

scholarly journals Basins of Attraction for Various Steffensen-Type Methods

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
Alicia Cordero ◽  
Fazlollah Soleymani ◽  
Juan R. Torregrosa ◽  
Stanford Shateyi
Keyword(s):  
Computer Algebra System ◽  
Dynamical Behavior ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Numerical Tests ◽  
Derivative Free ◽  
Iteration Functions ◽  
The Stability ◽  
Programming Package ◽  
The Basin Of Attraction

The dynamical behavior of different Steffensen-type methods is analyzed. We check the chaotic behaviors alongside the convergence radii (understood as the wideness of the basin of attraction) needed by Steffensen-type methods, that is, derivative-free iteration functions, to converge to a root and compare the results using different numerical tests. We will conclude that the convergence radii (and the stability) of Steffensen-type methods are improved by increasing the convergence order. The computer programming package MATHEMATICAprovides a powerful but easy environment for all aspects of numerics. This paper puts on show one of the application of this computer algebra system in finding fixed points of iteration functions.

scholarly journals Adaptive multimode signal reconstruction from time–frequency representations

2016 ◽  
Vol 374 (2065) ◽  
pp. 20150205 ◽  
Author(s):  
Sylvain Meignen ◽  
Thomas Oberlin ◽  
Philippe Depalle ◽  
Patrick Flandrin ◽  
Stephen McLaughlin
Keyword(s):  
Signal Reconstruction ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Alternative View ◽  
Short Time Fourier Transform ◽  
Time Frequency ◽  
Fm Signals ◽  
Adaptive Reconstruction ◽  
Short Time ◽  
The Basin Of Attraction

This paper discusses methods for the adaptive reconstruction of the modes of multicomponent AM–FM signals by their time–frequency (TF) representation derived from their short-time Fourier transform (STFT). The STFT of an AM–FM component or mode spreads the information relative to that mode in the TF plane around curves commonly called ridges . An alternative view is to consider a mode as a particular TF domain termed a basin of attraction . Here we discuss two new approaches to mode reconstruction. The first determines the ridge associated with a mode by considering the location where the direction of the reassignment vector sharply changes, the technique used to determine the basin of attraction being directly derived from that used for ridge extraction. A second uses the fact that the STFT of a signal is fully characterized by its zeros (and then the particular distribution of these zeros for Gaussian noise) to deduce an algorithm to compute the mode domains. For both techniques, mode reconstruction is then carried out by simply integrating the information inside these basins of attraction or domains.

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 Global Dynamics of Certain Homogeneous Second-Order Quadratic Fractional Difference Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
M. Garić-Demirović ◽  
M. R. S. Kulenović ◽  
M. Nurkanović
Keyword(s):  
Difference Equation ◽  
Unique Feature ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basins Of Attraction ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Minimal Period ◽  
Fractional Difference ◽  
The Difference

We investigate the basins of attraction of equilibrium points and minimal period-two solutions of the difference equation of the formxn+1=xn-12/(axn2+bxnxn-1+cxn-12),n=0,1,2,…,where the parameters  a,  b, and  c  are positive numbers and the initial conditionsx-1andx0are arbitrary nonnegative numbers. The unique feature of this equation is the coexistence of an equilibrium solution and the minimal period-two solution both of which are locally asymptotically stable.

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.

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