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.

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 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.

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 One Parameter Optimal Derivative-Free Family to Find the Multiple Roots of Algebraic Nonlinear Equations

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2223
Author(s):  
Munish Kansal ◽  
Ali Saleh Alshomrani ◽  
Sonia Bhalla ◽  
Ramandeep Behl ◽  
Mehdi Salimi
Keyword(s):  
Basin Of Attraction ◽  
Nonlinear Function ◽  
Second Order ◽  
Function Evaluation ◽  
Multiple Roots ◽  
Derivative Free ◽  
The One ◽  
Special Case ◽  
The Basin Of Attraction ◽  
Taylor’S Series

In this study, we construct the one parameter optimal derivative-free iterative family to find the multiple roots of an algebraic nonlinear function. Many researchers developed the higher order iterative techniques by the use of the new function evaluation or the first-order or second-order derivative of functions to evaluate the multiple roots of a nonlinear equation. However, the evaluation of the derivative at each iteration is a cumbersome task. With this motivation, we design the second-order family without the utilization of the derivative of a function and without the evaluation of the new function. The proposed family is optimal as it satisfies the convergence order of Kung and Traub’s conjecture. Here, we use one parameter a for the construction of the scheme, and for a=1, the modified Traub method is its a special case. The order of convergence is analyzed by Taylor’s series expansion. Further, the efficiency of the suggested family is explored with some numerical tests. The obtained results are found to be more efficient than earlier schemes. Moreover, the basin of attraction of the proposed and earlier schemes is also analyzed.

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.

scholarly journals Optimization and Stability of Heat Engines: The Role of Entropy Evolution

Entropy ◽  
2018 ◽  
Vol 20 (11) ◽  
pp. 865 ◽  
Author(s):  
Julian Gonzalez-Ayala ◽  
Moises Santillán ◽  
Maria Santos ◽  
Antonio Calvo Hernández ◽  
José Mateos Roco
Keyword(s):  
Local Stability ◽  
Production Efficiency ◽  
Basin Of Attraction ◽  
Heat Engine ◽  
Time Constraints ◽  
Thermal Gradients ◽  
Heat Engines ◽  
Low Dissipation ◽  
The Basin Of Attraction

Local stability of maximum power and maximum compromise (Omega) operation regimes dynamic evolution for a low-dissipation heat engine is analyzed. The thermodynamic behavior of trajectories to the stationary state, after perturbing the operation regime, display a trade-off between stability, entropy production, efficiency and power output. This allows considering stability and optimization as connected pieces of a single phenomenon. Trajectories inside the basin of attraction display the smallest entropy drops. Additionally, it was found that time constraints, related with irreversible and endoreversible behaviors, influence the thermodynamic evolution of relaxation trajectories. The behavior of the evolution in terms of the symmetries of the model and the applied thermal gradients was analyzed.

scholarly journals Time Delay and Feedback Control of an Inverted Pendulum With Stick Slip Friction

2007 ◽  
Author(s):  
Sue Ann Campbell ◽  
Stephanie Crawford ◽  
Kirsten Morris
Keyword(s):  
Time Delay ◽  
Basin Of Attraction ◽  
Critical Time ◽  
Viscous Friction ◽  
Experimental System ◽  
Friction Model ◽  
Stable Equilibrium Point ◽  
Feedback Controllers ◽  
Stick Slip ◽  
The Basin Of Attraction

We consider an experimental system consisting of a pendulum, which is free to rotate 360 degrees, attached to a cart which can move in one dimension. There is stick slip friction between the cart and the track on which it moves. Using two different models for this friction we design feedback controllers to stabilize the pendulum in the upright position. We show that controllers based on either friction model give better performance than one based on a simple viscous friction model. We then study the effect of time delay in this controller, by calculating the critical time delay where the system loses stability and comparing the calculated value with experimental data. Both models lead to controllers with similar robustness with respect to delay. Using numerical simulations, we show that the effective critical time delay of the experiment is much less than the calculated theoretical value because the basin of attraction of the stable equilibrium point is very small.

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