scholarly journals On Quasi-Periodic Perturbations of Elliptic Equilibrium Points

1996 ◽  
Vol 27 (6) ◽  
pp. 1704-1737 ◽  
Author(s):  
Àngel Jorba ◽  
Carles Simó
Keyword(s):  
Equilibrium Points ◽  
Periodic Perturbations ◽  
Elliptic Equilibrium

scholarly journals Nekhoroshev estimates for steep real-analytic elliptic equilibrium points

Nonlinearity ◽  
2019 ◽  
Vol 33 (1) ◽  
pp. 1-33 ◽  
Author(s):  
Abed Bounemoura ◽  
Bassam Fayad ◽  
Laurent Niederman
Keyword(s):  
Equilibrium Points ◽  
Elliptic Equilibrium ◽  
Real Analytic

Nonlinear Stability of Equilibrium Points in the Planar Equilateral Restricted Mass-Unequal Four-Body Problem

2021 ◽  
Vol 31 (11) ◽  
pp. 2130031
Author(s):  
José Alejandro Zepeda Ramírez ◽  
Martha Alvarez-Ramírez ◽  
Antonio García
Keyword(s):  
Nonlinear Stability ◽  
Mass Formula ◽  
Body Problem ◽  
Equilibrium Points ◽  
Constant Angular Velocity ◽  
Gravitational Attraction ◽  
Stability Of Equilibrium ◽  
Four Body Problem ◽  
Elliptic Equilibrium ◽  
The Stability

In this paper, we investigate the stability of equilibrium points for the planar restricted equilateral four-body problem in the case that one particle of negligible mass is moving under the Newtonian gravitational attraction of three positive masses [Formula: see text], [Formula: see text] and [Formula: see text] (called primaries). These always lie at the vertices of an equilateral triangle (Lagrangian configuration) and move with constant angular velocity in circular orbits around their center of masses. We consider the case where all the primaries have unequal masses, and investigate the nonlinear stability (in the sense of Lyapunov) of the elliptic equilibrium for the specific values of the mass [Formula: see text] and [Formula: see text] of the primary, fixed on the horizontal axis. Moreover, the [Formula: see text][Formula: see text]:[Formula: see text][Formula: see text] four-order resonant cases are determined and the stability is investigated. In this study, Markeev’s theorem and Arnold’s theorem become key ingredients.

scholarly journals Kolmogorov-Arnold-Moser Theory and Symmetries for a Polynomial Quadratic Second Order Difference Equation

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 790
Author(s):  
Tarek F. Ibrahim ◽  
Zehra Nurkanović
Keyword(s):  
Difference Equation ◽  
Equilibrium Points ◽  
Kam Theory ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Elliptic Equilibrium ◽  
Order Difference Equation ◽  
The Difference ◽  
The Stability ◽  
Theoretical Results

By using the Kolmogorov-Arnold-Moser (KAM) theory, we investigate the stability of two elliptic equilibrium points (zero equilibrium and negative equilibrium) of the difference equation t n + 1 = α t n + β t n 2 − t n − 1 , n = 0 , 1 , 2 , … , where are t − 1 , t 0 , α ∈ R , α ≠ 0 , β > 0 . By using the symmetries we find the periodic solutions with some periods. Finally, some numerical examples are given to verify our theoretical results.

scholarly journals Super-exponential stability for generic real-analytic elliptic equilibrium points

2020 ◽  
Vol 366 ◽  
pp. 107088 ◽  
Author(s):  
Abed Bounemoura ◽  
Bassam Fayad ◽  
Laurent Niederman
Keyword(s):  
Exponential Stability ◽  
Equilibrium Points ◽  
Elliptic Equilibrium ◽  
Real Analytic

Mathematical views of the fractional Chua's electrical circuit described by the Caputo-Liouville derivative

2021 ◽  
Vol 67 (1 Jan-Feb) ◽  
pp. 91
Author(s):  
N. Sene
Keyword(s):  
Caputo Derivative ◽  
Local Stability ◽  
Numerical Scheme ◽  
Equilibrium Points ◽  
Electrical Circuit ◽  
Maximal Lyapunov Exponent ◽  
Electrical Circuits ◽  
Adaptive Controls ◽  
Liouville Derivative

This paper revisits Chua's electrical circuit in the context of the Caputo derivative. We introduce the Caputo derivative into the modeling of the electrical circuit. The solutions of the new model are proposed using numerical discretizations. The discretizations use the numerical scheme of the Riemann-Liouville integral. We have determined the equilibrium points and study their local stability. The existence of the chaotic behaviors with the used fractional-order has been characterized by the determination of the maximal Lyapunov exponent value. The variations of the parameters of the model into the Chua's electrical circuit have been quantified using the bifurcation concept. We also propose adaptive controls under which the master and the slave fractional Chua's electrical circuits go in the same way. The graphical representations have supported all the main results of the paper.

Dynamics of a delayed competitive system affected by toxic substances with imprecise biological parameters

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5271-5293
Author(s):  
A.K. Pal ◽  
P. Dolai ◽  
G.P. Samanta
Keyword(s):  
Equilibrium Points ◽  
Stable Limit Cycle ◽  
Limit Cycle Oscillation ◽  
Toxic Substances ◽  
Stable Limit ◽  
Biological Parameters ◽  
Delay Model ◽  
Competitive System ◽  
Interval Numbers ◽  
Cycle Oscillation

In this paper we have studied the dynamical behaviours of a delayed two-species competitive system affected by toxicant with imprecise biological parameters. We have proposed a method to handle these imprecise parameters by using parametric form of interval numbers. We have discussed the existence of various equilibrium points and stability of the system at these equilibrium points. In case of toxic stimulatory system, the delay model exhibits a stable limit cycle oscillation. Computer simulations are carried out to illustrate our analytical findings.

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