scholarly journals Lyapunov stability of elliptic periodic solutions of nonlinear damped equations

2012 ◽  
Vol 396 (1) ◽  
pp. 294-301 ◽  
Author(s):  
Jifeng Chu ◽  
Jinhong Ding ◽  
Yongxin Jiang
Keyword(s):  
Periodic Solutions ◽  
Lyapunov Stability

scholarly journals Existence and Lyapunov Stability of Positive Periodic Solutions for a Third-Order Neutral Differential Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-28 ◽  
Author(s):  
Jingli Ren ◽  
Zhibo Cheng ◽  
Yueli Chen
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Periodic Solutions ◽  
Lyapunov Stability ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Neutral Differential Equation ◽  
Positive Periodic Solutions ◽  
Third Order

By applying Green's function of third-order differential equation and a fixed point theorem in cones, we obtain some sufficient conditions for existence, nonexistence, multiplicity, and Lyapunov stability of positive periodic solutions for a third-order neutral differential equation.

Lyapunov stability of periodic solutions of Brillouin type equations

2020 ◽  
Vol 101 ◽  
pp. 106057 ◽  
Author(s):  
Feng Wang ◽  
José Ángel Cid ◽  
Shengjun Li ◽  
Mirosława Zima
Keyword(s):  
Periodic Solutions ◽  
Lyapunov Stability

Lyapunov stability of periodic solutions of the quadratic Newtonian equation

2009 ◽  
Vol 282 (9) ◽  
pp. 1354-1366 ◽  
Author(s):  
Meirong Zhang ◽  
Jifeng Chu ◽  
Xiong Li
Keyword(s):  
Periodic Solutions ◽  
Lyapunov Stability ◽  
Newtonian Equation

On nearly-commensurable periods in the restricted problem of three bodies, with calculations of the long-period variations in the interior 2:1 case

1966 ◽  
Vol 25 ◽  
pp. 197-222 ◽  
Author(s):  
P. J. Message
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Large Amplitude ◽  
Restricted Problem ◽  
Small Amplitude ◽  
Minor Planets ◽  
Long Period ◽  
Numerical Integrations ◽  
Period Variations ◽  
The Mean

An analytical discussion of that case of motion in the restricted problem, in which the mean motions of the infinitesimal, and smaller-massed, bodies about the larger one are nearly in the ratio of two small integers displays the existence of a series of periodic solutions which, for commensurabilities of the typep+ 1:p, includes solutions of Poincaré'sdeuxième sortewhen the commensurability is very close, and of thepremière sortewhen it is less close. A linear treatment of the long-period variations of the elements, valid for motions in which the elements remain close to a particular periodic solution of this type, shows the continuity of near-commensurable motion with other motion, and some of the properties of long-period librations of small amplitude.To extend the investigation to other types of motion near commensurability, numerical integrations of the equations for the long-period variations of the elements were carried out for the 2:1 interior case (of which the planet 108 “Hecuba” is an example) to survey those motions in which the eccentricity takes values less than 0·1. An investigation of the effect of the large amplitude perturbations near commensurability on a distribution of minor planets, which is originally uniform over mean motion, shows a “draining off” effect from the vicinity of exact commensurability of a magnitude large enough to account for the observed gap in the distribution at the 2:1 commensurability.

Discontinuous Control and Stability Analysis of Step-Down DC-DC Voltage Converters

2020 ◽  
Vol 38 (3A) ◽  
pp. 446-456
Author(s):  
Bashar F. Midhat
Keyword(s):  
Stability Analysis ◽  
Lyapunov Stability ◽  
Control Method ◽  
Power Electronic ◽  
Electronic Circuits ◽  
Dc Voltage ◽  
Power Electronic Circuits ◽  
Lyapunov Stability Criterion ◽  
Control Step ◽  
Step Down

Step down DC-DC converters are power electronic circuits, which mainly used to convert voltage from a level to a lower level. In this paper, a discontinuous controller is proposed as a control method in order to control Step-Down DC-DC converters. A Lyapunov stability criterion is used to mathematically prove the ability of the proposed controller to give the desired voltage. Simulationsl1 are performedl1 in MATLABl1 software. The simulationl1 resultsl1 are presentedl1 for changesl1 in referencel1 voltagel1 and inputl1 voltagel1 as well as stepl1 loadl1 variations. The resultsl1 showl1 the goodl1 performancel1 of the proposedl1 discontinuousl1 controller.

Normal form method in search for periodic solutions of ordinary differential equations. Case of the fourth order

2018 ◽  
pp. 24-34
Author(s):  
V. F. Edneral ◽  
O. D. Timofeevskaya
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Numerical Calculations ◽  
Computational Time ◽  
Resonant Normal Form ◽  
Normal Form Method

Introduction:The method of resonant normal form is based on reducing a system of nonlinear ordinary differential equations to a simpler form, easier to explore. Moreover, for a number of autonomous nonlinear problems, it is possible to obtain explicit formulas which approximate numerical calculations of families of their periodic solutions. Replacing numerical calculations with their precalculated formulas leads to significant savings in computational time. Similar calculations were made earlier, but their accuracy was insufficient, and their complexity was very high.Purpose:Application of the resonant normal form method and a software package developed for these purposes to fourth-order systems in order to increase the calculation speed.Results:It has been shown that with the help of a single algorithm it is possible to study equations of high orders (4th and higher). Comparing the tabulation of the obtained formulas with the numerical solutions of the corresponding equations shows good quantitative agreement. Moreover, the speed of calculation by prepared approximating formulas is orders of magnitude greater than the numerical calculation speed. The obtained approximations can also be successfully applied to unstable solutions. For example, in the Henon — Heyles system, periodic solutions are surrounded by chaotic solutions and, when numerically integrated, the algorithms are often unstable on them.Practical relevance:The developed approach can be used in the simulation of physical and biological systems.

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