Centers and limit cycles in polynomial systems of ordinary differential equations

2019 ◽  
Author(s):  
Valery G. Romanovski ◽  
Douglas S. Shafer
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Limit Cycles ◽  
Polynomial Systems

Center-Focus Problem for Discontinuous Planar Differential Equations

2003 ◽  
Vol 13 (07) ◽  
pp. 1755-1765 ◽  
Author(s):  
Armengol Gasull ◽  
Joan Torregrosa
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Critical Point ◽  
Limit Cycles ◽  
New Method ◽  
Polar Coordinates ◽  
Mechanical Problem ◽  
Weak Focus ◽  
Elastic Walls

We study the center-focus problem as well as the number of limit cycles which bifurcate from a weak focus for several families of planar discontinuous ordinary differential equations. Our computations of the return map near the critical point are performed with a new method based on a suitable decomposition of certain one-forms associated with the expression of the system in polar coordinates. This decomposition simplifies all the expressions involved in the procedure. Finally, we apply our results to study a mathematical model of a mechanical problem, the movement of a ball between two elastic walls.

scholarly journals On the Polynomial Solutions and Limit Cycles of Some Generalized Polynomial Ordinary Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1139
Author(s):  
Claudia Valls
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Limit Cycles ◽  
Polynomial Solutions ◽  
Generalized Polynomial

We study equations of the form y d y / d x = P ( x , y ) where P ( x , y ) ∈ R [ x , y ] with degree n in the y-variable. We prove that this ordinary differential equation has at most n polynomial solutions (not necessarily constant but coprime among each other) and this bound is sharp. We also consider polynomial limit cycles and their multiplicity.

Periodic and Chaotic Dynamics of the Ehrhard–Müller System

2016 ◽  
Vol 26 (06) ◽  
pp. 1630015 ◽  
Author(s):  
Junho Park ◽  
Hyunho Lee ◽  
Jong-Jin Baik
Keyword(s):  
Natural Convection ◽  
Differential Equations ◽  
Asymptotic Analysis ◽  
Ordinary Differential Equations ◽  
Limit Cycles ◽  
Chaotic Dynamics ◽  
Single Phase ◽  
Chaotic Regime ◽  
Nonlinear Ordinary Differential Equations ◽  
Phase Loop

This paper investigates nonlinear ordinary differential equations of the Ehrhard–Müller system which describes natural convection in a single-phase loop in the presence of nonsymmetric heating. Stability and dynamics of periodic and chaotic behaviors of the equations are investigated and the periodicity diagram is obtained in wide ranges of parameters. Regimes of both periodic and chaotic solutions are observed with complex behaviors such that the periodic regimes enclose the chaotic regime while they are also immersed inside the chaotic regime with various shapes. An asymptotic analysis is performed for sufficiently large parameters to understand the enclosure by the periodic regimes and asymptotic limit cycles are obtained to compare with limit cycles obtained from numerical results.

scholarly journals A counterexample to the composition condition conjecture for polynomial Abel differential equations

2018 ◽  
Vol 39 (12) ◽  
pp. 3347-3352 ◽  
Author(s):  
JAUME GINÉ ◽  
MAITE GRAU ◽  
XAVIER SANTALLUSIA
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Model Problem ◽  
Polynomial Systems ◽  
Composition Condition ◽  
Abel Differential Equations

Polynomial Abel differential equations are considered a model problem for the classical Poincaré center–focus problem for planar polynomial systems of ordinary differential equations. In the last few decades, several works pointed out that all centers of the polynomial Abel differential equations satisfied the composition conditions (also called universal centers). In this work we provide a simple counterexample to this conjecture.

Calculation of focal values for first-order non-autonomous equation with algebraic and trigonometric coefficients

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 738-750
Author(s):  
Saima Akram ◽  
Allah Nawaz ◽  
Thabet Abdeljawad ◽  
Abdul Ghaffar ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Bifurcation Analysis ◽  
Limit Cycles ◽  
Higher Order ◽  
First Order ◽  
Autonomous Equation ◽  
Fine Focus ◽  
New Formula

AbstractThis article concerns with the development of the number of focal values. We analyzed periodic solutions for first-order cubic non-autonomous ordinary differential equations. Bifurcation analysis for periodic solutions from a fine focus {\mathfrak{z}}=0 is also examined. In particular, we are interested to detect the maximum number of periodic solutions for various classes of higher order in which a given solution can bifurcate under perturbation of the coefficients. We calculate the maximum number of periodic solutions for different classes, namely, {C}_{10,5} and {C}_{12,6} with trigonometric coefficients, and they are found with nine and eight multiplicities at most. The classes {C}_{8,3} and {C}_{8,4} with algebraic coefficients have at most eight limit cycles. The new formula {\varkappa }_{10} is developed by which we succeeded to find highest known multiplicity ten for class {C}_{\mathrm{9,3}} with polynomial coefficient. Periodicity is calculated for both trigonometric and algebraic coefficients. Few examples are also considered to explain the applicability and stability of the methods presented.

scholarly journals LIMIT CYCLES FOR GENERALIZED ABEL EQUATIONS

2006 ◽  
Vol 16 (12) ◽  
pp. 3737-3745 ◽  
Author(s):  
ARMENGOL GASULL ◽  
ANTONI GUILLAMON
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Limit Cycles ◽  
Upper Bounds ◽  
Periodic Functions ◽  
One Dimensional ◽  
Nonautonomous Differential Equations ◽  
Abel Equations ◽  
The Right

This paper deals with the problem of finding upper bounds on the number of periodic solutions of a class of one-dimensional nonautonomous differential equations: those with the right-hand sides being polynomials of degree n and whose coefficients are real smooth one-periodic functions. The case n = 3 gives the so-called Abel equations which have been thoroughly studied and are well understood. We consider two natural generalizations of Abel equations. Our results extend previous works of Lins Neto and Panov and try to step forward in the understanding of the case n > 3. They can be applied, as well, to control the number of limit cycles of some planar ordinary differential equations.

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