scholarly journals Bounds for the lengths and periods of closed orbits of two-dimensional autonomous systems of differential equations

1967 ◽  
Vol 3 (3) ◽  
pp. 330-342 ◽  
Author(s):  
Peter J Lau
Keyword(s):  
Differential Equations ◽  
Autonomous Systems ◽  
Two Dimensional ◽  
Systems Of Differential Equations ◽  
Closed Orbits

SUFFICIENT CONDITIONS FOR A CENTER AT A COMPLETELY DEGENERATE CRITICAL POINT

2002 ◽  
Vol 12 (07) ◽  
pp. 1659-1666 ◽  
Author(s):  
J. GINÉ
Keyword(s):  
Differential Equations ◽  
Critical Point ◽  
Autonomous Systems ◽  
Sufficient Conditions ◽  
Two Dimensional ◽  
Homogeneous Polynomials ◽  
Systems Of Differential Equations ◽  
Degenerate Critical Point

Consider the two-dimensional autonomous systems of differential equations of the form [Formula: see text] where P3(x, y) and Q3(x, y) are homogeneous polynomials of degree 3, and P4(x, y) and Q4(x, y) are homogeneous polynomials of degree 4. The origin is a completely degenerate critical point of this system. In this work we give sufficient conditions in order to have a center at the origin.

Isochronous Foci for Analytic Differential Systems

2003 ◽  
Vol 13 (06) ◽  
pp. 1617-1623 ◽  
Author(s):  
Jaume Giné
Keyword(s):  
Differential Equations ◽  
Analytic Functions ◽  
Autonomous Systems ◽  
Sufficient Conditions ◽  
Two Dimensional ◽  
Differential Systems ◽  
Weak Focus ◽  
Systems Of Differential Equations ◽  
Strong Focus

Consider the two-dimensional autonomous systems of differential equations of the form [Formula: see text] where P(x, y) and Q(x, y) are analytic functions. The origin is a strong focus of this system if λ ≠ 0 and is either a weak focus or its center if λ = 0. In this work we provide some sufficient conditions to have an isochronous focus at the origin.

Characterization of isochronous foci for planar analytic differential systems

2005 ◽  
Vol 135 (5) ◽  
pp. 985-998 ◽  
Author(s):  
Jaume Giné ◽  
Maite Grau
Keyword(s):  
Critical Point ◽  
Analytic Functions ◽  
Autonomous Systems ◽  
Sufficient Conditions ◽  
Two Dimensional ◽  
Differential Systems ◽  
Systems Of Differential Equations ◽  
Image Position ◽  
Necessary And Sufficient

We consider the two-dimensional autonomous systems of differential equations of the form where P(x,y) and Q(x,y) are analytic functions of order greater than or equal to 2. These systems have a focus at the origin if λ ≠ 0, and have either a centre or a weak focus if λ = 0. In this work we study the necessary and sufficient conditions for the existence of an isochronous critical point at the origin. Our result is, to the best of our knowledge, original when applied to weak foci and gives known results when applied to strong foci or to centres.

scholarly journals Ways for detection of the exact number of limit cycles of autonomous systems on the cylinder

2019 ◽  
Vol 55 (2) ◽  
pp. 182-194
Author(s):  
A. A. Hryn ◽  
S. V. Rudzevich
Keyword(s):  
Differential Equations ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Autonomous Systems ◽  
Type System ◽  
Transversality Condition ◽  
Second Step ◽  
Constant Sign ◽  
Exact Number ◽  
Systems Of Differential Equations

For real autonomous systems of differential equations with continuously differentiable right-hand sides, the problem of detecting the exact number and localization of the second-kind limit cycles on the cylinder is considered. To solve this problem in the absence of equilibria of the system on the cylinder, we have developed our previously proposed ways consisting in a sequential two-step application of the Dulac – Cherkas test or the Dulac test. Additionally, a new way has been worked out using the generalization of the Dulac – Cherkas or Dulac test at the second step, where the requirement of constant sign for divergence is replaced by the transversality condition of the curves on which the divergence vanishes. With the help of the developed ways, closed transversal curves are found that divide the cylinder into subdomains surrounding it, in each of which the system has exactly one second-kind limit cycle.The practical efficiency of the mentioned ways is demonstrated by the example of a pendulum-type system, for which, in the absence of equilibria, the existence of exactly three second-kind limit cycles on the entire phase cylinder is proved.

scholarly journals Exact stability and instability regions for two-dimensional linear autonomous multi-order systems of fractional-order differential equations

2021 ◽  
Vol 24 (1) ◽  
pp. 225-253
Author(s):  
Oana Brandibur ◽  
Eva Kaslik
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Fractional Order ◽  
Autonomous Systems ◽  
Sufficient Conditions ◽  
Two Dimensional ◽  
Fractional Order Differential Equations ◽  
Stability And Instability ◽  
Instability Regions ◽  
Necessary And Sufficient

Abstract Necessary and sufficient conditions are explored for the asymptotic stability and instability of linear two-dimensional autonomous systems of fractional-order differential equations with Caputo derivatives. Fractional-order-dependent and fractional-order-independent stability and instability properties are fully characterised, in terms of the main diagonal elements of the systems’ matrix, as well as its determinant.

scholarly journals Exact Solutions for Certain Nonlinear Autonomous Ordinary Differential Equations of the Second Order and Families of Two-Dimensional Autonomous Systems

2010 ◽  
Vol 2010 ◽  
pp. 1-13 ◽  
Author(s):  
M. P. Markakis
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Autonomous Systems ◽  
Closed Form Solution ◽  
First Integrals ◽  
Form Solution ◽  
Second Order ◽  
Two Dimensional ◽  
First Order ◽  
Abel Equations

Certain nonlinear autonomous ordinary differential equations of the second order are reduced to Abel equations of the first kind ((Ab-1) equations). Based on the results of a previous work, concerning a closed-form solution of a general (Ab-1) equation, and introducing an arbitrary function, exact one-parameter families of solutions are derived for the original autonomous equations, for the most of which only first integrals (in closed or parametric form) have been obtained so far. Two-dimensional autonomous systems of differential equations of the first order, equivalent to the considered herein autonomous forms, are constructed and solved by means of the developed analysis.

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