On the Poincaré problem and Liouvillian integrability of quadratic Liénard differential equations

2020 ◽  
Vol 150 (6) ◽  
pp. 3231-3251 ◽  
Author(s):  
Maria V. Demina ◽  
Claudia Valls
Keyword(s):  
Differential Equations ◽  
Algebraic Curves ◽  
First Integral ◽  
Complete Classification ◽  
Invariant Algebraic Curves ◽  
Liouvillian Integrability ◽  
Poincaré Problem

AbstractWe present the complete classification of irreducible invariant algebraic curves of quadratic Liénard differential equations. We prove that these equations have irreducible invariant algebraic curves of unbounded degrees, in contrast with what is wrongly claimed in the literature. In addition, we classify all the quadratic Liénard differential equations that admit a Liouvillian first integral.

Classification of Invariant Algebraic Curves and Nonexistence of Algebraic Limit Cycles in Quadratic Systems from Family (I) of the Chinese Classification

2020 ◽  
Vol 30 (04) ◽  
pp. 2050056 ◽  
Author(s):  
Maria V. Demina ◽  
Claudia Valls
Keyword(s):  
Differential System ◽  
Limit Cycles ◽  
Algebraic Curves ◽  
Complete Classification ◽  
Quadratic Systems ◽  
Correct Proof ◽  
Invariant Algebraic Curves ◽  
Algebraic Limit Cycles ◽  
Algebraic Limit

We give the complete classification of irreducible invariant algebraic curves in quadratic systems from family [Formula: see text] of the Chinese classification, that is, of differential system [Formula: see text] with [Formula: see text]. In addition, we provide a complete and correct proof of the nonexistence of algebraic limit cycles for these equations.

AN ALGORITHM FOR FINDING INVARIANT ALGEBRAIC CURVES OF A GIVEN DEGREE FOR POLYNOMIAL PLANAR VECTOR FIELDS

2005 ◽  
Vol 15 (03) ◽  
pp. 1033-1044 ◽  
Author(s):  
GRZEGORZ ŚWIRSZCZ
Keyword(s):  
Differential Equations ◽  
Algebraic Curve ◽  
Vector Fields ◽  
Algebraic Curves ◽  
Polynomial System ◽  
Polynomial Systems ◽  
Invariant Algebraic Curves ◽  
Invariant Algebraic Curve ◽  
Planar Vector Fields ◽  
Planar Vector

Given a system of two autonomous ordinary differential equations whose right-hand sides are polynomials, it is very hard to tell if any nonsingular trajectories of the system are contained in algebraic curves. We present an effective method of deciding whether a given system has an invariant algebraic curve of a given degree. The method also allows the construction of examples of polynomial systems with invariant algebraic curves of a given degree. We present the first known example of a degree 6 algebraic saddle-loop for polynomial system of degree 2, which has been found using the described method. We also present some new examples of invariant algebraic curves of degrees 4 and 5 with an interesting geometry.

Conservation laws and null divergences

1983 ◽  
Vol 94 (3) ◽  
pp. 529-540 ◽  
Author(s):  
Peter J. Olver
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Existence Of Solutions ◽  
Complete Classification ◽  
Complete Analysis ◽  
Partial Differential ◽  
Physical Systems

For a system of partial differential equations, the existence of appropriate conservation laws is often a key ingredient in the investigation of its solutions and their properties. Conservation laws can be used in proving existence of solutions, decay and scattering properties, investigation of singularities, analysis of integrability properties of the system and so on. Representative applications, and more complete bibliographies on conservation laws, can be found in references [7], [8], [12], [19]. The more conservation laws known for a given system, the more tools available for the above investigations. Thus a complete classification of all conservation laws of a given system is of great interest. Not many physical systems have been subjected to such a complete analysis, but two examples can be found in [11] and [14]. The present paper arose from investigations ([15], [16]) into the conservation laws of the equations of elasticity.

scholarly journals Non-linear differential equations with transcendental meromorphic solutions

2001 ◽  
Vol 70 (1) ◽  
pp. 88-118 ◽  
Author(s):  
Katsuya Ishizaki ◽  
Yuefei Wang
Keyword(s):  
Differential Equations ◽  
Complex Dynamics ◽  
Complete Classification ◽  
Linear Differential Equations ◽  
Meromorphic Solutions ◽  
Non Linear ◽  
Linear Differential

AbstractIn this paper we treat two non-linear differential equations which come from complex dynamics theory. We give a complete classification of the equations when they possess transcendental meromorphic solutions.

scholarly journals Symmetry Classification of First Integrals for Scalar Linearizable Second-Order ODEs

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
K. S. Mahomed ◽  
E. Momoniat
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Complete Classification ◽  
First Integrals ◽  
Second Order ◽  
Point Transformations ◽  
Maximal Algebra ◽  
The Relationship ◽  
Linear Odes

Symmetries of the fundamental first integrals for scalar second-order ordinary differential equations (ODEs) which are linear or linearizable by point transformations have already been obtained. Firstly we show how one can determine the relationship between the symmetries and the first integrals of linear or linearizable scalar ODEs of order two. Secondly, a complete classification of point symmetries of first integrals of such linear ODEs is studied. As a consequence, we provide a counting theorem for the point symmetries of first integrals of scalar linearizable second-order ODEs. We show that there exists the 0-, 1-, 2-, or 3-point symmetry cases. It is shown that the maximal algebra case is unique.

scholarly journals The method of Puiseux series and invariant algebraic curves

2021 ◽  
pp. 2150007
Author(s):  
Maria V. Demina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Algebraic Curve ◽  
Algebraic Curves ◽  
General Structure ◽  
First Integral ◽  
Puiseux Series ◽  
Polynomial Dynamical System ◽  
Invariant Algebraic Curves ◽  
Rational First Integral

An explicit expression for the cofactor related to an irreducible invariant algebraic curve of a polynomial dynamical system in the plane is derived. A sufficient condition for a polynomial dynamical system in the plane to have a finite number of irreducible invariant algebraic curves is obtained. All these results are applied to Liénard dynamical systems [Formula: see text], [Formula: see text] with [Formula: see text]. The general structure of their irreducible invariant algebraic curves and cofactors is found. It is shown that Liénard dynamical systems with [Formula: see text] can have at most two distinct irreducible invariant algebraic curves simultaneously and, consequently, are not integrable with a rational first integral.

scholarly journals Multiplicative automatic sequences

2021 ◽  
Author(s):  
Jakub Konieczny ◽  
Mariusz Lemańczyk ◽  
Clemens Müllner
Keyword(s):  
Complete Classification ◽  
Automatic Sequences ◽  
Complex Valued

AbstractWe obtain a complete classification of complex-valued sequences which are both multiplicative and automatic.

scholarly journals Limit Behavior of Solutions of Ordinary Linear Differential Equations

1994 ◽  
Vol 1 (3) ◽  
pp. 315-323
Author(s):  
František Neuman
Keyword(s):  
Differential Equations ◽  
Oscillatory Behavior ◽  
Linear Differential Equations ◽  
Limit Behavior ◽  
Limit Sets ◽  
Equivalent Linear ◽  
Linear Differential ◽  
Ordinary Linear Differential Equations

Abstract A classification of classes of equivalent linear differential equations with respect to ω-limit sets of their canonical representatives is introduced. Some consequences of this classification to the oscillatory behavior of solution spaces are presented.

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