Phase portraits of the quadratic polynomial Liénard differential systems (original) (raw)

2020, Proceedings

https://doi.org/10.1017/PRM.2020.10

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Abstract

We classify the global phase portraits in the Poincaré disc of the quadratic polynomial Liénard differential systemṡ x = y,ẏ = (ax + b)y + cx 2 + dx + e, where (x, y) ∈ R 2 are the variables and a, b, c, d, e are real parameters.

Global Phase Portraits of the Quadratic Systems Having a Singular and Irreducible Invariant Curve of Degree 3

International Journal of Bifurcation and Chaos, 2023

Any singular irreducible cubic curve (or simply, cubic) after an affine transformation can be written as either y 2 = x 3 , or y 2 = x 2 (x + 1), or y 2 = x 2 (x − 1). We classify the phase portraits of all quadratic polynomial differential systems having the invariant cubic y 2 = x 2 (x+1). We prove that there are 63 different topological phase portraits for such quadratic polynomial differential systems. We control all the bifurcations among these distinct topological phase portraits. These systems have no limit cycles. Only 3 phase portraits have a center, 19 of these phase portraits have one polycycle, 3 of these phase portraits have 2 polycycles. The maximum number of separartices that have these phase portraits is 26 and the minimum number is 9, the maximum number of canonical regions of these phase portraits is 7 and the minimum is 3.

On the Number of Limit Cycles for a Generalization of Liénard Polynomial Differential Systems

International Journal of Bifurcation and Chaos, 2013

We study the number of limit cycles of the polynomial differential systems of the form [Formula: see text] where g1(x) = εg11(x) + ε2g12(x) + ε3g13(x), g2(x) = εg21(x) + ε2g22(x) + ε3g23(x) and f(x) = εf1(x) + ε2 f2(x) + ε3 f3(x) where g1i, g2i, f2i have degree k, m and n respectively for each i = 1, 2, 3, and ε is a small parameter. Note that when g1(x) = 0 we obtain the generalized Liénard polynomial differential systems. We provide an upper bound of the maximum number of limit cycles that the previous differential system can have bifurcating from the periodic orbits of the linear center ẋ = y, ẏ = -x using the averaging theory of third order.

Limit Cycle Bifurcations of a Special Liénard Polynomial System

In this paper, applying a canonical system with field rotation parameters and using geometric properties of the spirals filling the interior and exterior domains of limit cycles, we solve first the problem on the maximum number of limit cycles surrounding a unique singular point for an arbitrary polynomial system. Then, by means of the same bifurcationally geometric approach, we solve the limit cycle problem for a Liénard system with cubic restoring and polynomial damping functions.

Phase portraits of quadratic polynomial vector fields having a rational first integral of degree 3

Nonlinear Analysis: Theory, Methods & Applications, 2009

We classify all the global phase portraits of the quadratic polynomial vector fields having a rational first integral of degree 2. In other words we characterize all the global phase portraits of the quadratic polynomial vector fields having all their orbits contained in conics. For such a vector field there are exactly 25 different global phase portraits in the Poincaré disc, up to a reversal of sense.

Bifurcation Analysis of a Multi-Parameter Liénard Polynomial System

IFAC-PapersOnLine

In this paper, we study a multi-parameter Liénard polynomial system carrying out its global bifurcation analysis. To control the global bifurcations of limit cycle in this systems, it is necessary to know the properties and combine the effects of all its field rotation parameters. It can be done by means of the development of our bifurcational geometric method based on the application of a canonical system with field rotation parameters. Using this method, we present a solution of Hilbert's Sixteenth Problem on the maximum number of limit cycles and their distribution for the Liénard polynomial system. We also conduct some numerical experiments to illustrate the obtained results.

Topological Classification of Quadratic Polynomial Differential Systems with a Finite Semi-Elemental Triple Saddle

International Journal of Bifurcation and Chaos

The study of planar quadratic differential systems is very important not only because they appear in many areas of applied mathematics but due to their richness in structure, stability and questions concerning limit cycles, for example. Even though many papers have been written on this class of systems, a complete understanding of this family is still missing. Classical problems, and in particular Hilbert’s 16th problem [Hilbert, 1900, 1902], are still open for this family. In this article, we make a global study of the family [Formula: see text] of all real quadratic polynomial differential systems which have a finite semi-elemental triple saddle (triple saddle with exactly one zero eigenvalue). This family modulo the action of the affine group and time homotheties is three-dimensional and we give its bifurcation diagram with respect to a normal form, in the three-dimensional real space of the parameters of this normal form. This bifurcation diagram yields 27 phase portraits for sy...

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The applied geometry of a general Liénard polynomial system

Applied Mathematics Letters, 2012

In this work, applying a canonical system with field rotation parameters and using geometric properties of the spirals filling the interior and exterior domains of limit cycles, we solve the limit cycle problem for a general Liénard polynomial system with an arbitrary (but finite) number of singular points.

Limit cycle bifurcations of the classical Lienard polynomial system

2011

In this paper, applying a canonical system with field rotation parameters and using geometric properties of the spirals filling the interior and exterior domains of limit cycles, we solve the problem on the maximum number of limit cycles for the classical Liénard polynomial system which is related to the solution of Smale's thirteenth problem.

Remarks on a class of generalized Liénard planar systems

2021

We continue the recent investigation [40] about the qualitative properties of the solutions for a class of generalized Liénard systems of the formẋ = y − F (x, y),ẏ = −g(x). We present some results on the existence/non-existence of limit cycles depending on different growth assumptions of F (•, y). The case of asymmetric conditions at infinity for g(x) and F (x, •) is also examined. In the second part of the article we consider also a bifurcation result for small limit cycles as well as we discuss the complex dynamics associated to a periodically perturbed reversible system.

Liénard systems and potential–Hamiltonian decomposition – Applications in biology

Comptes Rendus Biologies, 2007

We show here how to approach with an increasing precision the limit-cycles of Liénard systems, bifurcating from a stable stationary state, by contour lines of Hamiltonian systems derived from a potential-Hamiltonian decomposition of the Liénard flow. We evoked in a previous Note the case (non polynomial) of pure potential systems (n-switches) and pure Hamiltonian systems (2D Lotka-Volterra), and here we show that, with the proposed approximation, we can deal with the case of mixed systems (van der Pol or FitzHugh-Nagumo) frequently used for modelling oscillatory systems in biology. We suggest finally that the proposed algorithm, generic for PHdecomposition, can be used for estimating the isochronal fibration in some specific cases near the pure potential or Hamiltonian systems. In a following Note, we will give applications in biology of the potential-Hamiltonian decomposition. Résumé Nous montrons ici comment approcher, avec une précision croissante, les cycles limites des systèmes de Liénard par les courbes de niveau du système Hamiltonien obtenuà partir d'une décomposition potentielle-Hamiltonienne. Nous avons présenté, dans une Note précédente, des cas non polynomiaux de systèmes purement potentiels (type n-switch) et purement Hamiltoniens (type Lotka-Volterra 2D), puis nous donnons ici des exemples de décomposition potentielle-Hamiltonienne pour des systèmes de type van der Pol (mixtes) qui sont d'usage fréquent en modélisation des systèmes biologiques oscillants. Nous suggérons enfin que l'algorithme proposé, générique pour la décomposition potentielle-Hamiltonienne, soit utilisé pour estimer la fibration isochrone, dans le cas de systèmes voisins des systèmes purs (potentiels ou Hamiltoniens). Dans une Note future, nous décrirons des applications biologiques précises de la décomposition potentielle-Hamiltonienne.

Phase Portraits of Cubic Dynamic Systems in a Poincare Circle

Differential Equations - Theory and Current Research, 2018

In the proposed chapter, we are going to outline the results of a study on an arithmetical plane of a broad family of dynamic systems having polynomial right parts. Let these polynomials be of cubic and square reciprocal forms. The task of our investigation is to find out all the different (in the topological sense) phase portraits in a Poincare circle and indicate the coefficient criteria of their appearance. To achieve this goal, we use the Poincare method of central and orthogonal consecutive displays (or mappings). As a result of this thorough investigation, we have constructed more than 250 topologically different phase portraits in total. Every portrait we present using a special table called a descriptive phase portrait. Each line of such a special table corresponds to one invariant cell of the phase portrait and describes its boundaries, as well as a source of its phase flow and a sink of it.

A family of quadratic polynomial differential systems with algebraic solutions of arbitrary high degree

arXiv (Cornell University), 2014

We show that the algebraic curve a 0 (x)(y − r(x)) + p 2 (x)a ′ (x) = 0, where r(x) and p 2 (x) are polynomial of degree 1 and 2 respectively and a 0 (x) is a polynomial solution of the convenient Fucsh's equation, is an invariant curve of the quadratic planar differential system. We study the particular case when a 0 (x) is an orthogonal polynomials. We prove that that in this case the quadratic differential system is Liouvillian integrable.