The Simplest Polynomial Differential Systems in ℝ3 Having an Invariant Sphere

In the article Llibre and Vulpe (Rocky Mt J Math 38:1301–1373, 2006) the family of cubic polynomial differential systems possessing invariant straight lines of total multiplicity 9 was considered and 23 such classes of systems were detected. We recall that 9 invariant straight lines taking into account their multiplicities is the maximum number of straight lines that a cubic polynomial differential systems can have if this number is finite. Here we complete the classification given in Llibre and Vulpe (Rocky Mt J Math 38:1301–1373, 2006) by adding a new class of such cubic systems and for each one of these 24 such classes we perform the corresponding first integral as well as its phase portrait. Moreover we present necessary and sufficient affine invariant conditions for the realization of each one of the detected classes of cubic systems with maximum number of invariant straight lines when this number is finite.

This paper answers the following two questions: What are the easiest polynomial differential systems in R3 having an invariant hyperbolic, parabolic or elliptic cylinder?, and for such polynomial differential systems what are their phase portraits on such invariant cylinders?

Using Einstein’s notation [4]), the polynomial differential systems of finite dimension n and of degree at most k with coefficients in a field k of characteristic zero can be written as dx dt = a + ajα1x α1 + ajα1α2x 1x2 + ajα1···αrx α1 · · · xr (1) where j, α1, αr ∈ {1, . . . , n}, 1 ≤ r ≤ k, and for j = 1, . . . , n and for 2 ≤ r ≤ k, the tensor ajα1···αr (1 time contravariant and r times covariant) is symmetric with respect to the lower subscripts and we consider the action of the general linear group GL(n, k) on the phase space k. In this work, starting from a minimal system of generators of the ideal of algebraic invariants of polynomial differential systems (1) we will develop an algorithmic method to study the local stability of a given cubic differential system with the help of Grobner bases and using the GL(n, k)-invariants of this cubic system [2]. Our algorithmic method is motivated by the important role of the invariant theory in the survey of polynomial differential systems and the numerous works on Groebner basis one of the main practical tools for solving algebraic systems and computing algebraic varieties. Keywords Polynomial differential systems,linear transformations, cubic differential systems,singular points, invariant, covariant, Grobner bases.

Polynomial differential systems having a given Darbouxian first integral*1

The existence of limit cycles is interesting and very important in applications. It is a key to understand the dynamic of polynomial differential systems. The aim of this paper is to investigate a class of planar differential systems of degree seven. Under some suitable conditions, the existence of three limit cycles two of them are non-algebraic while the third is algebraic is proved. Furthermore, these limit cycles are explicitly given in polar coordinates. Some examples are presented in order to illustrate the applicability of our results.

We give sufficient conditions in terms of the Melnikov functions in order that an analytic or a polynomial differential system in the real plane has a period annulus. We study the first nonzero Melnikov function of the analytic differential systems in the real plane obtained by perturbing a Hamiltonian system having either a nondegenerate center, a heteroclinic cycle, a homoclinic cycle, or three cycles obtained connecting the four separatrices of a saddle. All the singular points of these cycles are hyperbolic saddles. Finally, using the first nonzero Melnikov function we give a new proof of a result of Roussarie on the finite cyclicity of the homoclinic orbit of the integrable system when we perturb it inside the class of analytic differential systems.

One of the classical problems of the differential systems in the plane R 2 is to distinguish between a focus and a centre, and when we have a centre to distinguish if it is global or not. Let ( x 0 , y 0 ) be an equilibrium point of a differential system in R 2 . If there is a neighbourhood V of ( x 0 , y 0 ) such that V ∖ { ( x 0 , y 0 ) } is filled with periodic orbits, then ( x 0 , y 0 ) is a center. When V can be chosen as R 2 , the centre is a global center. We deal with a family of cubic polynomial differential systems in R 2 studied by Devlin, Lloyd and Pearson in J. Devlin, N.G. Lloyd and J.M. Pearson, [Cubic systems and Abel equations, J. Differ. Equ. 147 (1997), pp. 435–454.]. These authors analyse when the equilibrium point at the origin of coordinates can be a centre. We show two mistakes in the paper (J. Devlin, N.G. Lloyd and J.M. Pearson, Cubic systems and Abel equations, J. Differ. Equ. 147 (1997), pp. 435–454.), and additionally we prove that when the origin of coordinates is a centre, it can not be a global centre.

First, doing a combination of analytical and algebraic computations, we determine by first time an explicit normal form depending only on three parameters for all cubic homogeneous polynomial differential systems having a center. After using the averaging method of first order we show that we can obtain at most one limit cycle bifurcating from the periodic orbits of the mentioned centers when they are perturbed inside the class of all cubic polynomial differential systems. Moreover, there are examples with one limit cycles.

On the number of algebraically independent Poincaré–Liapunov constants

In this paper, we deduce the normal forms of polynomial differential systems in ℝ3 having, Fermat cubic surface, Clebsch surface, Goursat surface, Racor surface and 2-dimensional tours as invariant objects. We prove if we have an odd number of polynomial differential equations with the determined inverse Jacobian multipliers then we can make a new polynomial differential system of the first systems so that it’s inverse Jacobian multiplier can be determined by the inverse Jacobian multipliers of the first systems. If new system which is made by them has a limit cycle then we determine it’s position.

In this paper we investigate the isochronicity and linearizability problem for a cubic polynomial differential system which can be considered as a generalization of the Riccati system. Conditions for isochronicity and linearizability are found. The global structure of systems of the family with an isochronous center is determined. Furthermore, we find the order of weak center and study the problem of local bifurcation of critical periods in a neighborhood of the center.

By using techniques of differential geometry we answer the following open problem proposed by Chavarriga, Giacomini, Giné, and Llibre in 1999. For a given two dimensional flow, what is the maximal order of differentiability of a first integral on a canonical region in function of the order of differentiability of the flow? Moreover, we prove that for every planar polynomial differential system there exist finitely many invariant curves and singular points <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma Subscript i Baseline comma i equals 1 comma 2 comma midline-horizontal-ellipsis comma l"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> γ </mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo> ⋯ </mml:mo> <mml:mo>,</mml:mo> <mml:mi>l</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\gamma _i,\,i=1,2,\cdots ,l</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R squared minus left-parenthesis union Underscript i equals 1 Overscript l Endscripts gamma Subscript i Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mi class="MJX-variant" mathvariant="normal"> ∖ </mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:munderover> <mml:mo> ⋃ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>l</mml:mi> </mml:mrow> </mml:munderover> <mml:msub> <mml:mi> γ </mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb R^2\backslash \left (\bigcup ^{l}_{i=1}\gamma _i\right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has finitely many connected open components, and that on each of these connected sets the system has an analytic first integral. For a homogeneous polynomial differential system in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R cubed"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb R^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , there exist finitely many invariant straight lines and invariant conical surfaces such that their complement in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R cubed"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb R^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the union of finitely many open connected components, and that on each of these connected open components the system has an analytic first integral.

In this work we consider the polynomial differential system $\dot x = -y + x y^{n-1}$, $\dot y =x + a y x^{n-1}$, where $a \in \mathbb{R}$ and $n \ge 2$ with $n \in \mathbb{N}$. This system is a certain generalization of the classical Liénard system. We study the center problem and consequently the analytic integrability problem for such family around the origin for any value of $n$.

In this article we summarize the results on algebraic aspects of integrability for polynomial differential systems and its application, which include the Darboux, elementary and Liouvelle integrability. Darboux theory of integrability was found by Darboux in 1878, and it becomes extremely useful in study of the center focus problem, of bifurcation, of limit cycle problem and of global dynamics. The importance of Darboux theory of integrability is also presented by the Singer's theorem for planar polynomial differential system. That is, if a polynomial system is Liouville integrable, then it is Darboux integrable, i.e. the system has a Darboux first integral or a Darboux integrating factor.

We obtain an explicit polynomial whose simple positive real roots provide the limit cycles which bifurcate from the periodic orbits of a family of cubic polynomial differential centers when it is perturbed inside the class of all cubic polynomial differential systems. The family considered is the unique family of weight-homogeneous polynomial differential systems of weight-degree 2 with a center. The computations has been done with the help of the algebraic manipulator Mathematica.