First Integral Of Degree Research Articles

On the Qualitative Analysis of a Family of Differential Equations with First Integrals of Degree More Then 2

On the Qualitative Analysis of a Family of Differential Equations with First Integrals of Degree More Then 2

Planar Kolmogorov Systems Coming from Spatial Lotka–Volterra Systems

In this paper, we classify the phase portraits in the Poincaré disc of all the Kolmogorov systems [Formula: see text] which depend on six parameters. We prove that these systems have [Formula: see text] topologically distinct phase portraits in the Poincaré disc. These systems are provided by a general three-dimensional Lotka–Volterra system with a rational first integral of degree two of the form [Formula: see text], restricted to each surface [Formula: see text] varying [Formula: see text], with the additional assumption that they have a Darboux invariant of the form [Formula: see text].

Phase portraits of a family of Kolmogorov systems depending on six parameters

We consider a general 3-dimensional Lotka-Volterra system with a rational first integral of degree two of the form \(H=x^i y^j z^k\). The restriction of this Lotka-Volterra system to each surface \(H(x,y,z)=h\) varying \(h\in \mathbb{R}\) provide Kolmogorov systems. With the additional assumption that they have a Darboux invariant of the form \(x^\ell y^m e^{st}\) they reduce to the Kolmogorov systems $$\displaylines{ \dot{x}=x\big(a_0- \mu (c_1 x + c_2 z^2 + c_3 z)\big),\cr \dot{z}=z\big(c_0+ c_1 x + c_2 z^2 + c_3 z\big). }$$ We classify the phase portraits in the Poincare disc of all these Kolmogorov systems which depend on six parameters. For more information see https://ejde.math.txstate.edu/Volumes/2021/35/abstr.html

Bi-homogeneity and integrability of rational potentials

In this paper we consider natural Hamiltonian systems with two degrees of freedom for which Hamiltonian function has the form H=12(p12+p22)+V(q1,q2) and potential V(q1,q2) is a rational function. Necessary conditions for the integrability of such systems are deduced from integrability of dominate term of the potential which usually is appropriately chosen homogeneous term of V. We show that introducing weights compatible with the canonical structure one can find new dominant terms which can give new necessary conditions for integrability. To deduce them we investigate integrability of a family of bi-homogeneous potentials which depend on two integer parameters. Unexpectedly systems with these potentials can be reduced to the Lotka–Volterra quadratic planar vector field. Then theorem of Jean Moulin Ollagnier, allows us to make complete classification of integrable cases. Moreover, the reduction is used for explicit integration of an exceptional case with a polynomial first integral of degree 4 in momenta.

Abelian integrals and limit cycles for a class of cubic polynomial vector fields of Lotka–Volterra type with a rational first integral of degree two

In this paper, we study the number of limit cycles which bifurcate from the periodic orbits of cubic polynomial vector fields of Lotka–Volterra type having a rational first integral of degree 2, under polynomial perturbations of degree n. The analysis is carried out by estimating the number of zeros of the corresponding Abelian integrals. Moreover, using Chebyshev criterion, we show that the sharp upper bound for the number of zeros of the Abelian integrals defined on each period annulus is 3 for n=3. The simultaneous bifurcation and distribution of limit cycles for the system with two period annuli under cubic polynomial perturbations are considered. All configurations (u,v) with 0≤u,v≤3, u+v≤5 are realizable.

Limit cycles of cubic polynomial differential systems with rational first integrals of degree 2

The main goal of this paper is to study the maximum number of limit cycles that bifurcate from the period annulus of the cubic centers that have a rational first integral of degree 2 when they are perturbed inside the class of all cubic polynomial differential systems using the averaging theory. The computations of this work have been made with Mathematica and Maple.

Remarks on polynomial integrals of higher degrees for reversible systems with toral configuration space

We consider problems related to the well-known conjecture on the degrees of irreducible polynomial integrals of a reversible Hamiltonian system with two degrees of freedom and toral position space. The main object of study is a special system arising in the analysis of irreducible polynomial integrals of degree 4. In a particular case we have the problem of the motion of two interacting particles on a circle in given potential fields. We prove that if the three potentials are smooth non-constant functions, then this problem has no non-trivial polynomial integrals of arbitrarily high degree. We prove the conjecture completely for systems with a polynomial first integral of degree 4 in the momenta.

First integrals for a charged particle moving on a plane under the action of a magnetic field orthogonal to this plane

We characterize the polynomial (respectively analytical) first integrals of degree one in the variables u and v of the differential systems of the form x′=u, y′=v, u′=B(x,y)v and v′=−B(x,y)u where B=B(x,y) is a polynomial (respectively analytic function) in the variables x and y. This differential system models a non-relativistic charge moving on a plane under the action of a magnetic field orthogonal to this plane.

Solving large linear algebraic systems in the context of integrable non-abelian Laurent ODEs

The paper reports on a computer algebra program LSSS (Linear Selective Systems Solver) for solving linear algebraic systems with rational coefficients. The program is especially efficient for very large sparse systems that have a solution in which many variables take the value zero. The program is applied to the symmetry investigation of a non-abelian Laurent ODE introduced recently by M. Kontsevich. The computed symmetries confirmed that a Lax pair found for this system earlier generates all first integrals of degree at least up to 14.

Phase portraits and invariant straight lines of cubic polynomial vector fields having a quadratic rational first integral

In this paper we classify all cubic polynomial dierential systems having a first integral of degree two. In other words we characterize all the global phase portraits of the cubic polynomial dierential systems having all their orbits contained in conics. We also determine their configurations of invariant straight lines. We show that there are exactly 36 topologically dierent phase

On algebraic construction of certain integrable and super-integrable systems

We propose a new construction of two-dimensional natural bi-Hamiltonian systems associated with a very simple Lie algebra. The presented construction allows us to distinguish three families of super-integrable monomial potentials for which one additional first integral is quadratic, and the second one can be of arbitrarily high degree with respect to the momenta. Many integrable systems with additional integrals of degree greater than two in momenta are given. Moreover, an example of a super-integrable system with first integrals of degree two, four and six in the momenta is found.

Quadratic systems with a rational first integral of degree three: a complete classification in the coefficient space ℝ12

A quadratic polynomial differential systemcan be identified with a single point of ℝ12 through its coefficients. The phase portrait of the quadratic systems having a rational first integral of degree 3 have been studied using normal forms. Here using the algebraic invariant theory, we characterize all the non-degenerate quadratic polynomial differential systems in ℝ12 having a rational first integral of degree 3. We show that there are only 31 different topological phase portraits in the Poincare disc associated to this family of quadratic systems up to a reversal of the sense of their orbits, and we provide representatives of every class modulo an affine change of variables and a rescaling of the time variable. Moreover, each one of these 31 representatives is determined by a set of algebraic invariant conditions and we provide for it a first integral.

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

In this paper, we classify all the global phase portraits of the quadratic polynomial vector fields having a rational first integral of degree 3.

Quadratic systems with a rational first integral of degree 2: A complete classification in the coefficient space $$\mathbb{R}^{12} $$

Quadratic systems with a rational first integral of degree 2: A complete classification in the coefficient space $$\mathbb{R}^{12} $$

Phase portraits of cubic polynomial vector fields of Lotka–Volterra type having a rational first integral of degree 2

We classify all the global phase portraits of the cubic polynomial vector fields of Lotka–Volterra type having a rational first integral of degree 2. For such vector fields there are exactly 28 different global phase portraits in the Poincaré disc up to a reversal of sense of all orbits.

Phase portraits of quadratic polynomial vector fields having a rational first integral of degree 2

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.

About a conjecture on quadratic vector fields

The Lotka–Volterra system of autonomous differential equations consists in three homogeneous polynomial equations of degree 2 in three variables. This system, or the corresponding vector field LV( A, B, C), depends on three non-zero (complex) parameters and may be written as LV( A, B, C)= V x ∂ x + V y ∂ y + V z ∂ z where V x=x(Cy+z), V y=y(Az+x), V z=z(Bx+y). As LV( A, B, C) is homogeneous, there is a foliation whose leaves are homogeneous surfaces in the three-dimensional space C 3 , or curves in the corresponding projective plane CP(2) , such that the trajectories of the vector field are completely contained in a leaf. An homogeneous first integral of degree 0 is then a non-constant function on the set of all leaves of the previous foliation. Trying to classify all values of the triple ( A, B, C) for which LV( A, B, C) has an homogeneous Liouvillian first integral of degree 0, we discovered the interesting family of Lotka–Volterra systems: SLV l=LV 2,B l=− 2l+1 2l−1 ,1/2 , l∈ N ★. This family provides a negative answer to the conjecture that there exists a uniform bound M 2 such that, if some homogeneous three-variable vector field of degree 2 has a particular solution (an irreducible Darboux polynomial in our words) of degree at least M 2, then the vector field is rationally integrable (has an homogeneous rational first integral of degree 0). Indeed: • SLV l has no homogeneous rational first integral of degree 0. • SLV l has an irreducible Darboux polynomial f l of degree m=2 l: (∗) x(y/2+z) ∂f l ∂ x +y(2z+x) ∂f l ∂ y +z(B lx+y) ∂f l ∂ z =((l−1)y+2lz)f l.

Liouvillian integration of the Lotka-Volterra system

The Lotka-Volterra system of autonomous differential equations consists in three homogeneous polynomial equations of degree 2 in three variables. This system, or the corresponding vector fieldLV(A, B, C), depends on three nonzero (complex) parameters and may be written asLV(A, B, C)=V x ∂ x +V y ∂ y +V z ∂ z whereV x=x(Cy+z), Vy=y(Az+x), Vz=z(Bx+y). Similar systems of equations have been studied by Volterra in his mathematical approach of the competition of species and this is the reason why this name has been given to such systems. In fact,LV(A, B, C) can be chosen as a normal form for most of factored quadratic systems; the study of its first integrals of degree 0 is thus of great mathematical interest. Given a homogeneous vector field, there is a foliation whose leaves are homogeneous surfaces in the three-dimensional space (or curves in the corresponding projective plane), such that the trajectories of the vector field are completely contained in a leaf. A first integral of degree 0 is then a function on the set of all leaves of the previous foliation. In the present paper, we give all values of the triple (A, B, C) of non-zero parameters for whichLV(A, B, C) has a homogeneous liouvillian first integral of degree 0. We also discuss the corresponding problem of liouvillian integration for quadratic factored vector fields that cannot be put in Lotka-Volterra normal form. Our proof essentially relies on combinatorics and elementary algebraic geometry, especially in proving that some conditions are necessary.

Rational integration of the Lotka–Volterra system

The Lotka–Volterra system of autonomous differential equations consists in three homogeneous polynomial equations of degree 2 in three variables. This system, or the corresponding vector field V(A,B,C), depends on three non-zero parameters and writes V(A,B,C)=V x∂ x+V y∂ y+V z∂ z where V x=x(Cy+z), V y=y(Az+x), V z=z(Bx+y). Similar systems of equations have been studied by Volterra in his mathematical approach of the competition of species. For us, V(A,B,C) is a normal form of a factorisable quadratic system and the study of its first integrals of degree 0 is of great mathematical interest. A first integral is a non-constant function f which satisfies the identity V x ∂f ∂x +V y ∂f ∂y +V z ∂f ∂z =0. As V(A,B,C) is homogeneous, there is a foliation whose leaves are homogeneous surfaces in the three-dimensional space (or curves in the corresponding projective plane), such that the trajectories of the vector field are completely contained in a leaf. A first integral of degree 0 is then a function on the set of all leaves of the previous foliation. In the present paper, we give all values of the triple (A,B,C) of parameters for which V(A,B,C) has an homogeneous rational first integral of degree 0. Our proof essentially relies on ideas of algebra and combinatorics, especially in proving that some conditions are necessary.