Quadratic Integrable System Research Articles

Bifurcation of limit cycles in a piecewise smooth near-integrable system

In this paper, we study the bifurcation of limit cycles in a class of piecewise smooth quadratic integrable systems under small polynomial perturbations of degree n. By using the first order Melnikov function, we derive a lower bound for the number of limit cycles which bifurcate from the period annulus.

Limit cycle bifurcations by perturbing a quadratic integrable system with a triangle

Finding cyclicity of homoclinic or heteroclinic loops in quadratic systems is an open problem, which is related to the second part of Hilbert's 16th problem. This paper mainly considers the problem of limit cycle bifurcation for quadratic polynomial systems near a triangle. It is proved that the cyclicity of the triangle in quadratic systems is at least three.

The Poincar ́e Bifurcation of the Non-Hamiltonian Quadratic Integrable System with One Center and One Unbounded Homoclinic Loop Formed by Cubics

In this paper, we investigate the Poincar ́e bifurcation of the non-Hamiltonian quadratic integrable system with one center and one unbounded homoclinic loop formed by cubics. It is proved that the cyclicity of the period annulus under quadratic perturbations is equal to two.

FOUR LIMIT CYCLES FROM PERTURBING QUADRATIC INTEGRABLE SYSTEMS BY QUADRATIC POLYNOMIALS

In this paper, we present four limit cycles in quadratic near-integrable polynomial systems. It is shown that when a quadratic integrable system has two centers and is perturbed by quadratic polynomials, it can generate at least four limit cycles with (3,1)-distribution. This result provides a positive answer to an open question in this research area.

FOUR LIMIT CYCLES IN QUADRATIC NEAR-INTEGRABLE SYSTEMS

In this note, we report of obtaining 4 limit cycles in quadratic near-integrable polynomial systems. It is shown that when a quadratic integrable system has two centers and is perturbed by quadratic polynomials, it can generate at least 4 limit cycles with (3.1) distribution. This result provides a positive answer to an open problem in this area.

Quantum superintegrable systems with quadratic integrals on a two dimensional manifold

There are two classes of quantum integrable systems on a manifold with quadratic integrals, the Liouville and the Lie integrable systems, as it happens in the classical case. The quantum Liouville quadratic integrable systems are defined on a Liouville manifold and the Schrödinger equation can be solved by separation of variables in one coordinate system. The Lie integrable systems are defined on a Lie manifold and are not generally separable ones but can be solved. Therefore, there are superintegrable systems with two quadratic integrals of motion not necessarily separable in two coordinate systems. The quantum analogs of the two dimensional superintegrable systems with quadratic integrals of motion on a manifold are classified by using the quadratic associative algebra of the integrals of motion. There are six general fundamental classes of quantum superintegrable systems corresponding to the classical ones. Analytic formulas for the involved integrals are calculated in all the cases. All the known quantum superintegrable systems are classified as special cases of these six general classes. The coefficients of the associative algebra of the general cases are calculated. These coefficients are the same as the coefficients of the classical case multiplied by −ℏ2 plus quantum corrections of orders ℏ4 and ℏ6.

Perturbations of non-Hamiltonian reversible quadratic systems with cubic orbits

This paper is concerned with degree n polynomial perturbations of a class of planar non-Hamiltonian reversible quadratic integrable system whose almost all orbits are cubics. We give an estimate of the number of limit cycles for such a system. If the first-order Melnikov function (Abelian integral) M 1 ( h ) is not identically zero, then the perturbed system has at most 5 for n = 3 and 3 n - 7 for n ⩾ 4 limit cycles on the finite plane. If M 1 ( h ) is identically zero but the second Melnikov function is not, then an upper bound for the number of limit cycles on the finite plane is 11 for n = 3 and 6 n - 13 for n ⩾ 4 , respectively. In the case when the perturbation is quadratic ( n = 2 ), there exists a neighborhood U of the initial non-Hamiltonian polynomial system in the space of all quadratic vector fields such that any system in U has at most two limit cycles on the finite plane. The bound for n = 2 is exact.

The period function for quadratic integrable systems with cubic orbits

This paper is concerned with the monotonicity of the period function for families of quadratic systems with centers whose orbits lie on cubic planar curves. It is proved that each such system has a period function with at most one critical point.

The cyclicity of the elliptic segment loops of the reversible quadratic Hamiltonian systems under quadratic perturbations

Denote by Q H and Q R the Hamiltonian class and reversible class of quadratic integrable systems. There are several topological types for systems belong to Q H ∩ Q R . One of them is the case that the corresponding system has two heteroclinic loops, sharing one saddle-connection, which is a line segment, and the other part of the loops is an ellipse. In this paper we prove that the maximal number of limit cycles, which bifurcate from the loops with respect to quadratic perturbations in a conic neighborhood of the direction transversal to reversible systems (called in reversible direction), is two. We also give the corresponding bifurcation diagram.

The cyclicity of the elliptic segment loops of the reversible quadratic Hamiltonian systems under quadratic perturbations

Denote by Q H and Q R the Hamiltonian class and reversible class of quadratic integrable systems. There are several topological types for systems belong to Q H ∩ Q R . One of them is the case that the corresponding system has two heteroclinic loops, sharing one saddle-connection, which is a line segment, and the other part of the loops is an ellipse. In this paper we prove that the maximal number of limit cycles, which bifurcate from the loops with respect to quadratic perturbations in a conic neighborhood of the direction transversal to reversible systems (called in reversible direction), is two. We also give the corresponding bifurcation diagram.

Unfolding of a Quadratic Integrable System with a Homoclinic Loop

In this paper, we make a complete study of the unfolding of a quadratic integrable system with a homoclinic loop. Making a Poincare transformation and using some new techniques to estimate the number of zeros of Abelian integrals, we obtain the complete bifurcation diagram and all phase portraits of systems corresponding to different regions in the parameter space. In particular, we prove that two is the maximal number of limit cycles bifurcation from the system under quadratic non-conservative perturbations.

Cyclicity of planar homoclinic loops and quadratic integrable systems

A general method for a homoclinic loop of planar Hamiltonian systems to bifurcate two or three limit cycles under perturbations is established.Certain conditions are given under which the cyclicity of a homoclinic loop equals 1 or 2.As an application to quadratic systems,it is proved that the cyclicity of homoclinic loops of quadratic in-tegrable and non-Hamiltonian systems equals 2 except for one case.

Unfolding of a Quadratic Integrable System with Two Centers and Two Unbounded Heteroclinic Loops

In this paper we present a complete study of quadratic 3-parameter unfoldings of some integrable system belonging to the classQR3, and having two centers and two unbounded heteroclinic loops. We restrict to unfoldings that are transverse toQR3, obtain a versal bifurcation diagram and all global phase portraits, including the precise number and configuration of the limit cycles. It is proved that 3 is the maximal number of limit cycles surrounding a single focus, and only the (1,1)-configuration can occur in case of simultaneous nests of limit cycles. Essentially the proof relies on a careful analysis of a related non-conservative Abelian integral.