Class Of Polynomial Differential Systems Research Articles

Solution of the Center Problem for a Class of Polynomial Differential Systems

Solution of the Center Problem for a Class of Polynomial Differential Systems

On the Local Stability of a Certain Class of Polynomial Differential System

We state and prove a condition for the local stability of a certain class of two dimensional system of polynomial differential equations. We give some examples of polynomial differential systems of equations to demonstrate that this local stability condition established for the trivial equilibrium point (0,0) is quite sharp and compare our result with the well known Lyapunov local stability criterion (Lyapunov's second method).

LIMIT CYCLES AND PHASE PORTRAIT FOR A CLASS OF DIFFERENTIAL SYSTEMS

We introduce an explicit expression of invariant algebraic curves for a class of polynomial differential systems and an explicit expression for its first integral. Moreover, we determine sufficient conditions for these systems to possess a limit cycle, which can be expressed by an explicit formula. Concrete examples exhibiting the applicability of our results are introduced.

Fifteen Limit Cycles Bifurcating from a Perturbed Cubic Center

In this work, we study the bifurcation of limit cycles from the period annulus surrounding the origin of a class of cubic polynomial differential systems; when they are perturbed inside the class of all polynomial differential systems of degree six, we obtain at most fifteenth limit cycles by using the averaging theory of first order.

Two Non Algebraic Limit Cycles of a Class of Polynomial Differential Systems with Non-Elementary Equilibrium Point

Abstract The problems of existence of limit cycles and their numbers are the most difficult problems in the dynamical planar systems. In this paper, we study the limit cycles for a family of polynomial differential systems of degree 6k + 1, k ∈ ℕ*, with the non-elementary singular point. Under some suitable conditions, we show our system exhibiting two non algebraic or two algebraic limit cycles explicitly given. To illustrate our results we present some examples.

The Center Problem and Time-Reversibility with Respect to a Quadratic Involution for a Class of Polynomial Differential Systems with Order 2 or 3

The Center Problem and Time-Reversibility with Respect to a Quadratic Involution for a Class of Polynomial Differential Systems with Order 2 or 3

On a class of polynomial differential systems of degree 4: phase portraits and limit cycles

In this paper we characterize the phase portraits in the Poincaré disc of the class of polynomial differential systems of the form \begin{equation*} \dot{x}=-y,\qquad \dot{y} =x+ax^{4}+bx^{2}y^{2}+cy^{4}, \end{equation*} with $a^2+b^2+c^2\neq0$, which are symmetric with respect to the $x$-axis. Such systems have a center at the origin of coordinates. Moreover, using the averaging theory of five order, we study the number of limit cycles which can bifurcate from the period annulus of this center when it is perturbed inside the class of all polynomial differential systems of degree $4$.

Generalized Analytic Integrability of a Class of Polynomial Differential Systems in $\mathbb{C}^{2}$

This paper study the type of integrability of differential systems with separable variables \(\dot{x}=h\left (x\right )f\left (y\right )\), \(\dot{y}= g\left (y\right )\), where \(h\), \(f\) and \(g\) are polynomials. We provide a criterion for the existence of generalized analytic first integrals of such differential systems. Moreover we characterize the polynomial integrability of all such systems.In the particular case \(h\left (x\right )=\left (ax+b\right )^{m}\) we provide necessary and sufficient conditions in order that this subclass of systems has a generalized analytic first integral. These results extend known results from Giné et al. (Discrete Contin. Dyn. Syst. 33:4531–4547, 2013) and Llibre and Valls (Discrete Contin. Dyn. Syst., Ser. B 20:2657–2661, 2015). Such differential systems of separable variables are important due to the fact that after a blow-up change of variables any planar quasi-homogeneous polynomial differential system can be transformed into a special differential system of separable variables \(\dot{x}=xf\left (y\right )\), \(\dot{y}=g\left (y\right )\), with \(f\) and \(g\) polynomials.

Limit Cycles of a Class of Polynomial Differential Systems Bifurcating from the Periodic Orbits of a Linear Center

In this paper, we study the number of limit cycles of a new class of polynomial differential systems, which is an extended work of two families of differential systems in systems considered earlier. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of a center using the averaging theory of first and second order.

The centers and their cyclicity for a class of polynomial differential systems of degree 7

We classify the global phase portraits in the Poincaré disc of the generalized Kukles systems ẋ=−y,ẏ=x+axy6+bx3y4+cx5y2+dx7,which are symmetric with respect to both axes of coordinates. Moreover using the averaging theory up to sixth order, we study the cyclicity of the center located at the origin of coordinates, i.e. how many limit cycles can bifurcate from the origin of coordinates of the previous differential system when we perturb it inside the class of all polynomial differential systems of degree 7.

On the limit cycles surrounding a diagonalizable linear node with homogeneous nonlinearities

In this paper we study the existence and non-existence of limit cycles for the class of polynomial differential systems of the form ẋ=λx+Pn(x,y),ẏ=μy+Qn(x,y),where Pn and Qn are homogeneous polynomials of degree n.

The Poincaré center‐focus problem for a class of higher order polynomial differential systems

In this paper, I have proved that for a class of polynomial differential systems of degree n + 1 (where n is an arbitrary positive integer), the composition conjecture is true. I give the sufficient and necessary conditions for these differential systems to have a center at origin point by using a different method from the previous references. By this, I can obtain all the focal values of these systems for an arbitrary n, and their expressions are succinct and beautiful. I believe that the idea and method of this article can be used to solve the center‐focus problem of more high‐order polynomial differential systems.

Bounding the number of limit cycles of discontinuous differential systems by using Picard–Fuchs equations

In this paper, by using Picard–Fuchs equations and Chebyshev criterion, we study the upper bounds of the number of limit cycles given by the first order Melnikov function for discontinuous differential systems, which can bifurcate from the periodic orbits of quadratic reversible centers of genus one (r19): x˙=y−12x2+16y2, y˙=−x−16xy, and (r20): x˙=y+4x2, y˙=−x+16xy, and the periodic orbits of the quadratic isochronous centers (S1):x˙=−y+x2−y2, y˙=x+2xy, and (S2):x˙=−y+x2, y˙=x+xy. The systems (r19) and (r20) are perturbed inside the class of polynomial differential systems of degree n and the system (S1) and (S2) are perturbed inside the class of quadratic polynomial differential systems. The discontinuity is the line y=0. It is proved that the upper bounds of the number of limit cycles for systems (r19) and (r20) are respectively 4n−3(n≥4) and 4n+3(n≥3) counting the multiplicity, and the maximum numbers of limit cycles bifurcating from the period annuluses of the isochronous centers (S1) and (S2) are exactly 5 and 6 (counting the multiplicity) on each period annulus respectively.

Explicit expression for a hyperbolic limit cycles of a class of polynomial differential systems

Explicit expression for a hyperbolic limit cycles of a class of polynomial differential systems

Limit Cycles Coming from Some Uniform Isochronous Centers

Abstract This article is about the weak 16th Hilbert problem, i.e. we analyze how many limit cycles can bifurcate from the periodic orbits of a given polynomial differential center when it is perturbed inside a class of polynomial differential systems. More precisely, we consider the uniform isochronous centers x ˙ = - y + x 2 ⁢ y ⁢ ( x 2 + y 2 ) n , y ˙ = x + x ⁢ y 2 ⁢ ( x 2 + y 2 ) n , $\dot{x}=-y+x^{2}y(x^{2}+y^{2})^{n},\quad\dot{y}=x+xy^{2}(x^{2}+y^{2})^{n},$ of degree 2 ⁢ n + 3 ${2n+3}$ and we perturb them inside the class of all polynomial differential systems of degree 2 ⁢ n + 3 ${2n+3}$ . For n = 0 , 1 ${n=0,1}$ we provide the maximum number of limit cycles, 3 and 8 respectively, that can bifurcate from the periodic orbits of these centers using averaging theory of first order, or equivalently Abelian integrals. For n = 2 ${n=2}$ we show that at least 12 limit cycles can bifurcate from the periodic orbits of the center.

Global Phase Portraits of Kukles Differential Systems with Homogeneous Polynomial Nonlinearities of Degree 6 Having a Center and Their Small Limit Cycles

We provide the nine topological global phase portraits in the Poincaré disk of the family of the centers of Kukles polynomial differential systems of the form [Formula: see text] [Formula: see text] where [Formula: see text] and [Formula: see text] are real parameters satisfying [Formula: see text] Using averaging theory up to sixth order we determine the number of limit cycles which bifurcate from the origin when we perturb this system first inside the class of all homogeneous polynomial differential systems of degree [Formula: see text] and second inside the class of all polynomial differential systems of degree [Formula: see text]

Limit cycles for a class of polynomial differential systems

Limit cycles for a class of polynomial differential systems

On the Limit Cycles of Linear Differential Systems with Homogeneous Nonlinearities

AbstractWe consider the class of polynomial differential systems of the form , where Pn and Qn are homogeneous polynomials of degree n. For this class of differential systems we summarize the known results for the existence of limit cycles, and we provide new results for their nonexistence and existence

Birth of limit cycles for a class of continuous and discontinuous differential systems in (d + 2)–dimension

ABSTRACTThe orbits of the reversible differential system , , , with and , are periodic with the exception of the equilibrium points (0, 0, z1,… , zd). We compute the maximum number of limit cycles which bifurcate from the periodic orbits of the system , , , using the averaging theory of first order, when this system is perturbed, first inside the class of all polynomial differential systems of degree n, and second inside the class of all discontinuous piecewise polynomial differential systems of degree n with two pieces, one in y > 0 and the other in y < 0. In the first case, this maximum number is nd(n − 1)/2, and in the second, it is nd + 1.

Centers and isochronous centers for generalized quintic systems

In this paper we classify the centers and the isochronous centers of certain polynomial differential systems in R2 of degree d≥5 odd that in complex notation are ż=(λ+i)z+(zz̄)d−52(Az5+Bz4z̄+Cz3z̄2+Dz2z̄3+Ezz̄4+Fz̄5), where z=x+iy, λ∈R and A,B,C,D,E,F∈C. Note that if d=5 we obtain the class of polynomial differential systems in the form of a linear system with homogeneous polynomial nonlinearities of degree 5.Due to the huge computations required for computing the necessary and sufficient conditions for the characterization of the centers and isochronous centers, our study uses algorithms of computational algebra based on the Gröbner basis theory and on modular arithmetics.