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In this paper, we study a class of perturbed nonlinear fractional differential systems with two control parameters. Under some appropriate conditions on the primitive function of nonlinear terms $F_{u}$ and $F_{v}$ , we prove the multiplicity of weak solutions by using the variational method and Ricceri’s critical points theorems.

This paper deals with the qualitative behavior of orbits at degenerate singular point with the method of quasi normal sector, which is a generalization of Frommer’s normal sectors. Several examples show that this method is more effective than the well-known methods of Z-sectors, normal sectors and generalized normal sector.

In this paper we study the existence of nontrivial solutions for an impulsive fractional differential equation with Dirichlet boundary conditions. By using Morse theory coupled with local linking arguments, we obtain some new criteria to guarantee that the impulsive fractional differential equations have at least one nontrivial solution.

This paper deals with bifurcation of limit cycles for piecewise smooth integrable non-Hamiltonian systems. We derive the first order Melnikov function, which plays an important role in the study of the number of limit cycles bifurcated from the periodic annulus of a center. As an application, we consider a class of cubic isochronous centers, which has a non-rational first integral. Using the first order Melnikov function, we obtain the sharp upper bound of the number of limit cycles which bifurcate from the periodic annulus of the center under piecewise smooth polynomial perturbations.

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.

In this paper, a coupled system of nonlinear fractional differential equations is considered. The existence of at least three distinct weak solutions is obtained by means of the variational methods and the critical point theory due to Bonanno and Marano. In addition, an example is presented to illustrate the feasibility and effectiveness of the main results.

In this paper we study a two-dimensional toxin-determined functional response model (TDFRM). The toxin-determined functional response explicitly takes into consideration the reduction in the consumption of plants by herbivore due to chemical defense, which generates more complex dynamics of the plant–herbivore interactions. The purpose of the present paper is to analyze the existence of limit cycles and bifurcations of the model. By applying the theories of rotated vector fields and the extended planar termination principle, we establish the conditions for the existence of limit cycles and homoclinic loop. It is shown that a limit cycle is generated in a supercritical Hopf bifurcation and terminated in a homoclinic bifurcation, as the parameters vary. Analytic proofs are provided for all results, which generalize the results presented in [11] .

In this paper, we consider the number of limit cycles for a class of discontinuous planar quadratic integrable non-Hamiltonian system under quadratic perturbation. Using the rst order averaging method, we obtain that there are at least 5 limit cycles which can bifurcate from the period annulus of the center for this system. Our result also shows that the discontinuous quadratic system can have at least 2 more limit cycles than the smooth one.

Based upon the hermite transformation and the extended sub-equation expansion method, soliton-like solutions of the stochastic Schrodinger equation in the white noise environment are obtained via Maple.

In this paper we study the quadratic homogeneous perturbations of the 3-dimensional May–Leonard system with α + β = 2 . It is shown that there are perturbed systems having exactly one or two limit cycles bifurcated from the periodic orbits of May–Leonard system. This is proved by estimating the number of zeros of the first and the second order Melnikov functions.

In this paper, we apply critical point theory and variational methods to study the multiple solutions of boundary value problems for an impulsive fractional differential equation with p-Laplacian. Some new criteria guaranteeing the existence of multiple solutions are established for the considered problem.

Let $$H_{pqm}$$ be the space of all planar $$(p,q)$$ -quasihomogeneous vector fields of weight degree $$m$$ endowed with the coefficient topology. In this paper we characterize the set $$\Omega _{pqm}$$ of all vector fields in $$H_{pqm}$$ which are structurally stable with respect to perturbations in $$H_{pqm}$$ in the Poincare disc, and determine the exact number of the topological equivalence classes in $$\Omega _{pqm}$$ in terms of $$p,q$$ and $$m$$ . This characterization is applied to give an extension of the Hartman–Grobmann Theorem at the origin for $$(p,q)$$ quasihomogeneous vector fields of weight degree greater than $$m$$ starting with a term $$X_m \in \Omega _{pqm}$$ . This work is an extension of the Llibre et al.’s paper (J Differ Equ 125:490–520, 1996), where the homogeneous case was considered.

During transients, the instability of the power system can attribute to the angle instability as well as the voltage instability. The angle and voltage stability are entangled with each other during transients. This paper proposed a direct approach to determine the types of instability of power systems during transients. Based on differential algebraic equations of the systems, the equilibriums and singular points of the system are defined. The singular surface consisting of singular points then divides the manifolds of the system trajectory into multiple algebraic manifolds, with each algebraic manifold containing one or more equilibriums. Three types of instability of the system are identified according to the distribution of equilibriums, i.e., the angle instability, voltage instability and angle-induced-voltage instability. The proposed method can easily determine the dominant instability mode of the system during transients and the case studies also demonstrate its effectiveness.

The present paper is devoted to the study of the maximum number of limit cycles bifurcated from the periodic orbits of the quadratic isochronous center x ˙ = − y + 16 3 x 2 − 4 3 y 2 , y ˙ = x + 8 3 x y by the averaging method of first order, when it is perturbed inside a class of discontinuous quadratic polynomial differential systems. The Chebyshev criterion is used to show that this maximum number is 5 and can be realizable. In some sense, the result and that in paper [6] also answer the questions left in the paper [9] .

In this paper, the existence of infinitely many solutions for perturbed systems of impulsive non-linear fractional differential equations including Lipschitz continuous non-linear terms is discussed. The approach is based on variational methods. In addition, examples are presented to illustrate the feasibility and effectiveness of the main results.

This paper is concerned with the quadratic perturbations of a one-parameter family of quadratic reversible system, having a center of genus one. The exact upper bound of the number of limit cycles emerging from the period annulus surrounding the center of the unperturbed system is given.

By means of the fixed point index theory of strict-set contraction operator, we study the existence of positive solutions for the multipoint singular boundary value problem(-1)n-kun(t)=f(t,ut),0

This paper is concerned with the bifurcation of limit cycles in general quadratic perturbations of quadratic codimension-four centres Q4. Gavrilov and Iliev set an upper bound of eight for the number of limit cycles produced from the period annuli around the centre. Based on Gavrilov?Iliev's proof, we prove in this paper that the perturbed system has at most five limit cycles which emerge from the period annuli around the centre. We also show that there exists a perturbed system with three limit cycles produced by the period annuli of Q4.

In this paper, we investigate a class of quadratic reversible Lotka–Volterra system of genus one z ˙ = i z + z 2 + b z ¯ 2 with b = 3 / 5 . It is proved that the cyclicity of the period annulus under quadratic perturbations is equal to two. Moreover, we prove that the period function of its period trajectories is monotone increasing.

This paper is concerned with limit cycles which bifurcate from periodic orbits of the cubic isochronous center. It is proved that in this situation, the cyclicity of the period annulus under cubic perturbations is equal to four. Moreover, for each k = 0,1, . . .,4, there are perturbations that give rise to exactly k limit cycles bifurcating from the period annulus.

In this paper, we study the existence of multiple positive solutions for boundary value problems based on second-order functional differential equations with the form {y″(t)+f(t,y(t−τ))=0,∀t∈(0,1)∖{τ},y(t)=η(t),∀t∈[−τ,0],y(1)=0 where 0

This paper deals with the relation between isochronicity and first integral for a class of reversible systems: x ˙ = − U ( x ) y , y ˙ = f ( x , y ) , which associates to the first integral of the form H ( x , y ) = F ( x ) y 2 + G ( x ) . Two necessary and sufficient conditions are given to characterize isochronicity for these systems. Moreover, we apply these results to show that there exists a class of polynomial reversible systems of degree n with isochronous center for any n.

This paper is concerned with the monotonicity of the period function for a class of reversible quadratic centers with their orbits inside quartics. It is proved that such a system has a period function with at most one critical point.

In this paper, we obtain some sufficient condition for the existence of at least three solutions to two-point boundary value problems for second-order impulsive differential equations by using Leray–Schauder degree theory. We assume the existence of two pairs of strict lower and upper solutions and the nonlinear term f satisfies a Nagumo’s growth condition with respect to the first order derivative.

Using elementary tools we construct cubic polynomial systems of differential equations with algebraic limit cycles of degrees 4, 5 and 6. We also construct a cubic polynomial system of differential equations having an algebraic homoclinic loop of degree 3. Moreover, we show that there are polynomial systems of differential equations of arbitrary degree that have algebraic limit cycles of degree 3, as well as give an example of a cubic polynomial system of differential equations with two algebraic limit cycles of degree 4.

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.

Up to now, most of the results on the tangential Hilbert 16th problem have been concerned with the Hamiltonian regular at infinity, i.e., its principal homogeneous part is a product of the pairwise different linear forms. In this paper, we study a polynomial Hamiltonian which is not regular at infinity. It is shown that the space of Abelian integral for this Hamiltonian is finitely generated as a R [ h ] module by several basic integrals which satisfy the Picard–Fuchs system of linear differential equations. Applying the bound meandering principle, an upper bound for the number of complex isolated zeros of Abelian integrals is obtained on a positive distance from critical locus. This result is a partial solution of tangential Hilbert 16th problem for this Hamiltonian. As a consequence, we get an upper bound of the number of limit cycles produced by the period annulus of the non-Hamiltonian integrable quadratic systems whose almost all orbits are algebraic curves of degree k + n , under polynomial perturbation of arbitrary degree.

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.

In this paper, we give an upper bound on the number of zeros of Abelian integrals for the quadratic centres having almost all their orbits formed by quartics, under polynomial perturbations of arbitrary degree n. The bound is linearly dependent on n.

The authors investigate a kind of degenerate quadratic Hamiltonian systems with saddle-loop. Under quadratic perturbations, it is proved that the perturbed system has at most two limit cycles in the finite plane. The proof relies on a careful analysis of a related Abelian integral.

By using a variational method and some critical points theorems, we establish some results on the multiplicity of solutions for second-order impulsive differential equation depending on two real parameters on the half-line. In addition, two examples to illustrate our results are given.

In this paper, we study the existence of multiple positive solutions of the second-order periodic boundary value problems for functional differential equations with impulse. The proof of our main results is based upon the fixed point index theorem in cones.

By using the fixed-point theorem of Leray-Schauder or Banach, we discuss the existence of solutions for third-order impulsive neutral differential equations with deviating arguments. Two examples are given to demonstrate our main results.

A class of planar cubic Kolmogorov systems with harvest and two positive equilibrium points is investigated. With the help of computer algebra system MATHEMATICA, we prove that five limit cycles can be bifurcated simultaneously from the two critical points (1, 1) and (2, 2), respectively, in the first quadrant. Moreover, the necessary conditions of centers are obtained.

The paper deals with the non-generic quadratic Hamiltonian vector fields with hyperbolic segment. It is proved that in this situation the cyclicity of period annulus under quadratic perturbation is equal to two.

In this paper, we investigate the quadratic Hamiltonian systems with non- Morsean point. It is proved that in this situation the cyclicity of the period annulus under quadratic perturbations is equal to two.

It is proved in this paper that the lowest upper bound of the number of the isolated zeros of the Abelian integral $$I\left( h \right) = \oint\limits_{\Gamma _h } {\left( {a + \beta x + \gamma x^2 } \right)ydx} $$ is two for h∈(−1/12, 0), where Γh is the compact component of H(x, y)=(1/2) y2+(1/3) x3+(1/4) x4=h, and α, β, γ are arbitrary constants.

In this paper, by using the fixed-point theorem in the cone of strict-set-contraction operators, we study a class of higher-order boundary value problems of nonlinear fractional differential equation in a Banach space. The sufficient conditions for the existence of at least two positive solutions is obtained. In addition, an example to illustrate the main results is given.

In this paper, we study a class of fractional q-difference equations with nonhomogeneous boundary conditions. By applying the classical tools from functional analysis, sufficient conditions for the existence of single and multiple positive solutions to the boundary value problem are obtained in term of the explicit intervals for the nonhomogeneous term. In addition, some examples to illustrate our results are given. MSC: 34A08; 34B18; 39A13

In this paper, we discuss the existence of positive solutions for nonlocal q-integral boundary value problems of fractional q-difference equations. By applying the generalized Banach contraction principle, the monotone iterative method, and Krasnoselskii’s fixed point theorem, some existence results of positive solutions are obtained. In addition, some examples to illustrate our results are given.

By using the fixed point index theorem, this paper investigates a class of singular semipositone integral boundary value problem for fractionalq-derivatives equations and obtains sufficient conditions for the existence of at least two and at least three positive solutions. Further, an example is given to illustrate the applications of our main results.

In this chapter, we study quadratic perturbations of a non-Hamiltonian quadratic integrable system with a homoclinic loop. We prove that the perturbed system has at most two limit cycles in the finite phase plane, and the bound is exact. The proof relies on an estimation of the number of zeros of related Abelian integrals.

By means of the fixed-point theorem in the cone of strict-set-contraction operators, we consider the existence of a nonlinear multi-point boundary value problem of fractional integro-differential equation in a Banach space. In addition, an example to illustrate the main results is given.

This paper is concerned with the limit cycles for planar semi-quasi-homogeneous polynomial systems. We give some explicit criteria for the nonexistence and existence of periodic orbits. Let $N=N(p,q,m,n)$ be the maximum number of limit cycles of such system. A lower bound is given for $N=N(p,q,m,n)$. The cyclicity and center problem are studied for some subfamilies of semi-quasi-homogeneous polynomial systems. Our results generalize those obtained for polynomial semi-homogeneous systems.

In this paper, we bound the number of limit cycles for a class of cubic reversible isochronous system inside the class of all cubic polynomial differential systems. By applying the averaging method of second order to this system, it is proved that at most eight limit cycles can bifurcate from the period annulus. Moreover, this bound is sharp.

In this paper, Taylor expansion approach is presented for solving (approximately) a class of linear fractional integro-differential equations including those of Fredholm and of Volterra types. By means of the mth-order Taylor expansion of the unknown function at an arbitrary point, the linear fractional integro-differential equation can be converted approximately to a system of equations for the unknown function itself and its m derivatives under initial conditions. This method gives a simple and closed form solution for a linear fractional integro-differential equation. In addition, illustrative examples are presented to demonstrate the efficiency and accuracy of the proposed method.

In this paper, by applying two pairs of lower and upper solutions method and the topological degree theory of strict-set-contractions, the existence of at least three positive solutions to m-point boundary value problems for second order ordinary differential equations in Banach spaces is obtained.

In this paper, the existence of positive solutions for the nonlinear Caputo fractional functional differential equation in the form {D"0"+^qy(t)+r(t)f(y"t)=0,@?t@?(0,1),q@?(n-1,n],y^(^i^)(0)=0,0@?i@?n-3,@ay^(^n^-^2^)(t)-@by^(^n^-^1^)(t)=@h(t),t@?[-@t,0],@cy^(^n^-^2^)(t)+@dy^(^n^-^1^)(t)=@x(t),t@?[1,1+a] is studied. By constructing a special cone and using Krasnosel'skii's fixed point theorem, various results on the existence of at least one or two positive solutions to the fractional functional differential equation are established. The main results improve and generalize the existing results.

This paper deals with the positive solutions of m-point nonlinear boundary value problem in Banach spaces. By using the fixed-point theorem of strict-set-contractions, some sufficient conditions for the existence of at least one or two positive solutions to m-point boundary value problem in Banach spaces are obtained. An example illustrating the main results is given.