In this paper, the authors consider limit cycle bifurcations for a kind of nonsmooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with a center at the origin and a heteroclinic l...In this paper, the authors consider limit cycle bifurcations for a kind of nonsmooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with a center at the origin and a heteroclinic loop around the origin. When the degree of perturbing polynomial terms is n(n ≥ 1), it is obtained that n limit cycles can appear near the origin and the heteroclinic loop respectively by using the first Melnikov function of piecewise near-Hamiltonian systems, and that there are at most n + [(n+1)/2] limit cycles bifurcating from the periodic annulus between the center and the heteroclinic loop up to the first order in ε. Especially, for n = 1, 2, 3 and 4, a precise result on the maximal number of zeros of the first Melnikov function is derived.展开更多
This paper investigates the equality-constrained minimization of polynomial functions. Let R be the field of real numbers, and R[x1,..., xn] the ring of polynomials over R in variables x1,..., xn. For an f ∈ R[x1,......This paper investigates the equality-constrained minimization of polynomial functions. Let R be the field of real numbers, and R[x1,..., xn] the ring of polynomials over R in variables x1,..., xn. For an f ∈ R[x1,..., xn] and a finite subset H of R[x1,..., xn], denote by V(f : H) the set {f( ˉα) | ˉα∈ Rn, and h( ˉα) =0, ? h ∈ H}. We provide an effective algorithm for computing a finite set U of non-zero univariate polynomials such that the infimum inf V(f : H) of V(f : H) is a root of some polynomial in U whenever inf V(f : H) = ±∞.The strategies of this paper are decomposing a finite set of polynomials into triangular chains of polynomials and computing the so-called revised resultants. With the aid of the computer algebraic system Maple, our algorithm has been made into a general program to treat the equality-constrained minimization of polynomials with rational coefficients.展开更多
基金supported by the National Natural Science Foundation of China(No.11271261)the Natural Science Foundation of Anhui Province(No.1308085MA08)the Doctoral Program Foundation(2012)of Anhui Normal University
文摘In this paper, the authors consider limit cycle bifurcations for a kind of nonsmooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with a center at the origin and a heteroclinic loop around the origin. When the degree of perturbing polynomial terms is n(n ≥ 1), it is obtained that n limit cycles can appear near the origin and the heteroclinic loop respectively by using the first Melnikov function of piecewise near-Hamiltonian systems, and that there are at most n + [(n+1)/2] limit cycles bifurcating from the periodic annulus between the center and the heteroclinic loop up to the first order in ε. Especially, for n = 1, 2, 3 and 4, a precise result on the maximal number of zeros of the first Melnikov function is derived.
基金supported by National Natural Science Foundation of China(Grant No.11161034)
文摘This paper investigates the equality-constrained minimization of polynomial functions. Let R be the field of real numbers, and R[x1,..., xn] the ring of polynomials over R in variables x1,..., xn. For an f ∈ R[x1,..., xn] and a finite subset H of R[x1,..., xn], denote by V(f : H) the set {f( ˉα) | ˉα∈ Rn, and h( ˉα) =0, ? h ∈ H}. We provide an effective algorithm for computing a finite set U of non-zero univariate polynomials such that the infimum inf V(f : H) of V(f : H) is a root of some polynomial in U whenever inf V(f : H) = ±∞.The strategies of this paper are decomposing a finite set of polynomials into triangular chains of polynomials and computing the so-called revised resultants. With the aid of the computer algebraic system Maple, our algorithm has been made into a general program to treat the equality-constrained minimization of polynomials with rational coefficients.
