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
Considering a spherical planet with a liquid core surrounded by a solid shell,we developed a quasi-static model to investigate the deformation of the double-layered planet with self-gravity and obtained the boundary v...Considering a spherical planet with a liquid core surrounded by a solid shell,we developed a quasi-static model to investigate the deformation of the double-layered planet with self-gravity and obtained the boundary value problem about radial equilibrium,which is solved by the numerical methods.The effects of governing parameters about geometry,density and bulk modulus on the deformation of the planet with self-gravity were discussed.In addition,we also developed the incremental equation theory to investigate the stability of the double-layered planet under its own gravity.It is concluded that instability is more likely to occur on the planet with smaller liquid cores when the outer radius and density of the planets are constant.Although we only study special double-layered planets,these methods can be conveniently extended to complex multi-layered planets.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 the Science Foundation of National Key Laboratory of Science and Technology on advanced composites in special environments,and Heilongjiang Touyan Innovation Team Program.
文摘Considering a spherical planet with a liquid core surrounded by a solid shell,we developed a quasi-static model to investigate the deformation of the double-layered planet with self-gravity and obtained the boundary value problem about radial equilibrium,which is solved by the numerical methods.The effects of governing parameters about geometry,density and bulk modulus on the deformation of the planet with self-gravity were discussed.In addition,we also developed the incremental equation theory to investigate the stability of the double-layered planet under its own gravity.It is concluded that instability is more likely to occur on the planet with smaller liquid cores when the outer radius and density of the planets are constant.Although we only study special double-layered planets,these methods can be conveniently extended to complex multi-layered planets.
