In this paper we consider a class of polynomial planar system with two small parameters,ε and λ,satisfying 0<ε《λ《1.The corresponding first order Melnikov function M_(1) with respect to ε depends on λ so tha...In this paper we consider a class of polynomial planar system with two small parameters,ε and λ,satisfying 0<ε《λ《1.The corresponding first order Melnikov function M_(1) with respect to ε depends on λ so that it has an expansion of the form M_(1)(h,λ)=∑k=0∞M_(1k)(h)λ^(k).Assume that M_(1k')(h) is the first non-zero coefficient in the expansion.Then by estimating the number of zeros of M_(1k')(h),we give a lower bound of the maximal number of limit cycles emerging from the period annulus of the unperturbed system for 0<ε《λ《1,when k'=0 or 1.In addition,for each k∈N,an upper bound of the maximal number of zeros of M_(1k)(h),taking into account their multiplicities,is presented.展开更多
In this paper,we address the stability of periodic solutions of piecewise smooth periodic differential equations.By studying the Poincarémap,we give a sufficient condition to judge the stability of a periodic sol...In this paper,we address the stability of periodic solutions of piecewise smooth periodic differential equations.By studying the Poincarémap,we give a sufficient condition to judge the stability of a periodic solution.We also present examples of some applications.展开更多
This paper deals with bifurcations of subharmonic solutions and invariant tori generated from limit cycles in the fast dynamics for a nonautonomous singularly perturbed system. Based on Poincare map, a series of blow-...This paper deals with bifurcations of subharmonic solutions and invariant tori generated from limit cycles in the fast dynamics for a nonautonomous singularly perturbed system. Based on Poincare map, a series of blow-up transformations and the theory of integral manifold, the conditions for the existence of invariant tori are obtained, and the saddle-node bifurcations of subharmonic solutions are studied.展开更多
This paper deals with a kind of fourth degree systems with perturbations. By using the method of multi-parameter perturbation theory and qualitative analysis, it is proved that the system can have six limit cycles.
The canard phenomenon occurring in planar fast-slow systems under non-generic conditions is investigated.When the critical manifold has a non-generic fold point,by using the method of asymptotic analysis combined with...The canard phenomenon occurring in planar fast-slow systems under non-generic conditions is investigated.When the critical manifold has a non-generic fold point,by using the method of asymptotic analysis combined with the recently developed blow-up technique,the existence of a canard is established and the asymptotic expansion of the parameter for which a canard exists is obtained.展开更多
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
In the study of the number of limit cycles of near-Hamiltonian systems,the first order Melnikov function plays an important role.This paper aims to generalize Horozov-Iliev’s method to estimate the upper bound of the...In the study of the number of limit cycles of near-Hamiltonian systems,the first order Melnikov function plays an important role.This paper aims to generalize Horozov-Iliev’s method to estimate the upper bound of the number of zeros of the function.展开更多
In this paper,we consider the first-order Melnikov functions and limit cycle bifurcations of a nearHamiltonian system near a cuspidal loop.By establishing relations between the coefficients in the expansions of the tw...In this paper,we consider the first-order Melnikov functions and limit cycle bifurcations of a nearHamiltonian system near a cuspidal loop.By establishing relations between the coefficients in the expansions of the two Melnikov functions,we give a general method to obtain the number of limit cycles near the cuspidal loop.As an application,we consider a kind of Liénard systems and obtain a new estimation on the lower bound of the maximum number of limit cycles.展开更多
基金The first author is supported by the National Natural Science Foundation of China(11671013)the second author is supported by the National Natural Science Foundation of China(11771296).
文摘In this paper we consider a class of polynomial planar system with two small parameters,ε and λ,satisfying 0<ε《λ《1.The corresponding first order Melnikov function M_(1) with respect to ε depends on λ so that it has an expansion of the form M_(1)(h,λ)=∑k=0∞M_(1k)(h)λ^(k).Assume that M_(1k')(h) is the first non-zero coefficient in the expansion.Then by estimating the number of zeros of M_(1k')(h),we give a lower bound of the maximal number of limit cycles emerging from the period annulus of the unperturbed system for 0<ε《λ《1,when k'=0 or 1.In addition,for each k∈N,an upper bound of the maximal number of zeros of M_(1k)(h),taking into account their multiplicities,is presented.
文摘In this paper,we address the stability of periodic solutions of piecewise smooth periodic differential equations.By studying the Poincarémap,we give a sufficient condition to judge the stability of a periodic solution.We also present examples of some applications.
基金Project supported by the National Natural Science Foundation of China (No. 10671214)the Chongqing Natural Science Foundation of China (No. 2005cc14)Shanghai Shuguang Genzong Project (No.04SGG05)
文摘This paper deals with bifurcations of subharmonic solutions and invariant tori generated from limit cycles in the fast dynamics for a nonautonomous singularly perturbed system. Based on Poincare map, a series of blow-up transformations and the theory of integral manifold, the conditions for the existence of invariant tori are obtained, and the saddle-node bifurcations of subharmonic solutions are studied.
基金Project supported by the National Natural Science Foundation of China (No.10371072)the New Century Excellent Ttdents in University (No.NCBT-04-038)the Shanghai Leading Academic Discipline (No.T0401).
文摘This paper deals with a kind of fourth degree systems with perturbations. By using the method of multi-parameter perturbation theory and qualitative analysis, it is proved that the system can have six limit cycles.
基金supported by the National Natural Science Foundation of China (No. 10701023)the E-Institutes of Shanghai Municipal Education Commission (No. N.E03004).
文摘The canard phenomenon occurring in planar fast-slow systems under non-generic conditions is investigated.When the critical manifold has a non-generic fold point,by using the method of asymptotic analysis combined with the recently developed blow-up technique,the existence of a canard is established and the asymptotic expansion of the parameter for which a canard exists is obtained.
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 Nos.11931016 and 11771296)Hunan Provincial Education Department(Grant No.19C1898).
文摘In the study of the number of limit cycles of near-Hamiltonian systems,the first order Melnikov function plays an important role.This paper aims to generalize Horozov-Iliev’s method to estimate the upper bound of the number of zeros of the function.
基金supported by National Natural Science Foundation of China(Grant No.11971145)supported by National Natural Science Foundation of China(Grant No.11931016)the National Key R&D Program of China(Grant No.2022YFA1005900)。
文摘In this paper,we consider the first-order Melnikov functions and limit cycle bifurcations of a nearHamiltonian system near a cuspidal loop.By establishing relations between the coefficients in the expansions of the two Melnikov functions,we give a general method to obtain the number of limit cycles near the cuspidal loop.As an application,we consider a kind of Liénard systems and obtain a new estimation on the lower bound of the maximum number of limit cycles.
