In this paper,the problem of brake orbits with minimal period estimates are considered for the first-order Hamiltonian systems with anisotropic growth,i.e.,the Hamiltonian functions may have super-quadratic,sub-quadra...In this paper,the problem of brake orbits with minimal period estimates are considered for the first-order Hamiltonian systems with anisotropic growth,i.e.,the Hamiltonian functions may have super-quadratic,sub-quadratic and quadratic behaviors simultaneously in different variable components.展开更多
In this paper, we consider the minimal period estimates for brake orbits of autonomous subquadratic Hamiltonian systems. We prove that if the Hamiltonian function H ∈ C2（R2n,R） is unbounded and not uniformly coerci...In this paper, we consider the minimal period estimates for brake orbits of autonomous subquadratic Hamiltonian systems. We prove that if the Hamiltonian function H ∈ C2（R2n,R） is unbounded and not uniformly coercive, there exists at least one nonconstant T-periodic brake orbit （z, T） with minimal period T or T/2 for every number T 〉 0.展开更多
In this paper,we consider the brake orbits of a reversible even Hamiltonian system near an equilibrium.Let the Hamiltonian system(H S)x=J H(x)satisfies H(0)=0,H(0)=0,reversible and even conditions H(Nx)=H(x)and H(-x)=...In this paper,we consider the brake orbits of a reversible even Hamiltonian system near an equilibrium.Let the Hamiltonian system(H S)x=J H(x)satisfies H(0)=0,H(0)=0,reversible and even conditions H(Nx)=H(x)and H(-x)=H(x)for all x∈R^(2n).Suppose the quadratic form Q(x)=1/2 is non-degenerate.Fixτ_(0)>0 and assume that R^(2n)=E⊕F decomposes into linear subspaces E and F which are invariant under the flow associated to the linear system x=J H''(0)x and such that each solution of the above linear system in E isτ_(0)-periodic whereas no solution in F{0}isτ_(0)-periodic.Writeσ(τ_(0))=σ_Q(τ_(0))for the signature of Q|E.Ifσ(τ_(0))≠=0,we prove that either there exists a sequence of brake orbits x_k→0 withτk-periodic on the hypersurface H^(-1)(0)whereτ_k→τ_(0);or for eachλclose to 0 withλ_(σ)(τ_(0))>0 the hypersurface H-1(λ)contains at least 1/2|σ(τ_(0))|distinct brake orbits of the Hamiltonian system(HS)near 0 with periods nearτ_(0).Such result for periodic solutions was proved by Bartsch in 1997.展开更多
Combining the dual least action principle with Mountain-pass lemma,we obtain the existence of brake orbits for first-order convex Hamiltonian systems with particular anisotropic growth.
In this paper,we establish a relationship between the Morse index at rest points in the saddle point reduction and the brake-orbit-type Maslov index at corresponding brake orbits.As an application,we give a criterion ...In this paper,we establish a relationship between the Morse index at rest points in the saddle point reduction and the brake-orbit-type Maslov index at corresponding brake orbits.As an application,we give a criterion to find brake orbits which are contractible and start at{0}×T^n■T^2n for even Hamiltonian on T^2 n by the methods of the Maslov-index theory and a critical point theorem formulated by Bartsch and Wang(1997).Explicitly,if all trivial solutions of a Hamiltonian are nondegenerate in the brake orbit boundary case,there are at least max{iL0(z0)}pairs of nontrivial 1-periodic brake orbits if iL0(z0)>0 or at least max{-iL0(z0)-n}pairs of nontrivial 1-periodic brake orbits if iL0(z0)<-n.In the end,we give an example to find brake orbits for certain Hamiltonian via this criterion.展开更多
In this paper,we study the Maslov-type index theory for linear Hamiltonian systems with brake orbits boundary value conditions and its applications to the existence of multiple brake orbits of nonlinear Hamiltonian sy...In this paper,we study the Maslov-type index theory for linear Hamiltonian systems with brake orbits boundary value conditions and its applications to the existence of multiple brake orbits of nonlinear Hamiltonian systems.展开更多
Via the homology link theorem and the L0-index theory,symmetric brake orbits with minimal period are ensured for first-order Hamiltonian systems under anisotropic growth assumptions,which are variant forms of sub-quad...Via the homology link theorem and the L0-index theory,symmetric brake orbits with minimal period are ensured for first-order Hamiltonian systems under anisotropic growth assumptions,which are variant forms of sub-quadratic growth conditions.展开更多
In this paper,we give a survey on the index iteration theory of an index theory for brake orbit type solutions and its applications in the study of brake orbit problems including the Seifert conjecture and the minimal...In this paper,we give a survey on the index iteration theory of an index theory for brake orbit type solutions and its applications in the study of brake orbit problems including the Seifert conjecture and the minimal period solution problems in brake orbit cases.展开更多
基金supported by the NSFC(12301138)the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi(2021L377)+1 种基金the Doctoral Scientific Research Foundation of Shanxi Datong University(2018-B-15)The second author’s work was supported by the NSFC(12171108).
文摘In this paper,the problem of brake orbits with minimal period estimates are considered for the first-order Hamiltonian systems with anisotropic growth,i.e.,the Hamiltonian functions may have super-quadratic,sub-quadratic and quadratic behaviors simultaneously in different variable components.
基金Supported by Beijing Natural Science Foundation(Grant No.1144012)Beijing Talents Found(Grant No.2014000020124G065)
文摘In this paper, we consider the minimal period estimates for brake orbits of autonomous subquadratic Hamiltonian systems. We prove that if the Hamiltonian function H ∈ C2（R2n,R） is unbounded and not uniformly coercive, there exists at least one nonconstant T-periodic brake orbit （z, T） with minimal period T or T/2 for every number T 〉 0.
基金Partially supported by the NSF of China(Grant Nos.17190271,11422103,11771341)National Key R&D Program of China(Grant No.2020YFA0713301)Nankai University。
文摘In this paper,we consider the brake orbits of a reversible even Hamiltonian system near an equilibrium.Let the Hamiltonian system(H S)x=J H(x)satisfies H(0)=0,H(0)=0,reversible and even conditions H(Nx)=H(x)and H(-x)=H(x)for all x∈R^(2n).Suppose the quadratic form Q(x)=1/2 is non-degenerate.Fixτ_(0)>0 and assume that R^(2n)=E⊕F decomposes into linear subspaces E and F which are invariant under the flow associated to the linear system x=J H''(0)x and such that each solution of the above linear system in E isτ_(0)-periodic whereas no solution in F{0}isτ_(0)-periodic.Writeσ(τ_(0))=σ_Q(τ_(0))for the signature of Q|E.Ifσ(τ_(0))≠=0,we prove that either there exists a sequence of brake orbits x_k→0 withτk-periodic on the hypersurface H^(-1)(0)whereτ_k→τ_(0);or for eachλclose to 0 withλ_(σ)(τ_(0))>0 the hypersurface H-1(λ)contains at least 1/2|σ(τ_(0))|distinct brake orbits of the Hamiltonian system(HS)near 0 with periods nearτ_(0).Such result for periodic solutions was proved by Bartsch in 1997.
基金supported by the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi(Grant No.2019L0766)the Doctoral Scientific Research Foundation of Shanxi Datong University+1 种基金supported by NNSF of China(Grant Nos.11471170 and 11790271)innovation and development project of Guangzhou University,and Nankai Zhide Foundation
文摘Combining the dual least action principle with Mountain-pass lemma,we obtain the existence of brake orbits for first-order convex Hamiltonian systems with particular anisotropic growth.
基金supported by National Natural Science Foundation of China(Grant Nos.11790271,11771341 and 11422103)and Nankai University。
文摘In this paper,we establish a relationship between the Morse index at rest points in the saddle point reduction and the brake-orbit-type Maslov index at corresponding brake orbits.As an application,we give a criterion to find brake orbits which are contractible and start at{0}×T^n■T^2n for even Hamiltonian on T^2 n by the methods of the Maslov-index theory and a critical point theorem formulated by Bartsch and Wang(1997).Explicitly,if all trivial solutions of a Hamiltonian are nondegenerate in the brake orbit boundary case,there are at least max{iL0(z0)}pairs of nontrivial 1-periodic brake orbits if iL0(z0)>0 or at least max{-iL0(z0)-n}pairs of nontrivial 1-periodic brake orbits if iL0(z0)<-n.In the end,we give an example to find brake orbits for certain Hamiltonian via this criterion.
基金partially supported by the National Natural Science Foundation of China(Grant No.20060390014)
文摘In this paper,we study the Maslov-type index theory for linear Hamiltonian systems with brake orbits boundary value conditions and its applications to the existence of multiple brake orbits of nonlinear Hamiltonian systems.
基金supported by the National Natural Science Foundation of China(Grant No.12301138)the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi(Grant No.2021L377)the Doctoral Scientific Research Foundation of Shanxi Datong University(Grant No.2018-B-15)。
文摘Via the homology link theorem and the L0-index theory,symmetric brake orbits with minimal period are ensured for first-order Hamiltonian systems under anisotropic growth assumptions,which are variant forms of sub-quadratic growth conditions.
基金The first author is partially supported by the NSFC Grants(No.11790271)Guangdong Basic and Applied basic Research Foundation(No.2020A1515011019)+4 种基金Innovation and Development Project of Guangzhou UniversityThe second author is partially supported by National Key R&D Program of China(No.2020YFA0713300)NSFC Grants Nos.11671215 and 11790271,LPMC of Ministry of Education of China,Nankai University,Nankai Zhide Foundation,Wenzhong Foundation,and the Beijing Center for Mathematics and Information Interdisciplinary Sciences at Capital Normal UniversityThe third author is partially supported by National Key R&D Program of China(No.2020YFA0713300)NSFC Garnts Nos.11790271 and 11171341,and LPMC of Nankai University.
文摘In this paper,we give a survey on the index iteration theory of an index theory for brake orbit type solutions and its applications in the study of brake orbit problems including the Seifert conjecture and the minimal period solution problems in brake orbit cases.
