In this paper, a sufficient condition for the existence of bifurcation points for discrete dynamical systems is presented. The relation between two families of systems is further discussed, and a sufficient condition ...In this paper, a sufficient condition for the existence of bifurcation points for discrete dynamical systems is presented. The relation between two families of systems is further discussed, and a sufficient condition for determining whether they may have the similar bifurcation points is given.展开更多
Static bifurcation theory is concerned with changes that occur in the structure of the set of zeros of a function as parameters varying in the function. Using a generalized Conley index theory, we develop a static bif...Static bifurcation theory is concerned with changes that occur in the structure of the set of zeros of a function as parameters varying in the function. Using a generalized Conley index theory, we develop a static bifurcation theory in the paper.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
基金Project supported by the National Natural Science Foundation of China (Grant No.10672146)the Shanghai Leading Academic Discipline Project (Grant No.S30104)
文摘In this paper, a sufficient condition for the existence of bifurcation points for discrete dynamical systems is presented. The relation between two families of systems is further discussed, and a sufficient condition for determining whether they may have the similar bifurcation points is given.
基金Research partially supported by the National Natural Science Foundation of China.
文摘Static bifurcation theory is concerned with changes that occur in the structure of the set of zeros of a function as parameters varying in the function. Using a generalized Conley index theory, we develop a static bifurcation theory in the paper.
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.
