中国科学院数学与系统科学研究院期刊网

15 January 2022, Volume 38 Issue 1
    

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  • Hua Gui DUAN, Hui LIU
    Acta Mathematica Sinica. 2022, 38(1): 1-21. https://doi.org/10.1007/s10114-021-0023-4
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    Let RPn be a bumpy and irreversible Finsler n-dimensional real projective space with reversibility λ and flag curvature K satisfying (λ/1+λ)2 < K ≤ 1 when n is odd, and K ≥ 0 when n is even. We show that if there exist exactly 2[n+1/2] prime closed geodesics on such RPn, then all of them are non-contractible, which coincides with the Katok's examples.
  • Xin LIU, Zhen Xin LIU
    Acta Mathematica Sinica. 2022, 38(1): 22-54. https://doi.org/10.1007/s10114-021-0107-1
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    In this paper, we use a unified framework to study Poisson stable (including stationary, periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent, almost recurrent in the sense of Bebutov, Levitan almost periodic, pseudo-periodic, pseudo-recurrent and Poisson stable) solutions for semilinear stochastic differential equations driven by infinite dimensional Lévy noise with large jumps. Under suitable conditions on drift, diffusion and jump coefficients, we prove that there exist solutions which inherit the Poisson stability of coefficients. Further we show that these solutions are globally asymptotically stable in square-mean sense. Finally, we illustrate our theoretical results by several examples.
  • Guan HUANG
    Acta Mathematica Sinica. 2022, 38(1): 55-67. https://doi.org/10.1007/s10114-022-0153-3
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    In this paper, we introduce a new notion of integrability for billiard tables, namely, integrability away from the boundary. One key feature of our notion is that the integrable region could be disjoint from the boundary with a positive distance. We prove that if a strictly convex billiard table, whose boundary is a small perturbation of an ellipse with small eccentricity, is integrable in this sense, then its boundary must be itself an ellipse.
  • Wen HUANG, Run Ju WEI, Tao YU, Xiao Min ZHOU
    Acta Mathematica Sinica. 2022, 38(1): 68-84. https://doi.org/10.1007/s10114-021-0179-y
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    In this paper we introduce two metrics: the max metric dn, q and the mean metric dn, q. We give an equivalent characterization of rigid measure preserving systems by the two metrics. It turns out that an invariant measure μ on a topological dynamical system (X, T) has bounded complexity with respect to dn, q if and only if μ has bounded complexity with respect to dn, q if and only if (X, BX, μ, T) is rigid. We also obtain computation formulas of the measure-theoretic entropy of an ergodic measure preserving system (resp. the topological entropy of a topological dynamical system) by the two metrics dn, q and dn, q.
  • Sai LIU, Wei WANG
    Acta Mathematica Sinica. 2022, 38(1): 85-96. https://doi.org/10.1007/s10114-022-0256-x
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    In this paper we review and systematize the index method in closed geodesic problem. As we know, the closed geodesic problem on compact Riemannian or Finsler manifold is a famous problem, and has far from been resolved. In recent years, the Maslov-type index theory for symplectic path has been applied to studying the closed geodesic problem, and has induced a great number of results on the multiplicity and stability of closed geodesics. We will systematically introduce these progresses in this review.
  • Shu Guan JI, Xiao Wan LI
    Acta Mathematica Sinica. 2022, 38(1): 97-106. https://doi.org/10.1007/s10114-022-0268-6
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    This paper is devoted to the study of the solitary wave solutions for the delayed coupled Higgs field equation

    We first establish the existence of solitary wave solutions for the corresponding equation without delay and perturbation by using the Hamiltonian system method. Then we consider the persistence of solitary wave solutions of the delayed coupled Higgs field equation by using the method of dynamical system, especially the geometric singular perturbation theory, invariant manifold theory and Fredholm theory. According to the relationship between solitary wave and homoclinic orbit, the coupled Higgs field equation is transformed into the ordinary differential equations with fast variables by using the variable substitution. It is proved that the equations with perturbation also possess homoclinic orbit, and thus we obtain the existence of solitary wave solutions of the delayed coupled Higgs field equation.
  • Tomasz DOWNAROWICZ, Guo Hua ZHANG
    Acta Mathematica Sinica. 2022, 38(1): 107-136. https://doi.org/10.1007/s10114-022-0311-7
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    In this survey we will present the symbolic extension theory in topological dynamics, which was built over the past twenty years.
  • Wen Meng ZHANG, Peng LIU, Xuan LEI
    Acta Mathematica Sinica. 2022, 38(1): 137-147. https://doi.org/10.1007/s10114-021-0411-9
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    The well-known Hartman–Grobman Theorem says that a C1 hyperbolic diffeomorphism F can be locally linearized by a homeomorphism Φ. For parameterized systems Fθ, known results show that the corresponding homeomorphisms Φθ exist uniquely in a functional space equipped with the supremum norm and depend continuously on the parameter θ. In this paper, we further extend the results to Hölder dependence of Φθ on θ by Pugh's strategy, but introducing a kind of special Hölder norm instead of the usual supremum norm in the proof to control the linear parts of Fθ. This requires a new Hölder linearization result for every Fθ.
  • Ji LI
    Acta Mathematica Sinica. 2022, 38(1): 148-160. https://doi.org/10.1007/s10114-022-0425-y
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    In this paper, we investigate the orbital stability of the peaked solitons (peakons) for the modified Camassa–Holm equation with cubic nonlinearity. We consider a minimization problem with an appropriately chosen constraint, from which we establish the orbital stability of the peakons under H1W1, 4 norm.
  • Xing LIANG, Tao ZHOU
    Acta Mathematica Sinica. 2022, 38(1): 161-178. https://doi.org/10.1007/s10114-022-0452-8
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    This paper is devoted to studying the asymptotic behavior of the solution to nonlocal Fisher-KPP type reaction diffusion equations in heterogeneous media. The kernel K is assumed to depend on the media. First, we give an estimate of the upper and lower spreading speeds by generalized principal eigenvalues. Second, we prove the existence of spreading speeds in the case where the media is periodic or almost periodic by showing that the upper and lower generalized principal eigenvalues are equal.
  • Guo Wei YU
    Acta Mathematica Sinica. 2022, 38(1): 179-204. https://doi.org/10.1007/s10114-022-0453-7
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    For a monotone twist map, under certain non-degenerate condition, we showed the existence of infinitely many homoclinic and heteroclinic orbits between two periodic neighboring minimal orbits with the same rotation number, which indicates chaotic dynamics. Our results also apply to geodesics of smooth Riemannian metrics on the two-dimension torus.
  • Lei YANG
    Acta Mathematica Sinica. 2022, 38(1): 205-224. https://doi.org/10.1007/s10114-022-0459-1
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    Let X = G/Γ be a homogeneous space with ambient group G containing the group H = (SO(n, 1))k and xX be such that Hx is dense in X. Given an analytic curve φ: I=[a, b] → H, we will show that if φ satisfies certain geometric condition, then for a typical diagonal subgroup A ={a(t): t ∈ R} ? H the translates {a(t)φ(I)x: t >0} of the curve φ(I)x will tend to be equidistributed in X as t → +∈∞. The proof is based on Ratner's theorem and linearization technique.
  • Da Wei YANG
    Acta Mathematica Sinica. 2022, 38(1): 225-248. https://doi.org/10.1007/s10114-022-0471-5
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    We give a brief survey on the dynamics of vector fields with singularities. The aim of this survey is not to list all results in this field, but only to introduce some results from several viewpoints and some technics.
  • Jian Jun LIU
    Acta Mathematica Sinica. 2022, 38(1): 249-262. https://doi.org/10.1007/s10114-022-0472-4
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    This paper is concerned with the derivative nonlinear Schr¨ odinger equation with periodic boundary conditions. We obtain complete Birkhoff normal form of order six. As an application, the long time stability for solutions of small amplitude is proved.
  • Zhong Jie LIU, Fan Jing WANG, Duan Zhi ZHANG
    Acta Mathematica Sinica. 2022, 38(1): 263-280. https://doi.org/10.1007/s10114-022-0473-3
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    In this paper, we consider the brake orbits of a reversible even Hamiltonian system near an equilibrium. Let the Hamiltonian system (HS) ?=JH'(x) satisfies H(0)=0, H'(0)=0, reversible and even conditions H(Nx)=H(x) and H(-x)=H(x) for all xR2n. Suppose the quadratic form Q(x)=1/2〈H"(0)x, x〉 is non-degenerate. Fix τ0>0 and assume that R2n=EF decomposes into linear subspaces E and F which are invariant under the flow associated to the linear system ?=JH"(0)x and such that each solution of the above linear system in E is τ0-periodic whereas no solution in F \ {0} is τ0-periodic. Write σ(τ0)=σQ(τ0) for the signature of QE. If σ(τ0)≠0, we prove that either there exists a sequence of brake orbits xk→ 0 with τk-periodic on the hypersurface H-1(0) where τkτ0; or for each λ close to 0 with λ σ(τ0)>0 the hypersurface H-1(λ) contains at least 1/2σ(τ0) distinct brake orbits of the Hamiltonian system (HS) near 0 with periods near τ0. Such result for periodic solutions was proved by Bartsch in 1997.
  • Wen HUANG, Zeng LIAN
    Acta Mathematica Sinica. 2022, 38(1): 281-290. https://doi.org/10.1007/s10114-022-0493-z
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    In this paper, we construct random horseshoes of Anosov systems driven by an equicontinuous system based on an ergodic measure with positive entropy.
  • Xi Jun HU, Li WU
    Acta Mathematica Sinica. 2022, 38(1): 291-310. https://doi.org/10.1007/s10114-022-0507-x
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    In this paper, we define mean index for non-periodic orbits in Hamiltonian systems and study its properties. In general, the mean index is an interval in R which is uniformly continuous on the systems. We show that the index interval is a point for a quasi-periodic orbit. The mean index can be considered as a generalization of rotation number defined by Johnson and Moser in the study of almost periodic Schr¨ odinger operators. Motivated by their works, we study the relation of Fredholm property of the linear operator and the mean index at the end of the paper.