In this paper, a class of discrete deterministic SIR epidemic model with vertical and horizontal transmission is studied. Based on the population assumed to be a constant size, we transform the discrete SIR epidemic m...In this paper, a class of discrete deterministic SIR epidemic model with vertical and horizontal transmission is studied. Based on the population assumed to be a constant size, we transform the discrete SIR epidemic model into a planar map. Then we find out its equilibrium points and eigenvalues. From discussing the influence of the coefficient parameters effected on the eigenvalues, we give the hyperbolicity of equilibrium points and determine which point is saddle, node or focus as well as their stability. Further, by deriving equations describing flows on the center manifolds, we discuss the transcritical bifurcation at the non-hyperbolic equilibrium point. Finally, we give some numerical simulation examples for illustrating the theoretical analysis and the biological explanation of our theorem.展开更多
In this paper,we define a generalized Lipschitz shadowing property for flows and prove that a flowΦgenerated by a C1vector field X on a closed Riemannian manifold M has this generalized Lipschitz shadowing property i...In this paper,we define a generalized Lipschitz shadowing property for flows and prove that a flowΦgenerated by a C1vector field X on a closed Riemannian manifold M has this generalized Lipschitz shadowing property if and only if it is structurally stable.展开更多
The main aim of this paper is to study whether the Gromov hyperbolicity is preserved under some transformations on Riemann surfaces （with their Poincare metrics）. We prove that quasiconformal maps between Riemann su...The main aim of this paper is to study whether the Gromov hyperbolicity is preserved under some transformations on Riemann surfaces （with their Poincare metrics）. We prove that quasiconformal maps between Riemann surfaces preserve hyperbolicity; however, we also show that arbitrary twists along simple closed geodesics do not preserve it, in general.展开更多
We prove that any C1-stable weakly shadowable volume-preserving diffeomorphism defined on a compact manifold displays a dominated splitting E ⊕ F. Moreover, both E and F are volume-hyperbolic. Finally, we prove the v...We prove that any C1-stable weakly shadowable volume-preserving diffeomorphism defined on a compact manifold displays a dominated splitting E ⊕ F. Moreover, both E and F are volume-hyperbolic. Finally, we prove the version of this result for divergence-free vector fields. As a consequence, in low dimensions, we obtain global hyperbolicity.展开更多
In this work, we obtain good upper bounds for the diameter of any graph in terms of its minimum degree and its order, improving a classical theorem due to Erdos, Pach, Pollack and Tuza. We use these bounds in order to...In this work, we obtain good upper bounds for the diameter of any graph in terms of its minimum degree and its order, improving a classical theorem due to Erdos, Pach, Pollack and Tuza. We use these bounds in order to study hyperbolic graphs （in the Gromov sense）. To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let H（n, δ0） be the set of graphs G with n vertices and minimum degree 50, and J（n, Δ） be the set of graphs G with n vertices and maximum degree A. We study the four following extremal problems on graphs： a（n,δ0） = min{δ（G） ｜ G ∈H（n, δ0）}, b（n, δ0） =- max{δ（G）｜ e ∈H（n, δ0）}, α（n, Δ） = min{δ（G） [ G ∈ J（n, Δ）} and β（n,Δ） = max{δ（G） ] G∈Π（n,Δ）}. In particular, we obtain bounds for b（n, δ0） and we compute the precise value of a（n, δ0）, α（n, Δ） and w（n, Δ） for all values of n, r0 and A, respectively.展开更多
The original version of the article was published in [1]. Unfortunately, the original version of this article contains a mistake： in Theorem 6.2 appears that β（n, △） = （n-△ ＋ 5）/4 but the correct statement is...The original version of the article was published in [1]. Unfortunately, the original version of this article contains a mistake： in Theorem 6.2 appears that β（n, △） = （n-△ ＋ 5）/4 but the correct statement is β（n, △） = （n -△ ＋ 4）/4. In this erratum we correct the theorem and give the correct proof.展开更多
Arnold diffusion was conjectured by Arnol’d(1964) as a typical phenomena of topological instability in classical mechanics. In this paper, we give a panorama of the researches on Arnold diffusion using the variationa...Arnold diffusion was conjectured by Arnol’d(1964) as a typical phenomena of topological instability in classical mechanics. In this paper, we give a panorama of the researches on Arnold diffusion using the variational approaches.展开更多
We present a well-posed and discretely stable perfectly matched layer for the anisotropic(and isotropic)elastic wave equations without first re-writing the governing equations as a first order system.The new model is ...We present a well-posed and discretely stable perfectly matched layer for the anisotropic(and isotropic)elastic wave equations without first re-writing the governing equations as a first order system.The new model is derived by the complex coordinate stretching technique.Using standard perturbation methods we show that complex frequency shift together with a chosen real scaling factor ensures the decay of eigen-modes for all relevant frequencies.To buttress the stability properties and the robustness of the proposed model,numerical experiments are presented for anisotropic elastic wave equations.The model is approximated with a stable node-centered finite difference scheme that is second order accurate both in time and space.展开更多
In the part 2, theorem 3.1 stut ed in part 1[15] is proved first. The proof is obtained via a way of changing variables to reduce the original system of differentialequations to a form concerning Standard systems of e...In the part 2, theorem 3.1 stut ed in part 1[15] is proved first. The proof is obtained via a way of changing variables to reduce the original system of differentialequations to a form concerning Standard systems of equations in the theory ofdifferentiable dynamical systems. Then by using theorem 3.1 together with thepreliminary theorem 2.l, foe main theorem of this paper announced in part 1 is proved.The definition of admissible perturbation is contained in the appendix of part 2. Themeanings of the main theorem is described in the introduction of part 1.展开更多
The study of linear and global. properties of linear dynamical systems on vector bundles appeared rather extensive already in the past.Presently we propose to study perturbations of this linear dynamics The perturbed...The study of linear and global. properties of linear dynamical systems on vector bundles appeared rather extensive already in the past.Presently we propose to study perturbations of this linear dynamics The perturbed dynamical system which we shallconsider is no longer linear.while the properties to be studied will be still global in general.Moreover.we are interested in the nonuniformly hyperbolic properties.In this paper,we set an appropriate definition for such perturbations.Though it appearssome what not quite usual yet has deeper root in standard systens of differential equations in the theory of differentiable dynamical systens The general problen is to see which property of the original given by the dynamical system is persistent when a perturbation takes place.The whole contenl of the paper is deyoted to establishinga theorem of this sort.展开更多
Let （X, G（X）, m） be a probability space with a-algebra G（X） and probability measure m. The set V in G is called P-admissible, provided that for any positive integer n and positive-measure set Vn∈ contained in V...Let （X, G（X）, m） be a probability space with a-algebra G（X） and probability measure m. The set V in G is called P-admissible, provided that for any positive integer n and positive-measure set Vn∈ contained in V, there exists a Zn∈G such that Zn belong to Vn and 0 〈 m（Zn） 〈 1/n. Let T be an ergodic automorphism of （X, G） preserving m, and A belong to the space of linear measurable symplectic cocycles展开更多
This paper provides a new approach to study the solutions of a class of generalized Jazobi equations associated with the linearization of certain singular flows on Riemannian manifolds with dimension n ＋ 1. A new cla...This paper provides a new approach to study the solutions of a class of generalized Jazobi equations associated with the linearization of certain singular flows on Riemannian manifolds with dimension n ＋ 1. A new class of generalized differential operators is defined. We investigate the kernel of the corresponding maximal operators by applying operator theory. It is shown that all nontrivial solutions to the generalized Jacobi equation are hyperbolic, in which there are n dimension solutions with exponential-decaying amplitude.展开更多
Let γ be a hyperbolic closed orbit of a C1 vector field X on a compact C∞ manifold M of dimension n ≥ 3, and let Hx（γ） be the homoclinic class of X containing γ. In this paper, we provethat Cl-generically, if ...Let γ be a hyperbolic closed orbit of a C1 vector field X on a compact C∞ manifold M of dimension n ≥ 3, and let Hx（γ） be the homoclinic class of X containing γ. In this paper, we provethat Cl-generically, if Hx （γ） is expansive and isolated, then it is hyperbolic.展开更多
To decide when a graph is Gromov hyperbolic is,in general,a very hard problem.In this paper,we solve this problem for the set of short graphs（in an informal way,a graph G is r-short if the shortcuts in the cycles of ...To decide when a graph is Gromov hyperbolic is,in general,a very hard problem.In this paper,we solve this problem for the set of short graphs（in an informal way,a graph G is r-short if the shortcuts in the cycles of G have length less than r）：an r-short graph G is hyperbolic if and only if S9r（G）is finite,where SR（G）：=sup{L（C）：C is an R-isometric cycle in G}and we say that a cycle C is R-isometric if dC（x,y）≤dG（x,y）＋R for every x,y∈C.展开更多
We generalize Bangert＇s non-hyperbolicity result for uniformly tamed almost complex structures on standard symplectic R^2n to asymptotically standard symplectic manifolds.
A standard conservation form is derived in this paper.The hyperbolicity of Helbing＇s fluid dynamic traffic flow model is proved,which is essential to the general analytical and numerical study of this model.On the ba...A standard conservation form is derived in this paper.The hyperbolicity of Helbing＇s fluid dynamic traffic flow model is proved,which is essential to the general analytical and numerical study of this model.On the basis of this conservation form,a local discontinuous Galerkin scheme is designed to solve the resulting system efficiently.The evolution of an unstable equilibrium traffic state leading to a stable stop-and-go traveling wave is simulated.This simulation also verifies that the model is truly improved by the introduction of the modified diffusion coefficients,and thus helps to protect vehicles from collisions and avoide the appearance of the extremely large density.展开更多
In this paper we give a classification of special endomorphisms of nil-manifolds:Let f:N/Γ→N/Γbe a covering map of a nil-manifold and denote by A:N/Γ→N/Γthe nil-endomorphism which is homotopic to f.If f is a spe...In this paper we give a classification of special endomorphisms of nil-manifolds:Let f:N/Γ→N/Γbe a covering map of a nil-manifold and denote by A:N/Γ→N/Γthe nil-endomorphism which is homotopic to f.If f is a special TA-map,then A is a hyperbolic nil-endomorphism and f is topologically conjugate to A.展开更多
文摘In this paper, a class of discrete deterministic SIR epidemic model with vertical and horizontal transmission is studied. Based on the population assumed to be a constant size, we transform the discrete SIR epidemic model into a planar map. Then we find out its equilibrium points and eigenvalues. From discussing the influence of the coefficient parameters effected on the eigenvalues, we give the hyperbolicity of equilibrium points and determine which point is saddle, node or focus as well as their stability. Further, by deriving equations describing flows on the center manifolds, we discuss the transcritical bifurcation at the non-hyperbolic equilibrium point. Finally, we give some numerical simulation examples for illustrating the theoretical analysis and the biological explanation of our theorem.
基金supported by National Natural Science Foundation of China(12071018)Fundamental Research Funds for the Central Universitiessupported by the National Research Foundation of Korea(NRF)funded by the Korea government(MIST)(2020R1F1A1A01051370)。
文摘In this paper,we define a generalized Lipschitz shadowing property for flows and prove that a flowΦgenerated by a C1vector field X on a closed Riemannian manifold M has this generalized Lipschitz shadowing property if and only if it is structurally stable.
文摘The main aim of this paper is to study whether the Gromov hyperbolicity is preserved under some transformations on Riemann surfaces （with their Poincare metrics）. We prove that quasiconformal maps between Riemann surfaces preserve hyperbolicity; however, we also show that arbitrary twists along simple closed geodesics do not preserve it, in general.
基金supported by National Funds through FCT-"Fundao para a Ciênciae a Tecnologia"(Grant No.PEst-OE/MAT/UI0212/2011)supported by Basic Science Research Program through the National Research Foundation of Korea(NRF)funded by the Ministry of Education,Science and Technology,Korea(Grant No.2011-0007649)
文摘We prove that any C1-stable weakly shadowable volume-preserving diffeomorphism defined on a compact manifold displays a dominated splitting E ⊕ F. Moreover, both E and F are volume-hyperbolic. Finally, we prove the version of this result for divergence-free vector fields. As a consequence, in low dimensions, we obtain global hyperbolicity.
基金Supported in part by two grants from Ministerio de Economía y Competitividad,Spain:MTM2013-46374-P and MTM2015-69323-REDT
文摘In this work, we obtain good upper bounds for the diameter of any graph in terms of its minimum degree and its order, improving a classical theorem due to Erdos, Pach, Pollack and Tuza. We use these bounds in order to study hyperbolic graphs （in the Gromov sense）. To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let H（n, δ0） be the set of graphs G with n vertices and minimum degree 50, and J（n, Δ） be the set of graphs G with n vertices and maximum degree A. We study the four following extremal problems on graphs： a（n,δ0） = min{δ（G） ｜ G ∈H（n, δ0）}, b（n, δ0） =- max{δ（G）｜ e ∈H（n, δ0）}, α（n, Δ） = min{δ（G） [ G ∈ J（n, Δ）} and β（n,Δ） = max{δ（G） ] G∈Π（n,Δ）}. In particular, we obtain bounds for b（n, δ0） and we compute the precise value of a（n, δ0）, α（n, Δ） and w（n, Δ） for all values of n, r0 and A, respectively.
基金Supported by two grants from Ministerio de Economía y Competitividad,Spain(Grant Nos.MTM2013-46374-P and MTM2015-69323-REDT)
文摘The original version of the article was published in [1]. Unfortunately, the original version of this article contains a mistake： in Theorem 6.2 appears that β（n, △） = （n-△ ＋ 5）/4 but the correct statement is β（n, △） = （n -△ ＋ 4）/4. In this erratum we correct the theorem and give the correct proof.
基金supported by National Natural Science Foundation of China (Grant Nos. 11790272 and No.11631006)supported by National Natural Science Foundation of China (Grant No. 11790273)Beijing Natural Science Foundation (Grant No. Z180003)
文摘Arnold diffusion was conjectured by Arnol’d(1964) as a typical phenomena of topological instability in classical mechanics. In this paper, we give a panorama of the researches on Arnold diffusion using the variational approaches.
文摘We present a well-posed and discretely stable perfectly matched layer for the anisotropic(and isotropic)elastic wave equations without first re-writing the governing equations as a first order system.The new model is derived by the complex coordinate stretching technique.Using standard perturbation methods we show that complex frequency shift together with a chosen real scaling factor ensures the decay of eigen-modes for all relevant frequencies.To buttress the stability properties and the robustness of the proposed model,numerical experiments are presented for anisotropic elastic wave equations.The model is approximated with a stable node-centered finite difference scheme that is second order accurate both in time and space.
文摘In the part 2, theorem 3.1 stut ed in part 1[15] is proved first. The proof is obtained via a way of changing variables to reduce the original system of differentialequations to a form concerning Standard systems of equations in the theory ofdifferentiable dynamical systems. Then by using theorem 3.1 together with thepreliminary theorem 2.l, foe main theorem of this paper announced in part 1 is proved.The definition of admissible perturbation is contained in the appendix of part 2. Themeanings of the main theorem is described in the introduction of part 1.
文摘The study of linear and global. properties of linear dynamical systems on vector bundles appeared rather extensive already in the past.Presently we propose to study perturbations of this linear dynamics The perturbed dynamical system which we shallconsider is no longer linear.while the properties to be studied will be still global in general.Moreover.we are interested in the nonuniformly hyperbolic properties.In this paper,we set an appropriate definition for such perturbations.Though it appearssome what not quite usual yet has deeper root in standard systens of differential equations in the theory of differentiable dynamical systens The general problen is to see which property of the original given by the dynamical system is persistent when a perturbation takes place.The whole contenl of the paper is deyoted to establishinga theorem of this sort.
文摘Let （X, G（X）, m） be a probability space with a-algebra G（X） and probability measure m. The set V in G is called P-admissible, provided that for any positive integer n and positive-measure set Vn∈ contained in V, there exists a Zn∈G such that Zn belong to Vn and 0 〈 m（Zn） 〈 1/n. Let T be an ergodic automorphism of （X, G） preserving m, and A belong to the space of linear measurable symplectic cocycles
基金supported by the National Natural Science Foundation of USA(NSF-DMS 0901448)
文摘This paper provides a new approach to study the solutions of a class of generalized Jazobi equations associated with the linearization of certain singular flows on Riemannian manifolds with dimension n ＋ 1. A new class of generalized differential operators is defined. We investigate the kernel of the corresponding maximal operators by applying operator theory. It is shown that all nontrivial solutions to the generalized Jacobi equation are hyperbolic, in which there are n dimension solutions with exponential-decaying amplitude.
基金supported by Math Vision 2020 Project by the National Research Foundation of Korea(NRF)
文摘Let γ be a hyperbolic closed orbit of a C1 vector field X on a compact C∞ manifold M of dimension n ≥ 3, and let Hx（γ） be the homoclinic class of X containing γ. In this paper, we provethat Cl-generically, if Hx （γ） is expansive and isolated, then it is hyperbolic.
基金Supported by Ministerio de Ciencia e Innovación of Spain(Grant No.MTM 2009-07800)a grant from Consejo Nacional De Ciencia Y Tecnologia of México(Grant No.CONACYT-UAG I0110/62/10)
文摘To decide when a graph is Gromov hyperbolic is,in general,a very hard problem.In this paper,we solve this problem for the set of short graphs（in an informal way,a graph G is r-short if the shortcuts in the cycles of G have length less than r）：an r-short graph G is hyperbolic if and only if S9r（G）is finite,where SR（G）：=sup{L（C）：C is an R-isometric cycle in G}and we say that a cycle C is R-isometric if dC（x,y）≤dG（x,y）＋R for every x,y∈C.
文摘We generalize Bangert＇s non-hyperbolicity result for uniformly tamed almost complex structures on standard symplectic R^2n to asymptotically standard symplectic manifolds.
基金supported by the National Natural Science Foundation of China (No. 11072141)the Shanghai Program for Innovative Research Team in Universities+1 种基金the University Research Committee of the University of Hong Kong (No. 201007176059)the Outstanding Researcher Award from the University of Hong Kong
文摘A standard conservation form is derived in this paper.The hyperbolicity of Helbing＇s fluid dynamic traffic flow model is proved,which is essential to the general analytical and numerical study of this model.On the basis of this conservation form,a local discontinuous Galerkin scheme is designed to solve the resulting system efficiently.The evolution of an unstable equilibrium traffic state leading to a stable stop-and-go traveling wave is simulated.This simulation also verifies that the model is truly improved by the introduction of the modified diffusion coefficients,and thus helps to protect vehicles from collisions and avoide the appearance of the extremely large density.
文摘In this paper we give a classification of special endomorphisms of nil-manifolds:Let f:N/Γ→N/Γbe a covering map of a nil-manifold and denote by A:N/Γ→N/Γthe nil-endomorphism which is homotopic to f.If f is a special TA-map,then A is a hyperbolic nil-endomorphism and f is topologically conjugate to A.
