A three-degree-of-freedom vibro-bounce system is considered. The disturbed map of period one single-impact motion is derived analytically. A center manifold theorem technique is applied to reduce the Poincaré map...A three-degree-of-freedom vibro-bounce system is considered. The disturbed map of period one single-impact motion is derived analytically. A center manifold theorem technique is applied to reduce the Poincaré map to a three-dimensional one, and the normal form map associated with Hopf-flip bifurcation is obtained. Dynamical behavior of the system, near the point of codimension two bifurcation, is investigated by using qualitative analysis and numerical simulation. It is found that near the point of Hopf-flip bifurcation there exists not only Hopf bifurcation of period one single-impact motion, but also Hopf bifurcation of period two double-impact motion. The results from simulation show that there exists an interesting torus doubling bifurcation near the codimension two bifurcation. The torus doubling bifurcation makes the quasi-periodic attractor associated with period one single-impact motion transform to the other quasi-periodic attractor represented by two attracting closed circles. The torus bifurcation is qualitatively different from the typical torus doubling bifurcation occurring in the vibro-impact systems. Different routes from period one single-impact motion to chaos are observed by numerical simulation.展开更多
A vibro-impact forming machine with double masses is considered. The components of the vibrating system collide with each other. Such models play an important role in the studies of dynamics of mechanical systems with...A vibro-impact forming machine with double masses is considered. The components of the vibrating system collide with each other. Such models play an important role in the studies of dynamics of mechanical systems with impacting components. The Poincaré section associated with the state of the impact-forming system, just immediately after the impact, is chosen, and the period n single-impact motion and its disturbed map are derived analytically. A center manifold theorem technique is applied to reduce the Poincaré map to a two-dimensional map, and the normal form map associated with codimension two bifurcation of 1：2 resonance is obtained, Unfolding of the normal form map is analyzed. Dynamical behavior of the impact-forming system, near the point of codimension two bifurcation, is investigated by using qualitative analyses and numerical simulation. Near the point of codimension two bifurcation there exists not only Neimark-Sacker bifurcation associated with period one single-impact motion, but also Neimark-Sacker bifurcation of period two double-impact motion. Transition of different forms of fixed points of single-impact periodic orbits, near the bifurcation point, is demonstrated, and different routes from periodic impact motions to chaos are also discussed.展开更多
This paper deals with the optimal control problems of systems governed by a parabolic variational inequality coupled with a semilinear parabolic differential equations. The maximum principle and some kind of approxima...This paper deals with the optimal control problems of systems governed by a parabolic variational inequality coupled with a semilinear parabolic differential equations. The maximum principle and some kind of approximate controllability are studied.展开更多
Bifurcation problems of a spring-mass system vibrating against an infinite large plane are studied in this paper. It is shown that there exist phenomena of codimension two bifurcations when the ratios of frequencies a...Bifurcation problems of a spring-mass system vibrating against an infinite large plane are studied in this paper. It is shown that there exist phenomena of codimension two bifurcations when the ratios of frequencies are in the neigborhood of the same special values and the coefficient of restitution approach unity. By theory of normal forms, we reduce Poincare maps to normal forms.and find flip bifurcations, Hopf bifurcations of fixed points and that of period two points The theoretical solutions are verified by numerical computations.展开更多
The existence and stability ol periodic solutions for the two-dimensional system x' = f(x)+?g(x ,a), 0<ε<<1 ,a?R whose unperturbed systemis Hamiltonian can be decided by using the signs of Melnikov's...The existence and stability ol periodic solutions for the two-dimensional system x' = f(x)+?g(x ,a), 0<ε<<1 ,a?R whose unperturbed systemis Hamiltonian can be decided by using the signs of Melnikov's function. The results can be applied to the construction of phase portraits in the bifurcation set of codimension two bifurcations of flows with doublezero eigenvalues.展开更多
In this paper,we study the determinacy of real smooth map-germs and obtain severalresults about the sufficient condition and necessary condition of a special determinacy .and aboutthe exact order to determinacy.We als...In this paper,we study the determinacy of real smooth map-germs and obtain severalresults about the sufficient condition and necessary condition of a special determinacy .and aboutthe exact order to determinacy.We also obtain a result on the Zeeman's conjecture of smooth function-germs. This resultis a generalization of Siersma's lemma.展开更多
Normal form theory is a very effective method when we study degenerate bifurcations of nonlinear dynamical systems. In this paper by using adjoint operator method, normal forms of order 3 and 4 for nonlinear dynamical...Normal form theory is a very effective method when we study degenerate bifurcations of nonlinear dynamical systems. In this paper by using adjoint operator method, normal forms of order 3 and 4 for nonlinear dynamical system with nilpotent linear part and Z(2)-asymmetry are computed. According to normal forms obtained, universal unfoldings for some degenerate bifurcation cases of codimension 3 and simple global characterizations, are studied.展开更多
By virtue of the singular point theory for one-dimension diffusionprocess and the stochastic averaging approach of energy envelop, thebifurcation behavior of a homoclinic bifurcation system, which is inthe presence of...By virtue of the singular point theory for one-dimension diffusionprocess and the stochastic averaging approach of energy envelop, thebifurcation behavior of a homoclinic bifurcation system, which is inthe presence of parametric white noise and is concealed behind acodimension two bifurcation point, is investigated in this paper.展开更多
In this paper,we derive global bounds for the H?lder norms of the gradient of solutions of graphic mean curvature flows with boundaries of arbitrary codimension.
The Morris-Lecar (ML) neuronal model with current-feedback control is considered as a typical fast-slow dynamical system to study the combined influences of the reversal potential VCa of Ca2+ and the feedback current ...The Morris-Lecar (ML) neuronal model with current-feedback control is considered as a typical fast-slow dynamical system to study the combined influences of the reversal potential VCa of Ca2+ and the feedback current I on the generation and transition of different bursting oscillations. Two-parameter bifurcation analysis of the fast subsystem is performed in the parameter (I, VCa)-plane at first. Three important codimension-2 bifurcation points and some codimension-1 bifurcation curves are obtained which enable one to determine the parameter regions for different types of bursting. Next, we further divide the control parameter (V0, VCa)-plane into five different bursting regions, namely, the "fold/fold" bursting region R1, the "fold/Hopf" bursting region R2, the "fold/homoclinic" bursting region R3, the "subHopf/homoclinic" bursting region R4 and the "subHopf/subHopf" bursting region R5, as well as a silence region R6. Codimension-1 and -2 bifurcations are responsible for explanation of transition mechanisms between different types of bursting. The results are instructive for further understanding the dynamical behavior and mechanisms of complex firing activities and information processing in biological nervous systems.展开更多
First it is proved that both the integral of the divergence and the Melnikov function are invariants of the C2 transformation. Then, the problem of the planar homoclinic bifurcation with codimension 3 is considered. I...First it is proved that both the integral of the divergence and the Melnikov function are invariants of the C2 transformation. Then, the problem of the planar homoclinic bifurcation with codimension 3 is considered. It is proved that, in a small neighborhood of the origin in the parameter space of a Cr (r≥5) system, there exist exactly two Cr-1 semi- stable- limit- cycle branching surfaces, and their common boundary is a unique Cr-1 three-multiple- limit-cycle branching curve. The bifurcation pictures and the asymptotic expansions of the bifurcation functions are given. The stability criterion for the homoclinic loop is also obtained when the integral of the divergence is zero. The proof of the auxiliary theorems will be presented in [16].展开更多
We prove that if A is a finite-dimensional associative H-comodule algebra over a field F for some involutory Hopf algebra H not necessarily finite-dimensional, where either char F = 0 or char F 〉 dim A, then the Jaco...We prove that if A is a finite-dimensional associative H-comodule algebra over a field F for some involutory Hopf algebra H not necessarily finite-dimensional, where either char F = 0 or char F 〉 dim A, then the Jacobson radical J（A） is an H-subcomodule of A. In particular, if A is a finite-dimensional associative Mgebra over such a field F, graded by any group, then the Jacobson radical J（A） is a graded ideal of A. Analogous results hold for nilpotent and solvable radicals of finite-dimensional Lie algebras over a field of characteristic 0. We use the results obtained to prove the analog of Amitsur＇s conjecture for graded polynomial identities of finite-dimensional associative algebras over a field of characteristic 0, graded by any group. In addition, we provide a criterion for graded simplicity of an associative algebra in terms of graded codimensions.展开更多
Let M be an approximately finite codimensional quasi-invariant subspace of the Fock space. This paper gives a formula to calculate the codimension of such spaces and uses this formula to study the structure of quasi-i...Let M be an approximately finite codimensional quasi-invariant subspace of the Fock space. This paper gives a formula to calculate the codimension of such spaces and uses this formula to study the structure of quasi-invariant subspaces of the Fock space. Especially, as one of applications, it is showed that the analogue of Beurling's theorem is not true for the Fock space L_a^2 in the case of n > 2.展开更多
Ⅰ. INTRODUCTION We are interested in the following problem of differential topology: Let W be a manifold with boundary M. When does an immersion f: M→X extend to an immersion of
Ⅰ. INTRODUCTIONThroughout this note, all manifolds are assumed to be smooth and closed. We refer to [1]—[3] for the definitions and basic properties of higher order tangent bundles and pth order nonsingular immersio...Ⅰ. INTRODUCTIONThroughout this note, all manifolds are assumed to be smooth and closed. We refer to [1]—[3] for the definitions and basic properties of higher order tangent bundles and pth order nonsingular immersions (p-immersions). Let M be an n-manifold. We denote展开更多
Denote by V<sub>f</sub><sup>p</sup> the set of homotopy classes of codimensiom m framings of the p thorder stable normal bundle of a map f:M<sup>n</sup>→N<sup>v（n,p）+m</...Denote by V<sub>f</sub><sup>p</sup> the set of homotopy classes of codimensiom m framings of the p thorder stable normal bundle of a map f:M<sup>n</sup>→N<sup>v（n,p）+m</sup>.Where 3≤m=n-1 or m=n-2 butn=0,1 mod 4.We determine in this paper the set V<sub>f</sub><sup>p</sup> and then the set of p-homotopyclasses of p-immersions homotopic to f for some special cases.展开更多
The recent years have seen a beautiful breakthrough culminating in a comprehensive understanding of certain scale-invariant properties of n-1 dimensional sets across analysis, geometric measure theory, and PDEs. The p...The recent years have seen a beautiful breakthrough culminating in a comprehensive understanding of certain scale-invariant properties of n-1 dimensional sets across analysis, geometric measure theory, and PDEs. The present paper surveys the first steps of a program recently launched by the authors and aimed at the new PDE approach to sets with lower dimensional boundaries. We define a suitable class of degenerate elliptic operators, explain our intuition, motivation, and goals, and present the first results regarding absolute continuity of the emerging elliptic measure with respect to the surface measure analogous to the classical theorems of C. Kenig and his collaborators in the case of co-dimension one.展开更多
Let X C P^NC be an n-dimensional nondegenerate smooth projective variety containing an mdimensional subvariety Y.Assume that either m〉n/2 and X is a complete intersection or that m≥ N2.We show deg（X）｜deg（Y）and ...Let X C P^NC be an n-dimensional nondegenerate smooth projective variety containing an mdimensional subvariety Y.Assume that either m〉n/2 and X is a complete intersection or that m≥ N2.We show deg（X）｜deg（Y）and codim Y Y ≥codimPN X,where Y is the linear span of Y.These bounds are sharp.As an application,we classify smooth projective n-dimensional quadratic varieties swept out by m≥[n/2]＋1 dimensional quadrics passing through one point.展开更多
基金The project supported by the National Natural Scicnce Foundation of China(10172042,50475109)the Natural Science Foundation of Gansu Province Government of China(ZS-031-A25-007-Z(key item))
文摘A three-degree-of-freedom vibro-bounce system is considered. The disturbed map of period one single-impact motion is derived analytically. A center manifold theorem technique is applied to reduce the Poincaré map to a three-dimensional one, and the normal form map associated with Hopf-flip bifurcation is obtained. Dynamical behavior of the system, near the point of codimension two bifurcation, is investigated by using qualitative analysis and numerical simulation. It is found that near the point of Hopf-flip bifurcation there exists not only Hopf bifurcation of period one single-impact motion, but also Hopf bifurcation of period two double-impact motion. The results from simulation show that there exists an interesting torus doubling bifurcation near the codimension two bifurcation. The torus doubling bifurcation makes the quasi-periodic attractor associated with period one single-impact motion transform to the other quasi-periodic attractor represented by two attracting closed circles. The torus bifurcation is qualitatively different from the typical torus doubling bifurcation occurring in the vibro-impact systems. Different routes from period one single-impact motion to chaos are observed by numerical simulation.
基金The project supported by the National Natural Science Foundation of China （10572055, 50475109） and the Natural Science Foundation of Gansu Province Government of China （3ZS051-A25-030（key item）） The English text was polished by Keren Wang.
文摘A vibro-impact forming machine with double masses is considered. The components of the vibrating system collide with each other. Such models play an important role in the studies of dynamics of mechanical systems with impacting components. The Poincaré section associated with the state of the impact-forming system, just immediately after the impact, is chosen, and the period n single-impact motion and its disturbed map are derived analytically. A center manifold theorem technique is applied to reduce the Poincaré map to a two-dimensional map, and the normal form map associated with codimension two bifurcation of 1：2 resonance is obtained, Unfolding of the normal form map is analyzed. Dynamical behavior of the impact-forming system, near the point of codimension two bifurcation, is investigated by using qualitative analyses and numerical simulation. Near the point of codimension two bifurcation there exists not only Neimark-Sacker bifurcation associated with period one single-impact motion, but also Neimark-Sacker bifurcation of period two double-impact motion. Transition of different forms of fixed points of single-impact periodic orbits, near the bifurcation point, is demonstrated, and different routes from periodic impact motions to chaos are also discussed.
基金This work was partially supported by the NutionalNatural Science Foundation of China
文摘This paper deals with the optimal control problems of systems governed by a parabolic variational inequality coupled with a semilinear parabolic differential equations. The maximum principle and some kind of approximate controllability are studied.
文摘Bifurcation problems of a spring-mass system vibrating against an infinite large plane are studied in this paper. It is shown that there exist phenomena of codimension two bifurcations when the ratios of frequencies are in the neigborhood of the same special values and the coefficient of restitution approach unity. By theory of normal forms, we reduce Poincare maps to normal forms.and find flip bifurcations, Hopf bifurcations of fixed points and that of period two points The theoretical solutions are verified by numerical computations.
基金The project is supported by the National Natural Science Foundation of China
文摘The existence and stability ol periodic solutions for the two-dimensional system x' = f(x)+?g(x ,a), 0<ε<<1 ,a?R whose unperturbed systemis Hamiltonian can be decided by using the signs of Melnikov's function. The results can be applied to the construction of phase portraits in the bifurcation set of codimension two bifurcations of flows with doublezero eigenvalues.
文摘In this paper,we study the determinacy of real smooth map-germs and obtain severalresults about the sufficient condition and necessary condition of a special determinacy .and aboutthe exact order to determinacy.We also obtain a result on the Zeeman's conjecture of smooth function-germs. This resultis a generalization of Siersma's lemma.
文摘Normal form theory is a very effective method when we study degenerate bifurcations of nonlinear dynamical systems. In this paper by using adjoint operator method, normal forms of order 3 and 4 for nonlinear dynamical system with nilpotent linear part and Z(2)-asymmetry are computed. According to normal forms obtained, universal unfoldings for some degenerate bifurcation cases of codimension 3 and simple global characterizations, are studied.
基金Supported by the National Science Foundation of China under Grant No.19602016
文摘By virtue of the singular point theory for one-dimension diffusionprocess and the stochastic averaging approach of energy envelop, thebifurcation behavior of a homoclinic bifurcation system, which is inthe presence of parametric white noise and is concealed behind acodimension two bifurcation point, is investigated in this paper.
基金supported by National Natural Science Foundation of China(Grant No.12371053)。
文摘In this paper,we derive global bounds for the H?lder norms of the gradient of solutions of graphic mean curvature flows with boundaries of arbitrary codimension.
基金Supported by the National Natural Science Foundation of China (Grant Nos. 10872014 and 10702002)
文摘The Morris-Lecar (ML) neuronal model with current-feedback control is considered as a typical fast-slow dynamical system to study the combined influences of the reversal potential VCa of Ca2+ and the feedback current I on the generation and transition of different bursting oscillations. Two-parameter bifurcation analysis of the fast subsystem is performed in the parameter (I, VCa)-plane at first. Three important codimension-2 bifurcation points and some codimension-1 bifurcation curves are obtained which enable one to determine the parameter regions for different types of bursting. Next, we further divide the control parameter (V0, VCa)-plane into five different bursting regions, namely, the "fold/fold" bursting region R1, the "fold/Hopf" bursting region R2, the "fold/homoclinic" bursting region R3, the "subHopf/homoclinic" bursting region R4 and the "subHopf/subHopf" bursting region R5, as well as a silence region R6. Codimension-1 and -2 bifurcations are responsible for explanation of transition mechanisms between different types of bursting. The results are instructive for further understanding the dynamical behavior and mechanisms of complex firing activities and information processing in biological nervous systems.
文摘First it is proved that both the integral of the divergence and the Melnikov function are invariants of the C2 transformation. Then, the problem of the planar homoclinic bifurcation with codimension 3 is considered. It is proved that, in a small neighborhood of the origin in the parameter space of a Cr (r≥5) system, there exist exactly two Cr-1 semi- stable- limit- cycle branching surfaces, and their common boundary is a unique Cr-1 three-multiple- limit-cycle branching curve. The bifurcation pictures and the asymptotic expansions of the bifurcation functions are given. The stability criterion for the homoclinic loop is also obtained when the integral of the divergence is zero. The proof of the auxiliary theorems will be presented in [16].
文摘We prove that if A is a finite-dimensional associative H-comodule algebra over a field F for some involutory Hopf algebra H not necessarily finite-dimensional, where either char F = 0 or char F 〉 dim A, then the Jacobson radical J（A） is an H-subcomodule of A. In particular, if A is a finite-dimensional associative Mgebra over such a field F, graded by any group, then the Jacobson radical J（A） is a graded ideal of A. Analogous results hold for nilpotent and solvable radicals of finite-dimensional Lie algebras over a field of characteristic 0. We use the results obtained to prove the analog of Amitsur＇s conjecture for graded polynomial identities of finite-dimensional associative algebras over a field of characteristic 0, graded by any group. In addition, we provide a criterion for graded simplicity of an associative algebra in terms of graded codimensions.
文摘Let M be an approximately finite codimensional quasi-invariant subspace of the Fock space. This paper gives a formula to calculate the codimension of such spaces and uses this formula to study the structure of quasi-invariant subspaces of the Fock space. Especially, as one of applications, it is showed that the analogue of Beurling's theorem is not true for the Fock space L_a^2 in the case of n > 2.
文摘Ⅰ. INTRODUCTION We are interested in the following problem of differential topology: Let W be a manifold with boundary M. When does an immersion f: M→X extend to an immersion of
文摘In this paper the following problem has been completely solved:when is a map f:P（m‘n）→CP<sup>?</sup> homotopic to an immersion with codimension one or
文摘Ⅰ. INTRODUCTIONThroughout this note, all manifolds are assumed to be smooth and closed. We refer to [1]—[3] for the definitions and basic properties of higher order tangent bundles and pth order nonsingular immersions (p-immersions). Let M be an n-manifold. We denote
文摘Denote by V<sub>f</sub><sup>p</sup> the set of homotopy classes of codimensiom m framings of the p thorder stable normal bundle of a map f:M<sup>n</sup>→N<sup>v（n,p）+m</sup>.Where 3≤m=n-1 or m=n-2 butn=0,1 mod 4.We determine in this paper the set V<sub>f</sub><sup>p</sup> and then the set of p-homotopyclasses of p-immersions homotopic to f for some special cases.
基金partially supported by the ANR,programme blanc GEOMETRYA ANR-12-BS01-0014the European Community Marie Curie grant MANET 607643 and H2020 grant GHAIA 777822+5 种基金the Simons Collaborations in MPS grant 601941,GDsupported by the NSF INSPIRE Award DMS1344235NSF CAREER Award DMS 1220089the NSF RAISE-TAQ grant DMS 1839077the Simons Fellowshipthe Simons Foundation grant 563916,SM
文摘The recent years have seen a beautiful breakthrough culminating in a comprehensive understanding of certain scale-invariant properties of n-1 dimensional sets across analysis, geometric measure theory, and PDEs. The present paper surveys the first steps of a program recently launched by the authors and aimed at the new PDE approach to sets with lower dimensional boundaries. We define a suitable class of degenerate elliptic operators, explain our intuition, motivation, and goals, and present the first results regarding absolute continuity of the emerging elliptic measure with respect to the surface measure analogous to the classical theorems of C. Kenig and his collaborators in the case of co-dimension one.
文摘Let X C P^NC be an n-dimensional nondegenerate smooth projective variety containing an mdimensional subvariety Y.Assume that either m〉n/2 and X is a complete intersection or that m≥ N2.We show deg（X）｜deg（Y）and codim Y Y ≥codimPN X,where Y is the linear span of Y.These bounds are sharp.As an application,we classify smooth projective n-dimensional quadratic varieties swept out by m≥[n/2]＋1 dimensional quadrics passing through one point.
