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
The coefficients of the simplest normal forms of both high-dimensional generalized Hopf and high-dimensional Hopf bifurcation systems were discussed using the adjoint operator method. A particular nonlinear scaling an...The coefficients of the simplest normal forms of both high-dimensional generalized Hopf and high-dimensional Hopf bifurcation systems were discussed using the adjoint operator method. A particular nonlinear scaling and an inner product were introduced in the space of homogeneous polynomials. Theorems were established for the explicit expression of the simplest normal forms in terms of the coefficients of both the conventional normal forms of Hopf and generalized Hopf bifurcation systems. A symbolic manipulation was designed to perform the calculation of the coefficients of the simplest normal forms using Mathematica. The original ordinary differential equation was required in the input and the simplest normal form could be obtained as the output. Finally, the simplest normal forms of 6-dimensional generalized Hopf singularity of type 2 and 5-dimensional Hopf bifurcation system were discussed by executing the program. The output showed that the 5th- and 9th-order terms remained in 6-dimensional generalized Hopf singularity of type 2 and the 3rd- and 5th-order terms remained in 5-dimensional Hopf bifurcation system.展开更多
The simplest normal form of resonant double Hopf bifurcation was studied based on Lie operator. The coefficients of the simplest normal forms of resonant double Hopf bifurcation and the nonlinear transformations in te...The simplest normal form of resonant double Hopf bifurcation was studied based on Lie operator. The coefficients of the simplest normal forms of resonant double Hopf bifurcation and the nonlinear transformations in terms of the original system coefficients were given explicitly. The nonlinear transformations were used for reducing the lower- and higher-order normal forms, and the rank of system matrix was used to determine the coefficient of normal form which could be reduced. These make the gained normal form simpler than the traditional one. A general program was compiled with Mathematica. This program can compute the simplest normal form of resonant double Hopf bifurcation and the non-resonant form up to the 7th order.展开更多
The normal forms of generalized Neimark-Sacker bifurcation are extensively studied using normal form theory of dynamic system. It is well known that if the normal forms of the generalized Neimark-Sacker bifurcation ar...The normal forms of generalized Neimark-Sacker bifurcation are extensively studied using normal form theory of dynamic system. It is well known that if the normal forms of the generalized Neimark-Sacker bifurcation are expressed in polar coordinates, then all odd order terms must, in general, remain in the normal forms. In this paper, five theorems are presented to show that the conventional Neimark-Sacker bifurcation can be further simplified. The simplest normal forms of generalized Neimark-Sacker bifurcation are calculated. Based on the conventional normal form, using appropriate nonlinear transformations, it is found that the generalized Neimark-Sacker bifurcation has at most two nonlinear terms remaining in the amplitude equations of the simplest normal forms up to any order. There are two kinds of simplest normal forms. Their algebraic expression formulas of the simplest normal forms in terms of the coefficients of the generalized Neimark-Sacker bifurcation systems are given.展开更多
In this paper we first summarize our results published in recent years and their sketch proofs on local integrability,which are on the characterization of local integrability and on the existence of analytic normaliza...In this paper we first summarize our results published in recent years and their sketch proofs on local integrability,which are on the characterization of local integrability and on the existence of analytic normalization of analytically integrable differential systems. Then we present a new result on the equivalent characterization of the existence of the first integrals of an analytic differential systems near a nonhyperbolic singularity. Finally we pose some open problems on this subject.展开更多
The following problem is called the everywhere-cover problem:“Given a set of dependencies over a database scheme,is the set of dependencies explicitly given for each relation scheme equivalent to the dependencies imp...The following problem is called the everywhere-cover problem:“Given a set of dependencies over a database scheme,is the set of dependencies explicitly given for each relation scheme equivalent to the dependencies implied for that relation scheme?”It is shown that when the everywhere-cover problem has a ‘yes’ answer,examining only the dependencies explicitly given will suffice to test 3NF,BCNF and 4NF of a database scheme.But this does not hold for 2NF.Consequently,in such cases,tests of BCNF and 4NF all take polynomial time.Then a proof is given that test of 3NF of a database scheme is Co-NP-complete,and from this result it is shown that everywhere-cover is also Co-NP-complete when only functional dependencies are allowed.These results lead to doubt the truth of the well believed conjec- ture that no polynomial time algorithm for designing a Iossless BCNF database scheme is likely to exist.展开更多
In this paper the concept of absolute observability of nonlinear control systems is advanced.Different from the linear time-invariant version, there are different definitions of absolute observability for nonlinear co...In this paper the concept of absolute observability of nonlinear control systems is advanced.Different from the linear time-invariant version, there are different definitions of absolute observability for nonlinear control systems.Two algorithms for maximal absolutely observable subsystems are given.Correspondingly,there are two relevant normal forms.The relations with the largest controlled invariant distribution contained in kerdh,zero dynamics etc.,are discussed from the view point of maximal absolute observabilities.展开更多
In this paper,we prove that for every symplectic matrix M possessing eigenvalues on the unit circle,there exists a symplectic matrix P such that P<sup>-1</sup> MP is a symplectic matrix of the normal forms...In this paper,we prove that for every symplectic matrix M possessing eigenvalues on the unit circle,there exists a symplectic matrix P such that P<sup>-1</sup> MP is a symplectic matrix of the normal forms defined in this paper.展开更多
The paper is devoted to the theory of normal forms of main symbols for linear second order partial differential equations on the plane.We discuss the results obtained in the last decades and some problems,which are im...The paper is devoted to the theory of normal forms of main symbols for linear second order partial differential equations on the plane.We discuss the results obtained in the last decades and some problems,which are important both for the development of this theory and the applications.The reduction theorem,which was used to obtain many of recent results in the theory,is included in the paper in the parametric form together with proof.There is a feeling that the theorem still has potential to get progress in the solution of open problems in the theory.展开更多
In this paper, we develop an efficient approach to compute the equivariant normal form of delay differential equations with parameters in the presence of symmetry. We present and justify a process that involves center...In this paper, we develop an efficient approach to compute the equivariant normal form of delay differential equations with parameters in the presence of symmetry. We present and justify a process that involves center manifold reduction and normalization preserving the symmetry, and that yields normal forms explicitly in terms of the coefficients of the original system. We observe that the form of the reduced vector field relies only on the information of the linearized system at the critical point and on the inherent symmetry, and the normal forms give critical information about not only the existence but also the stability and direction of bifurcated spatiotemporal patterns. We illustrate our general results by some applications to fold bifurcation, equivariant Hopf bifurcation and Hopf-Hopf interaction, with a detailed case study of additive neurons with delayed feedback.展开更多
We consider normal forms of real vector fields near periodic orbits and provide sufficient conditions for their smooth linearization. In addition, the main results also assert the existence of vertical local foliation...We consider normal forms of real vector fields near periodic orbits and provide sufficient conditions for their smooth linearization. In addition, the main results also assert the existence of vertical local foliations, whose leaves are all transversal to the periodic orbit.展开更多
A brief introduction is given to the topic of Smith normal forms of incidence matrices. A general discussion of techniques is illustrated by some classical examples. Some recent advances are described and the limits o...A brief introduction is given to the topic of Smith normal forms of incidence matrices. A general discussion of techniques is illustrated by some classical examples. Some recent advances are described and the limits of our current understanding are indicated.展开更多
In this paper, based on the invariant subspace theory and adjoint operator concept of linear operator, a new matrix representation method is proposed to calculate the normal forms of n order general nonlinear dyna...In this paper, based on the invariant subspace theory and adjoint operator concept of linear operator, a new matrix representation method is proposed to calculate the normal forms of n order general nonlinear dynamic systems. In the method, there is no need to determine the structure of the class of normal forms in advance. Because the subspace is not related to the dimensions of the system and the order of the normal forms directly, it is determined only by a given vector field. So the normal forms with high orders and dimensions can be calculated by the method without difficulties. In this paper, is used the method for selecting the minimal subspace and solving homological equations in the subspace, the examples show that the method is very effective.展开更多
This paper puts forward a complex inner product averaging method for calculating normal form of ODE. Compared with conventional averaging method, the theoretic analytical process has such simple forms as to realize co...This paper puts forward a complex inner product averaging method for calculating normal form of ODE. Compared with conventional averaging method, the theoretic analytical process has such simple forms as to realize computer program easily. Results can be applied in both autonomous and non-autonomous systems. At last, an example is resolved to verify the method.展开更多
By using the Smith normal form of polynomial matrix and algebraic methods, this paper discusses the solvability for the linear matrix equation A iXB i=C over a field, and obtains the explicit formulas of general sol...By using the Smith normal form of polynomial matrix and algebraic methods, this paper discusses the solvability for the linear matrix equation A iXB i=C over a field, and obtains the explicit formulas of general solution or unique solution.展开更多
The modified normal form approach presented by ZHANG Wei-yi, K Huseyin and CHEN Yu-shu is further extended and a different procedure is introduced which lends itself readily to symbolic calculations, like MAPLE. This ...The modified normal form approach presented by ZHANG Wei-yi, K Huseyin and CHEN Yu-shu is further extended and a different procedure is introduced which lends itself readily to symbolic calculations, like MAPLE. This provides a number of significant advantages over the previous approach, and facilitates the associated calculations. To illustrate the new approach, three examples are presented.展开更多
On the basis of the method proposed in [1], the paper gives the method for finding the normal form of nonsemi-simple bifurcation problems. As an example, it analyses the normal form of a general nonlinear dynamical sy...On the basis of the method proposed in [1], the paper gives the method for finding the normal form of nonsemi-simple bifurcation problems. As an example, it analyses the normal form of a general nonlinear dynamical system with the nonsemi-simple double zero eigenvalues, and gives out the expression for the coefficients in the normal form by using those in the original system.展开更多
We present a new method to calculate the focal value of ordinary differential equation by applying the theorem defined the relationship between the normal form and focal value,with the help of a symbolic computation l...We present a new method to calculate the focal value of ordinary differential equation by applying the theorem defined the relationship between the normal form and focal value,with the help of a symbolic computation language M ATHEMATICA,and extending the matrix representation method.This method can be used to calculate the focal value of any high order terms.This method has been verified by an example.The advantage of this method is simple and more readily applicable.the result is directly obtained by substitution.展开更多
Using the normally ordered Gaussian form of the Wigner operator we recapitulate the quantum phase space representation, we derive a new formula for searching for the classical correspondence of quantum mechanical oper...Using the normally ordered Gaussian form of the Wigner operator we recapitulate the quantum phase space representation, we derive a new formula for searching for the classical correspondence of quantum mechanical operators; we also show that if there exists the eigenvector ｜q〉λ,v of linear combination of the coordinate and momentum operator, （λQ ＋ vP）, where λ,v are real numbers, and ｜q〉λv is complete, then the projector ｜q〉λ,vλ,v〈q｜ must be the Radon transform of Wigner operator. This approach seems concise and physical appealing.展开更多
The reaction diffusion Gray-Scott model with time delay is put forward with the assumption of Neumann boundary condition is satisfied. Based on the Turing bifurcation condition, the Turing curves on two parameter plan...The reaction diffusion Gray-Scott model with time delay is put forward with the assumption of Neumann boundary condition is satisfied. Based on the Turing bifurcation condition, the Turing curves on two parameter plane are discussed without time delay. The normal form is computed via applying Lyapunov-Schmidt reduction method in system of PDE, and the bifurcating direction of pitchfork bifurcation underlying codimension-1 singularity of Turing point is computed. The continuation of Pitchfork bifurcation is simulated with varying free parameter continuously near the turing point, which is in coincidence with the theoritical analysis results. The wave pattern formation in the case of turing instability is also simulated which discover Turing oscillation phenomena from periodicity to irregularity.展开更多
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National Natural Science Foundation of China (No 10372068)
文摘The coefficients of the simplest normal forms of both high-dimensional generalized Hopf and high-dimensional Hopf bifurcation systems were discussed using the adjoint operator method. A particular nonlinear scaling and an inner product were introduced in the space of homogeneous polynomials. Theorems were established for the explicit expression of the simplest normal forms in terms of the coefficients of both the conventional normal forms of Hopf and generalized Hopf bifurcation systems. A symbolic manipulation was designed to perform the calculation of the coefficients of the simplest normal forms using Mathematica. The original ordinary differential equation was required in the input and the simplest normal form could be obtained as the output. Finally, the simplest normal forms of 6-dimensional generalized Hopf singularity of type 2 and 5-dimensional Hopf bifurcation system were discussed by executing the program. The output showed that the 5th- and 9th-order terms remained in 6-dimensional generalized Hopf singularity of type 2 and the 3rd- and 5th-order terms remained in 5-dimensional Hopf bifurcation system.
基金Supported by National Natural Science Foundation of China(No. 10372068).
文摘The simplest normal form of resonant double Hopf bifurcation was studied based on Lie operator. The coefficients of the simplest normal forms of resonant double Hopf bifurcation and the nonlinear transformations in terms of the original system coefficients were given explicitly. The nonlinear transformations were used for reducing the lower- and higher-order normal forms, and the rank of system matrix was used to determine the coefficient of normal form which could be reduced. These make the gained normal form simpler than the traditional one. A general program was compiled with Mathematica. This program can compute the simplest normal form of resonant double Hopf bifurcation and the non-resonant form up to the 7th order.
基金Supported by National Natural Science Foundation of China (No10872141)Doctoral Foundation of Ministry of Education of China (No20060056005)Natural Science Foundation of Tianjin University of Science and Technology (No20070210)
文摘The normal forms of generalized Neimark-Sacker bifurcation are extensively studied using normal form theory of dynamic system. It is well known that if the normal forms of the generalized Neimark-Sacker bifurcation are expressed in polar coordinates, then all odd order terms must, in general, remain in the normal forms. In this paper, five theorems are presented to show that the conventional Neimark-Sacker bifurcation can be further simplified. The simplest normal forms of generalized Neimark-Sacker bifurcation are calculated. Based on the conventional normal form, using appropriate nonlinear transformations, it is found that the generalized Neimark-Sacker bifurcation has at most two nonlinear terms remaining in the amplitude equations of the simplest normal forms up to any order. There are two kinds of simplest normal forms. Their algebraic expression formulas of the simplest normal forms in terms of the coefficients of the generalized Neimark-Sacker bifurcation systems are given.
基金supported by the NNSF of China Grant 11271252the RFDP of Higher Education of China grant 20110073110054the FP7-PEOPLE-2012-IRSES-316338 of Europe
文摘In this paper we first summarize our results published in recent years and their sketch proofs on local integrability,which are on the characterization of local integrability and on the existence of analytic normalization of analytically integrable differential systems. Then we present a new result on the equivalent characterization of the existence of the first integrals of an analytic differential systems near a nonhyperbolic singularity. Finally we pose some open problems on this subject.
基金Supported by the National Natural Science Foundation of China.
文摘The following problem is called the everywhere-cover problem:“Given a set of dependencies over a database scheme,is the set of dependencies explicitly given for each relation scheme equivalent to the dependencies implied for that relation scheme?”It is shown that when the everywhere-cover problem has a ‘yes’ answer,examining only the dependencies explicitly given will suffice to test 3NF,BCNF and 4NF of a database scheme.But this does not hold for 2NF.Consequently,in such cases,tests of BCNF and 4NF all take polynomial time.Then a proof is given that test of 3NF of a database scheme is Co-NP-complete,and from this result it is shown that everywhere-cover is also Co-NP-complete when only functional dependencies are allowed.These results lead to doubt the truth of the well believed conjec- ture that no polynomial time algorithm for designing a Iossless BCNF database scheme is likely to exist.
文摘In this paper the concept of absolute observability of nonlinear control systems is advanced.Different from the linear time-invariant version, there are different definitions of absolute observability for nonlinear control systems.Two algorithms for maximal absolutely observable subsystems are given.Correspondingly,there are two relevant normal forms.The relations with the largest controlled invariant distribution contained in kerdh,zero dynamics etc.,are discussed from the view point of maximal absolute observabilities.
基金Partially supported by the NSF,MCSEC of China the Qiu Shi Sci.Tech.Foundation
文摘In this paper,we prove that for every symplectic matrix M possessing eigenvalues on the unit circle,there exists a symplectic matrix P such that P<sup>-1</sup> MP is a symplectic matrix of the normal forms defined in this paper.
基金supported by the Ministry of Education and Science of the Russian Federation (Grant No. 1.638.2016/FPM)Open access funding provided by International Institute for Applied Systems Analysis (IIASA)
文摘The paper is devoted to the theory of normal forms of main symbols for linear second order partial differential equations on the plane.We discuss the results obtained in the last decades and some problems,which are important both for the development of this theory and the applications.The reduction theorem,which was used to obtain many of recent results in the theory,is included in the paper in the parametric form together with proof.There is a feeling that the theorem still has potential to get progress in the solution of open problems in the theory.
基金supported by NSFC(Grant No.10971057)the Key Project of Chinese Ministry of Education(Grant No.[2009]41)+4 种基金by Hu'nan Provincial Natural Science Foundation(Grant No.10JJ1001)by the Fundamental Research Funds for the Central Universities,Hu'nan Universitysupported by NSERC of Canadaby ERA Program of Ontariosupported in part by MITACS,CRC,and NSERC of Canada
文摘In this paper, we develop an efficient approach to compute the equivariant normal form of delay differential equations with parameters in the presence of symmetry. We present and justify a process that involves center manifold reduction and normalization preserving the symmetry, and that yields normal forms explicitly in terms of the coefficients of the original system. We observe that the form of the reduced vector field relies only on the information of the linearized system at the critical point and on the inherent symmetry, and the normal forms give critical information about not only the existence but also the stability and direction of bifurcated spatiotemporal patterns. We illustrate our general results by some applications to fold bifurcation, equivariant Hopf bifurcation and Hopf-Hopf interaction, with a detailed case study of additive neurons with delayed feedback.
基金NSF Grant No.10531010 NSF Grant NNSF of China (No.10525104)
文摘We consider normal forms of real vector fields near periodic orbits and provide sufficient conditions for their smooth linearization. In addition, the main results also assert the existence of vertical local foliations, whose leaves are all transversal to the periodic orbit.
基金supported by the Simons Foundation(Grant No.#204181)
文摘A brief introduction is given to the topic of Smith normal forms of incidence matrices. A general discussion of techniques is illustrated by some classical examples. Some recent advances are described and the limits of our current understanding are indicated.
文摘In this paper, based on the invariant subspace theory and adjoint operator concept of linear operator, a new matrix representation method is proposed to calculate the normal forms of n order general nonlinear dynamic systems. In the method, there is no need to determine the structure of the class of normal forms in advance. Because the subspace is not related to the dimensions of the system and the order of the normal forms directly, it is determined only by a given vector field. So the normal forms with high orders and dimensions can be calculated by the method without difficulties. In this paper, is used the method for selecting the minimal subspace and solving homological equations in the subspace, the examples show that the method is very effective.
文摘This paper puts forward a complex inner product averaging method for calculating normal form of ODE. Compared with conventional averaging method, the theoretic analytical process has such simple forms as to realize computer program easily. Results can be applied in both autonomous and non-autonomous systems. At last, an example is resolved to verify the method.
基金the NSF of Hunan Province and the Science and Technology Development Foundation of Xiangtan Polytechnic University
文摘By using the Smith normal form of polynomial matrix and algebraic methods, this paper discusses the solvability for the linear matrix equation A iXB i=C over a field, and obtains the explicit formulas of general solution or unique solution.
文摘The modified normal form approach presented by ZHANG Wei-yi, K Huseyin and CHEN Yu-shu is further extended and a different procedure is introduced which lends itself readily to symbolic calculations, like MAPLE. This provides a number of significant advantages over the previous approach, and facilitates the associated calculations. To illustrate the new approach, three examples are presented.
文摘On the basis of the method proposed in [1], the paper gives the method for finding the normal form of nonsemi-simple bifurcation problems. As an example, it analyses the normal form of a general nonlinear dynamical system with the nonsemi-simple double zero eigenvalues, and gives out the expression for the coefficients in the normal form by using those in the original system.
文摘We present a new method to calculate the focal value of ordinary differential equation by applying the theorem defined the relationship between the normal form and focal value,with the help of a symbolic computation language M ATHEMATICA,and extending the matrix representation method.This method can be used to calculate the focal value of any high order terms.This method has been verified by an example.The advantage of this method is simple and more readily applicable.the result is directly obtained by substitution.
基金Supported by National Natural Science Foundation of China under Grant Nos. 10874174 and 10775097
文摘Using the normally ordered Gaussian form of the Wigner operator we recapitulate the quantum phase space representation, we derive a new formula for searching for the classical correspondence of quantum mechanical operators; we also show that if there exists the eigenvector ｜q〉λ,v of linear combination of the coordinate and momentum operator, （λQ ＋ vP）, where λ,v are real numbers, and ｜q〉λv is complete, then the projector ｜q〉λ,vλ,v〈q｜ must be the Radon transform of Wigner operator. This approach seems concise and physical appealing.
文摘The reaction diffusion Gray-Scott model with time delay is put forward with the assumption of Neumann boundary condition is satisfied. Based on the Turing bifurcation condition, the Turing curves on two parameter plane are discussed without time delay. The normal form is computed via applying Lyapunov-Schmidt reduction method in system of PDE, and the bifurcating direction of pitchfork bifurcation underlying codimension-1 singularity of Turing point is computed. The continuation of Pitchfork bifurcation is simulated with varying free parameter continuously near the turing point, which is in coincidence with the theoritical analysis results. The wave pattern formation in the case of turing instability is also simulated which discover Turing oscillation phenomena from periodicity to irregularity.
