In the present paper we prove that a planar analytic system with a resonant saddle is analytically linearizable at the saddle if and only if it has an analytic first integral in a neighborhood of the saddle.
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
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
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
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
In this paper, we use some programing tools and algorithms for solving system of word equation for regular languages. There are many possibilities for presentation of regular languages such as grammars, finite automat...In this paper, we use some programing tools and algorithms for solving system of word equation for regular languages. There are many possibilities for presentation of regular languages such as grammars, finite automata, rewriting systems and so on. Some of these systems is presented by system of computational discrete algebra GAP and the possibilities of presentation now in some systems interactive theorem provers (Isabelle, Coq). This computer system can give to detailed understanding of solution of system of word equation, compared the languages and regular expressions of the languages.展开更多
A cavitated bifurcation problem is examined for a sphere composed of a class of generalized Valanis-Landel materials subjected to a uniform radial tensile dead-load. A cavitated bifurcation equation is obtained. An ...A cavitated bifurcation problem is examined for a sphere composed of a class of generalized Valanis-Landel materials subjected to a uniform radial tensile dead-load. A cavitated bifurcation equation is obtained. An explicit formula for the critical value associated with the vari- ation of the imperfection parameters is presented. The distinguishing between the left-bifurcation and right-bifurcation of the nontrivial solution of the cavitated bifurcation equation at the critical point is made. It is proved that there exists a secondary turning bifurcation point on the nontrivial solution branch, which bifurcates locally to the left. It is shown that the dimensionless cavitated bifurcation equation is equivalent to normal forms with single-sided constraint conditions at the critical point by using the singularity theory. The stability and catastrophe of the solutions of the cavitated bifurcation equation are discussed.展开更多
The wash-out filter （WF） technique is used to control the flutter of a two dimensional airfoil with cubic non-linearity in incompressible flow. Firstly, Hopf bifurcation theory is used to determine the point at whic...The wash-out filter （WF） technique is used to control the flutter of a two dimensional airfoil with cubic non-linearity in incompressible flow. Firstly, Hopf bifurcation theory is used to determine the point at which the nonlinear controller is introduced. The system is then transformed into Jordan canonical form, based on analysis of linearized eigenvalues of the system. Secondly, for the introduced WF controller, the linear control gain is determined according to Hopf bifurcation condition. The sym- bolic computing program of normal form direct method （NFDM） is also used to obtain the normal form of the controlled system. The non-linear control gain can be determined based on the relation of the type of bifurcation and the parameters of the normal form, to transform sub-critical Hopf bifurcation to be su- per-critical one. Lastly, numerical simulations are used to certify the validity of theoretical analysis, in which the amplitude of flutter or limit cycle of the controlled system is reduced greatly, comparing to the original system.展开更多
The 1:2 internal resonance of coupled dynamic system with quadratic and cubic nonlinearities is studied. The normal forms of this system in 1 :2 internal resonance were derived by using the direct method of normal for...The 1:2 internal resonance of coupled dynamic system with quadratic and cubic nonlinearities is studied. The normal forms of this system in 1 :2 internal resonance were derived by using the direct method of normal form. In the normal,forms, quadratic and cubic nonlinearities were remained. Based on a new convenient transformation technique, the 4-dimension bifurcation equations were reduced to 3-dimension. A bifurcation equation with one-dimension was obtained. Then the bifurcation behaviors of a universal unfolding were studied by using the singularity theory. The method of this paper can be applied to analyze the bifurcation behavior in strong internal resonance on 4-dimension center manifolds.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
A Jeffcott rotor-magnetic bearing with time delays is investigated in this paper. Firstly, it is found that the characteristic equation of the system satisfies the conditions of the singularity. Secondly, the center m...A Jeffcott rotor-magnetic bearing with time delays is investigated in this paper. Firstly, it is found that the characteristic equation of the system satisfies the conditions of the singularity. Secondly, the center manifold reduction and normal form are employed to study the bifurcation from simple zero and zero-purely imaginary singularities. The results of this paper will help to understand the influence of the time delays in feedback loop on the dynamics of rotor-magnetic bearing system.展开更多
A food chain made up of two typical algae and a zooplankton was considered. Based on ecological eutrophication, interaction of the algal and the prey of the zooplankton, a nutrient nonlinear dynamic system was constru...A food chain made up of two typical algae and a zooplankton was considered. Based on ecological eutrophication, interaction of the algal and the prey of the zooplankton, a nutrient nonlinear dynamic system was constructed. Using the methods of the modern nonlinear dynamics, the bifurcation behaviors and stability of the model equations by changing the control parameter r were discussed. The value of r for bifurcation point was calculated, and the stability of the limit cycle was also discussed. The result shows that through quasi-periodicity bifurcation the system is lost in chaos.展开更多
文摘In the present paper we prove that a planar analytic system with a resonant saddle is analytically linearizable at the saddle if and only if it has an analytic first integral in a neighborhood of the saddle.
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.
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 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.
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In this paper, we use some programing tools and algorithms for solving system of word equation for regular languages. There are many possibilities for presentation of regular languages such as grammars, finite automata, rewriting systems and so on. Some of these systems is presented by system of computational discrete algebra GAP and the possibilities of presentation now in some systems interactive theorem provers (Isabelle, Coq). This computer system can give to detailed understanding of solution of system of word equation, compared the languages and regular expressions of the languages.
基金Project supported by the National Natural Science Foundation of China (No.10272069) and Shanghai Key Subject Program.
文摘A cavitated bifurcation problem is examined for a sphere composed of a class of generalized Valanis-Landel materials subjected to a uniform radial tensile dead-load. A cavitated bifurcation equation is obtained. An explicit formula for the critical value associated with the vari- ation of the imperfection parameters is presented. The distinguishing between the left-bifurcation and right-bifurcation of the nontrivial solution of the cavitated bifurcation equation at the critical point is made. It is proved that there exists a secondary turning bifurcation point on the nontrivial solution branch, which bifurcates locally to the left. It is shown that the dimensionless cavitated bifurcation equation is equivalent to normal forms with single-sided constraint conditions at the critical point by using the singularity theory. The stability and catastrophe of the solutions of the cavitated bifurcation equation are discussed.
文摘The wash-out filter （WF） technique is used to control the flutter of a two dimensional airfoil with cubic non-linearity in incompressible flow. Firstly, Hopf bifurcation theory is used to determine the point at which the nonlinear controller is introduced. The system is then transformed into Jordan canonical form, based on analysis of linearized eigenvalues of the system. Secondly, for the introduced WF controller, the linear control gain is determined according to Hopf bifurcation condition. The sym- bolic computing program of normal form direct method （NFDM） is also used to obtain the normal form of the controlled system. The non-linear control gain can be determined based on the relation of the type of bifurcation and the parameters of the normal form, to transform sub-critical Hopf bifurcation to be su- per-critical one. Lastly, numerical simulations are used to certify the validity of theoretical analysis, in which the amplitude of flutter or limit cycle of the controlled system is reduced greatly, comparing to the original system.
文摘The 1:2 internal resonance of coupled dynamic system with quadratic and cubic nonlinearities is studied. The normal forms of this system in 1 :2 internal resonance were derived by using the direct method of normal form. In the normal,forms, quadratic and cubic nonlinearities were remained. Based on a new convenient transformation technique, the 4-dimension bifurcation equations were reduced to 3-dimension. A bifurcation equation with one-dimension was obtained. Then the bifurcation behaviors of a universal unfolding were studied by using the singularity theory. The method of this paper can be applied to analyze the bifurcation behavior in strong internal resonance on 4-dimension center manifolds.
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 Heilongjiang Province Department of Education Science and Technology Project (11544048)
文摘A Jeffcott rotor-magnetic bearing with time delays is investigated in this paper. Firstly, it is found that the characteristic equation of the system satisfies the conditions of the singularity. Secondly, the center manifold reduction and normal form are employed to study the bifurcation from simple zero and zero-purely imaginary singularities. The results of this paper will help to understand the influence of the time delays in feedback loop on the dynamics of rotor-magnetic bearing system.
基金Project supported by the National Natural Science Foundation of China (No. 10472077)the Science and Technology Development Project of Tianjin of China (No. 023111811)
文摘A food chain made up of two typical algae and a zooplankton was considered. Based on ecological eutrophication, interaction of the algal and the prey of the zooplankton, a nutrient nonlinear dynamic system was constructed. Using the methods of the modern nonlinear dynamics, the bifurcation behaviors and stability of the model equations by changing the control parameter r were discussed. The value of r for bifurcation point was calculated, and the stability of the limit cycle was also discussed. The result shows that through quasi-periodicity bifurcation the system is lost in chaos.
