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 applied the rational formal expansion method to construct a series of sofiton-like and period-form solutions for nonlinear differential-difference equations. Compared with most existing methods, the ...In this paper, we applied the rational formal expansion method to construct a series of sofiton-like and period-form solutions for nonlinear differential-difference equations. Compared with most existing methods, the proposed method not only recovers some known solutions, but also finds some new and more general solutions. The efficiency of the method can be demonstrated on Toda Lattice and Ablowitz-Ladik Lattice.展开更多
The non-linear flux equation, the non-linear Fokker-Planck equation （or Smoluchowski equation）, and the non-linear Langiven equation are the basic equations for describing particle diffusion in non-ideal system subj...The non-linear flux equation, the non-linear Fokker-Planck equation （or Smoluchowski equation）, and the non-linear Langiven equation are the basic equations for describing particle diffusion in non-ideal system subjected to time-dependent external fields. Nevertheless, the exact solution of those equations is still a challenge because of their inherent complexity of the non-linear mathematics. Li et al. found that, based on the defined apparent variables, the nonlinear Fokker-Planck equation and the non-linear flux equation could be transformed to linear forms under the condition of strong friction limit or loeal equilibrium assumption. In this paper, some new features of the theory were found： （i） The linear flux equation for describing non-linear diffusion can be obtained from the irreversible thermodynamic theory; （ii） The linear non-steady state diffusion equation for describing non-linear diffusion of the non-steady state, which was described by the non-linear Fokker-Planek equation, can be derived more consistently from the microscopic molecular statistical theory; （iii） In the theory, the non-linear Langiven equation also bears a linear form; （iv） For some special cases, e.g. diffusion in a periodic total potential system, the local equilibrium assumption or the strong friction limit is not required in establishing the linear theory for describing non-linear diffusion, so the linear theory may be important in the study of Brown motor.展开更多
The concept of general argumentation has expanded the family of logic so that it incorporates the logic of other cultures besides modern culture. Based on reports of fieldwork among the Azande and the fruits of resear...The concept of general argumentation has expanded the family of logic so that it incorporates the logic of other cultures besides modern culture. Based on reports of fieldwork among the Azande and the fruits of research on ancient Chinese logic and the logic of Buddhism, this paper attempts to provide a factual foundation for the proposition ＂the cultural relativity of logic＂ from a descriptive perspective. Adopting deductive argument as a meta-method, this paper argues for the existence of the cultural relativity of logic in modern culture and of the translated version of the logic of other cultures in modem culture. With the aid of ethnography and the historical research findings, we show that the logic of other cultures also has its own cultural relativity. We also seek to show through the concepts of language games and life forms that deductive argumentation as a meta-method likewise possesses cultural relativity.展开更多
Differential-difference equations are considered to be hybrid systems because the spatial variable n is discrete while the time t is usually kept continuous.Although a considerable amount of research has been carried ...Differential-difference equations are considered to be hybrid systems because the spatial variable n is discrete while the time t is usually kept continuous.Although a considerable amount of research has been carried out in the field of nonlinear differential-difference equations,the majority of the results deal with polynomial types.Limited research has been reported regarding such equations of rational type.In this paper we present an adaptation of the(G /G)-expansion method to solve nonlinear rational differential-difference equations.The procedure is demonstrated using two distinct equations.Our approach allows one to construct three types of exact traveling wave solutions(hyperbolic,trigonometric,and rational) by means of the simplified form of the auxiliary equation method with reduced parameters.Our analysis leads to analytic solutions in terms of topological solitons and singular periodic functions as well.展开更多
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 Leading Academic Discipline Program211 Project for Shanghai University of Finance and Economics(the 3rd Phase)
文摘In this paper, we applied the rational formal expansion method to construct a series of sofiton-like and period-form solutions for nonlinear differential-difference equations. Compared with most existing methods, the proposed method not only recovers some known solutions, but also finds some new and more general solutions. The efficiency of the method can be demonstrated on Toda Lattice and Ablowitz-Ladik Lattice.
基金Supported by the National Natural Science Foundation of China under Grant Nos.40671090 and 40740420660
文摘The non-linear flux equation, the non-linear Fokker-Planck equation （or Smoluchowski equation）, and the non-linear Langiven equation are the basic equations for describing particle diffusion in non-ideal system subjected to time-dependent external fields. Nevertheless, the exact solution of those equations is still a challenge because of their inherent complexity of the non-linear mathematics. Li et al. found that, based on the defined apparent variables, the nonlinear Fokker-Planck equation and the non-linear flux equation could be transformed to linear forms under the condition of strong friction limit or loeal equilibrium assumption. In this paper, some new features of the theory were found： （i） The linear flux equation for describing non-linear diffusion can be obtained from the irreversible thermodynamic theory; （ii） The linear non-steady state diffusion equation for describing non-linear diffusion of the non-steady state, which was described by the non-linear Fokker-Planek equation, can be derived more consistently from the microscopic molecular statistical theory; （iii） In the theory, the non-linear Langiven equation also bears a linear form; （iv） For some special cases, e.g. diffusion in a periodic total potential system, the local equilibrium assumption or the strong friction limit is not required in establishing the linear theory for describing non-linear diffusion, so the linear theory may be important in the study of Brown motor.
文摘The concept of general argumentation has expanded the family of logic so that it incorporates the logic of other cultures besides modern culture. Based on reports of fieldwork among the Azande and the fruits of research on ancient Chinese logic and the logic of Buddhism, this paper attempts to provide a factual foundation for the proposition ＂the cultural relativity of logic＂ from a descriptive perspective. Adopting deductive argument as a meta-method, this paper argues for the existence of the cultural relativity of logic in modern culture and of the translated version of the logic of other cultures in modem culture. With the aid of ethnography and the historical research findings, we show that the logic of other cultures also has its own cultural relativity. We also seek to show through the concepts of language games and life forms that deductive argumentation as a meta-method likewise possesses cultural relativity.
文摘Differential-difference equations are considered to be hybrid systems because the spatial variable n is discrete while the time t is usually kept continuous.Although a considerable amount of research has been carried out in the field of nonlinear differential-difference equations,the majority of the results deal with polynomial types.Limited research has been reported regarding such equations of rational type.In this paper we present an adaptation of the(G /G)-expansion method to solve nonlinear rational differential-difference equations.The procedure is demonstrated using two distinct equations.Our approach allows one to construct three types of exact traveling wave solutions(hyperbolic,trigonometric,and rational) by means of the simplified form of the auxiliary equation method with reduced parameters.Our analysis leads to analytic solutions in terms of topological solitons and singular periodic functions as well.
