The stochastic bifurcation of a generalized Duffing–van der Pol system with fractional derivative under color noise excitation is studied. Firstly, fractional derivative in a form of generalized integral with time-de...The stochastic bifurcation of a generalized Duffing–van der Pol system with fractional derivative under color noise excitation is studied. Firstly, fractional derivative in a form of generalized integral with time-delay is approximated by a set of periodic functions. Based on this work, the stochastic averaging method is applied to obtain the FPK equation and the stationary probability density of the amplitude. After that, the critical parameter conditions of stochastic P-bifurcation are obtained based on the singularity theory. Different types of stationary probability densities of the amplitude are also obtained. The study finds that the change of noise intensity, fractional order, and correlation time will lead to the stochastic bifurcation.展开更多
In this article, we study the blow-up solutions for a case of b-family equations.Using the qualitative theory of differential equations and the bifurcation method of dynamical systems, we obtain five types of blow-up ...In this article, we study the blow-up solutions for a case of b-family equations.Using the qualitative theory of differential equations and the bifurcation method of dynamical systems, we obtain five types of blow-up solutions: the hyperbolic blow-up solution, the fractional blow-up solution, the trigonometric blow-up solution, the first elliptic blow-up solution, and the second elliptic blow-up solution. Not only are the expressions of these blow-up solutions given, but also their relationships are discovered. In particular, it is found that two bounded solitary solutions are bifurcated from an elliptic blow-up solution.展开更多
In this paper, the Lie symmetry analysis and generalized symmetry method are performed for a short-wave model. The symmetries for this equation are given, and the phase portraits of the traveling wave systems are anal...In this paper, the Lie symmetry analysis and generalized symmetry method are performed for a short-wave model. The symmetries for this equation are given, and the phase portraits of the traveling wave systems are analyzed using the bifurcation theory of dynamical systems. The exact parametric representations of four types of traveling wave solutions are obtained.展开更多
The nonlinear dynamic behaviors of nonlinear viscoelastic shallow arches sub- jected to external excitation are investigated. Based on the d'Alembert principle and the Euler-Bernoulli assumption, the governing equati...The nonlinear dynamic behaviors of nonlinear viscoelastic shallow arches sub- jected to external excitation are investigated. Based on the d'Alembert principle and the Euler-Bernoulli assumption, the governing equation of a shallow arch is obtained, where the Leaderman constitutive relation is applied. The Galerkin method and numerical in- tegration are used to study the nonlinear dynamic properties of the viscoelastic shallow arches. Moreover, the effects of the rise, the material parameter and excitation on the nonlinear dynamic behaviors of the shallow arch viscoelastic shallow arches may appear to have a are investigated. The results show that chaotic motion for certain conditions.展开更多
The bifurcation analysis of a simple electric power system involving two synchronous generators connected by a transmission network to an infinite-bus is carried out in this paper. In this system, the infinite-bus vol...The bifurcation analysis of a simple electric power system involving two synchronous generators connected by a transmission network to an infinite-bus is carried out in this paper. In this system, the infinite-bus voltage are considered to maintain two fluctuations in the amplitude and phase angle. The case of 1:3 internal resonance between the two modes in the presence of parametric principal resonance is considered and examined. The method of multiple scales is used to obtain the bifurcation equations of this system. Then, by employing the singularity method, the transition sets determining different bifurcation patterns of the system are obtained and analyzed, which reveal the effects of the infinite-bus voltage amplitude and phase fluctuations on bifurcation patterns of this system. Finally, the bifurcation patterns are all examined by bifurcation diagrams. The results obtained in this paper will contribute to a better understanding of the complex nonlinear dynamic behaviors in a two-machine infinite-bus (TMIB) power system.展开更多
In this paper, the Klein-Gordon equation (KGE) with power law nonlinearity will be considered. The bifurcation analysis as well as the ansatz method of integration will be applied to extract soliton and other wave s...In this paper, the Klein-Gordon equation (KGE) with power law nonlinearity will be considered. The bifurcation analysis as well as the ansatz method of integration will be applied to extract soliton and other wave solutions. In particular, the second approach to integration will lead to a singular soliton solution. However, the bifurcation analysis will reveal several other solutions that are of prime importance in relativistic quantum mechanics where this equation appears.展开更多
A new detecting method for the conditions of existence of the Hopf bifurcation is given in terms of the coefficients of the characteristic polynomial at the equilibrium by using the Hopf bifurcation theory and matrix ...A new detecting method for the conditions of existence of the Hopf bifurcation is given in terms of the coefficients of the characteristic polynomial at the equilibrium by using the Hopf bifurcation theory and matrix theory. The method is available and important for the study of the existence of the Hopf bifurcation for higher differential equations which often occur in biological models, chemical models, epidemiological models, and models of AIDS.展开更多
We are concerned with determining the values of λ, for which there exist nodal solutions of the fourth-order boundary value problem y″″=λa(x)f(y),0〈x〈1,y(0)=y(1)=y″(0)=y″(1)=0where λ is a positive...We are concerned with determining the values of λ, for which there exist nodal solutions of the fourth-order boundary value problem y″″=λa(x)f(y),0〈x〈1,y(0)=y(1)=y″(0)=y″(1)=0where λ is a positive parameter, a ∈ C([0, 1], (0, ∞), f ∈C(R,R) satisfies f(u)u 〉 0 for all u ≠ 0. We give conditions on the ratio f(s)/s, at infinity and zero, that guarantee the existence of nodal solutions.The proof of our main results is based upon bifurcation techniques.展开更多
In this paper, we use the bifurcation method of dynamical systems to study the periodic wave solutions and their limits for the modified KdV-KP equations. Some explicit periodic wave solutions are obtained. These solu...In this paper, we use the bifurcation method of dynamical systems to study the periodic wave solutions and their limits for the modified KdV-KP equations. Some explicit periodic wave solutions are obtained. These solutions contain smooth periodic wave solutions and periodic blow-up solutions. Their limits contain solitary wave solutions, periodic wave solutions, kink wave solutions and unbounded solutions.展开更多
In this paper, we use phase plane analysis to study the compactons of the nonlinear equation. Four new implicit expressions of the compactons are obtained. These new implicit expressions are given by inverse tangent f...In this paper, we use phase plane analysis to study the compactons of the nonlinear equation. Four new implicit expressions of the compactons are obtained. These new implicit expressions are given by inverse tangent functions. Our work extends previous results. For two sets of the data, the graphs of the implicit functions are drawn and numerical simulations are given to test the correctness of our theoretical results.展开更多
In this paper, we consider the following nonlinear equation ut+2kux-uxxt+au^2ux=2uxuxx+uuxxx,which is a modified form of the Camassa-Holm equation. We construct four new explicit periodic wave solutions by bifurcat...In this paper, we consider the following nonlinear equation ut+2kux-uxxt+au^2ux=2uxuxx+uuxxx,which is a modified form of the Camassa-Holm equation. We construct four new explicit periodic wave solutions by bifurcation method of dynamical systems. We also obtain two explicit solitary wave solutions via the limits of the explicit periodic wave solutions. One of the two solitary wave solutions is new.展开更多
基金Project supported by the National Natural Science Foundation of China(Grant No.11302157)the Natural Science Basic Research Plan in Shaanxi Province of China(Grant No.2015JM1028)+1 种基金the Fundamental Research Funds for the Central Universities,China(Grant No.JB160706)Chinese–Serbian Science and Technology Cooperation for the Years 2015-2016(Grant No.3-19)
文摘The stochastic bifurcation of a generalized Duffing–van der Pol system with fractional derivative under color noise excitation is studied. Firstly, fractional derivative in a form of generalized integral with time-delay is approximated by a set of periodic functions. Based on this work, the stochastic averaging method is applied to obtain the FPK equation and the stationary probability density of the amplitude. After that, the critical parameter conditions of stochastic P-bifurcation are obtained based on the singularity theory. Different types of stationary probability densities of the amplitude are also obtained. The study finds that the change of noise intensity, fractional order, and correlation time will lead to the stochastic bifurcation.
基金partially supported by the National Natural Science Foundation of China(11771152,11971176)Guangdong Basic and Applied Basic Research Foundation(2019B151502062)the Fundamental Research Founds for the Central Universities(2019MS111)。
文摘In this article, we study the blow-up solutions for a case of b-family equations.Using the qualitative theory of differential equations and the bifurcation method of dynamical systems, we obtain five types of blow-up solutions: the hyperbolic blow-up solution, the fractional blow-up solution, the trigonometric blow-up solution, the first elliptic blow-up solution, and the second elliptic blow-up solution. Not only are the expressions of these blow-up solutions given, but also their relationships are discovered. In particular, it is found that two bounded solitary solutions are bifurcated from an elliptic blow-up solution.
基金Project supported by the Foundation of Guangxi Key Laboratory of Trusted Software, the Guangxi Natural Science Foundation, China (Grant No. 2011GXNSFA018134)the National Natural Science Foundation of China (Grant Nos. 11161013 and 61004101)
文摘In this paper, the Lie symmetry analysis and generalized symmetry method are performed for a short-wave model. The symmetries for this equation are given, and the phase portraits of the traveling wave systems are analyzed using the bifurcation theory of dynamical systems. The exact parametric representations of four types of traveling wave solutions are obtained.
基金supported by the National Natural Science Foundation of China (Nos. 10502020, 10772065)
文摘The nonlinear dynamic behaviors of nonlinear viscoelastic shallow arches sub- jected to external excitation are investigated. Based on the d'Alembert principle and the Euler-Bernoulli assumption, the governing equation of a shallow arch is obtained, where the Leaderman constitutive relation is applied. The Galerkin method and numerical in- tegration are used to study the nonlinear dynamic properties of the viscoelastic shallow arches. Moreover, the effects of the rise, the material parameter and excitation on the nonlinear dynamic behaviors of the shallow arch viscoelastic shallow arches may appear to have a are investigated. The results show that chaotic motion for certain conditions.
基金Project supported by the National Natural Science Foundation of China(No.10632040)
文摘The bifurcation analysis of a simple electric power system involving two synchronous generators connected by a transmission network to an infinite-bus is carried out in this paper. In this system, the infinite-bus voltage are considered to maintain two fluctuations in the amplitude and phase angle. The case of 1:3 internal resonance between the two modes in the presence of parametric principal resonance is considered and examined. The method of multiple scales is used to obtain the bifurcation equations of this system. Then, by employing the singularity method, the transition sets determining different bifurcation patterns of the system are obtained and analyzed, which reveal the effects of the infinite-bus voltage amplitude and phase fluctuations on bifurcation patterns of this system. Finally, the bifurcation patterns are all examined by bifurcation diagrams. The results obtained in this paper will contribute to a better understanding of the complex nonlinear dynamic behaviors in a two-machine infinite-bus (TMIB) power system.
文摘In this paper, the Klein-Gordon equation (KGE) with power law nonlinearity will be considered. The bifurcation analysis as well as the ansatz method of integration will be applied to extract soliton and other wave solutions. In particular, the second approach to integration will lead to a singular soliton solution. However, the bifurcation analysis will reveal several other solutions that are of prime importance in relativistic quantum mechanics where this equation appears.
文摘A new detecting method for the conditions of existence of the Hopf bifurcation is given in terms of the coefficients of the characteristic polynomial at the equilibrium by using the Hopf bifurcation theory and matrix theory. The method is available and important for the study of the existence of the Hopf bifurcation for higher differential equations which often occur in biological models, chemical models, epidemiological models, and models of AIDS.
基金the NSFC (No.10671158)the NSF of Gansu Province (No.3ZS051-A25-016)+3 种基金NWNUKJCXGC-03-17the Spring-Sun Program (No.Z2004-1-62033)SRFDP (No.20060736001)the SRF for ROCS,SEM (2006[311])
文摘We are concerned with determining the values of λ, for which there exist nodal solutions of the fourth-order boundary value problem y″″=λa(x)f(y),0〈x〈1,y(0)=y(1)=y″(0)=y″(1)=0where λ is a positive parameter, a ∈ C([0, 1], (0, ∞), f ∈C(R,R) satisfies f(u)u 〉 0 for all u ≠ 0. We give conditions on the ratio f(s)/s, at infinity and zero, that guarantee the existence of nodal solutions.The proof of our main results is based upon bifurcation techniques.
基金Supported by National Natural Science Foundation of China(Grant Nos.11361069 and 11171115)
文摘In this paper, we use the bifurcation method of dynamical systems to study the periodic wave solutions and their limits for the modified KdV-KP equations. Some explicit periodic wave solutions are obtained. These solutions contain smooth periodic wave solutions and periodic blow-up solutions. Their limits contain solitary wave solutions, periodic wave solutions, kink wave solutions and unbounded solutions.
基金Supported by the National Natural Science Foundation of China (No.10571062 10371037).Acknowledgment The first author thanks to the Department of Science and Technology of Yuxi City for its support for doing this work.
文摘In this paper, we use phase plane analysis to study the compactons of the nonlinear equation. Four new implicit expressions of the compactons are obtained. These new implicit expressions are given by inverse tangent functions. Our work extends previous results. For two sets of the data, the graphs of the implicit functions are drawn and numerical simulations are given to test the correctness of our theoretical results.
基金Supported by the National Natural Science Foundation of China (No. 10871073)Guangdong Province(No.07006552)
文摘In this paper, we consider the following nonlinear equation ut+2kux-uxxt+au^2ux=2uxuxx+uuxxx,which is a modified form of the Camassa-Holm equation. We construct four new explicit periodic wave solutions by bifurcation method of dynamical systems. We also obtain two explicit solitary wave solutions via the limits of the explicit periodic wave solutions. One of the two solitary wave solutions is new.