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On Canard homoclinic of a Liénard Perturbation System

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Applied Mathematics 2011年第10期2卷 1221-1224页

作者： Makoto Hayashi 不详

The classification on the orbits of some Liénard perturbation system with several parameters, which is relation to the example in [1] or [2], is discussed. The conditions for the parameters in order that the system has a unique limit cycle, homoclinic orbits, canards or the unique equilibrium point is globally asymptotic stable are given. The methods in our previous papers are used for the proofs.

关键词： Liénard System Canards Limit Cycles homoclinic Orbits Global Asymptotic Stability

Bifurcations of nontwisted heteroclinic loop

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Science China Mathematics 2000年第8期43卷 818-828页

作者：田清平朱德明 Department of Mathematics East China Normal University Shanghai 200062 China

Bifurcations of nontwisted and fine heteroclinic loops are studied for higher dimensional systems. The existence and its associated existing regions are given for the 1-hom orbit and the 1-per orbit, respectively, and bifurcation surfaces of the two-fold periodic orbit are also obtained. At last, these bifurcation results are applied to the fine heteroclinic loop for the planar system, which leads to some new and interesting results.

关键词： heteroclinic orbit homoclinic orbit periodic orbit inside stability.

Melnikov Vector and Heteroclinic Manifolds

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Science China Mathematics 1994年第6期37卷 673-682页

作者：朱德明 Department of Mathematics East China Normal UniversityShanghai 200062PRC

Using the exponential dichotomies,the transversality theory and the generalized Melnikov method,we consider the conditions for the persistence and the transversality of the singular orbit,with high degeneracy,situated on the heteroclinic or homoclinic manifold under *** results obtained extend,include and improve the corresponding ones given in certain papers well known in this area.

关键词： invariant manifold heteroclinic orbit homoclinic orbit transversality Melnikov vector

Equivalence of two Melnikov functions

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Chinese Science Bulletin 1996年第8期41卷 626-630页

作者：孙建华 Nanjing University Nanjing 210008 China

The research of homoclinic and heteroclinic bifurcations for high-dimension systems havedrawn much attention. It is often difficult to study such bifurcation problems ofhigh-dimension. We considered the

关键词： homoclinic orbit bifurcation Melnikov function equivalence.

ON NONLINEAR GRAVITY TRAVELLING WAVE

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Science China Chemistry 1992年第12期35卷 1471-1484页

作者：黄思训张铭 LASG Academia Sinica Nanjing 211101 PRC

In this paper, a two--layer model with dispersive and dissipative effects but without Co-?iolis effect is investigated. It is proved that in the model, under certain parameter condi-tions, there exists monotone travelling wave as well as oscillation travelling wave in additionto the nonlinear periodic solution and solitary wave. The conditions for their existence areprovided. It is particularly pointed out that the pattern of the oscillation travelling wave issimilar to that of the pressure upwelling wave of a squall line which passes certain region.

关键词： squall line homoclinic orbit monotone travelling wave oseillation travelling wave

Exact Solutions to the Bidirectional SK-Ramani Equation

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Communications in Mathematical Research 2019年第4期35卷 289-300页

作者： Zou Jia-hui Wang Yun-hu Gao Wen-jie College of Science University of Shanghai for Science and TechnologyShanghai200093 College of Art and Sciences Shanghai Maritime UniversityShanghai201306 不详

In this paper,the bidirectional SK-Ramani equation is investigated by means of the extended homoclinic test approach and Riemann theta function method,*** on the Hirota bilinear method,exact solutions including one-soliton wave solution are obtained by using the extended homoclinic approach and one-periodic wave solution is constructed by using the Riemann theta function method.A limiting procedure is presented to analyze in detail the relations between the one periodic wave solution and one-soliton solution.

关键词： Hirota bilinear method bidirectional Sawada-Kotera equation extended homoclinic test approach Riemann theta function

Rogue Wave for the Benjamin Ono Equation

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Advances in Pure Mathematics 2015年第2期5卷 82-87页

作者： Lili Song Wei Chen Zhenhui Xu Hanlin Chen School of Science Southwest University of Science and Technology Mianyang China Applied Technology College Southwest University of Science and Technology Mianyang China

In the paper, the homoclinic (hateroclinic) breather limit method (HBLM) is applied to seek rogue wave solution of the Benjamin Ono equation. We find that the rational breather wave solution is just a rogue wave solution. This result shows that rogue wave can come from the extreme behavior of the breather solitary wave for (1+1)-dimensional nonlinear wave fields.

关键词： Benjamin Ono Equation Extended homoclinic Test Method homoclinic (Hateroclinic) Breather Limit Method Rogue Wave Solution

Generalization of the Global Error Minimization for Constructing Analytical Solutions to Nonlinear Evolution Equations

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Journal of Applied Mathematics and Physics 2015年第9期3卷 1151-1158页

作者： Serge Bruno Yamgoué Bonaventure Nana Department of Physics Higher Teacher Training College-Bambili The University of Bamenda Bamenda Cameroon

The global error minimization is a variational method for obtaining approximate analytical solutions to nonlinear oscillator equations which works as follows. Given an ordinary differential equation, a trial solution containing unknowns is selected. The method then converts the problem to an equivalent minimization problem by averaging the squared residual of the differential equation for the selected trial solution. Clearly, the method fails if the integral which defines the average is undefined or infinite for the selected trial. This is precisely the case for such non-periodic solutions as heteroclinic (front or kink) and some homoclinic (dark-solitons) solutions. Based on the fact that these types of solutions have vanishing velocity at infinity, we propose to remedy to this shortcoming of the method by averaging the product of the residual and the derivative of the trial solution. In this way, the method can apply for the approximation of all relevant type of solutions of nonlinear evolution equations. The approach is simple, straightforward and accurate as its original formulation. Its effectiveness is demonstrated using a Helmholtz-Duffing oscillator.

关键词： Global Error Minimization Heteroclinic Solution homoclinic Solution Front/Kink Dark/Anti-Dark Soliton

Heteroclinic cycles in lattice dynamical systems

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Science China Mathematics 1998年第1期41卷 1-7页

作者：秦文新钱敏 LNM Institute of Mechanics Chinese Academy of Sciences Beijing Department of Mathematics Suzhou University Suzhou China Department of Mathematics Peking University Beijing China

A criterion of spatial chaos occurring in lattice dynamical systems--heteroclinic cycle--is discussed. It is proved that if the system has asymptotically stable heteroclinic cycle, then it has asymptotically stable ho... 详细信息

关键词： lattice dynamical system heteroclinic cycle homoclinic point.