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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

Homoclinic, heteroclinic and periodic orbits of singularly perturbed systems

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Science China Mathematics 2019年第9期62卷 1687-1704页

作者： Xiang Zhang School of Mathematical Sciences Key Laboratory of Scientific and Engineering Computing (Ministry of Education)Shanghai Jiao Tong University

The main aims of this paper are to study the persistence of homoclinic and heteroclinic orbits of the reduced systems on normally hyperbolic critical manifolds, and also the limit cycle bifurcations either from the homoclinic loop of the reduced systems or from a family of periodic orbits of the layer systems. For the persistence of homoclinic and heteroclinic orbits, and the limit cycles bifurcating from a homolinic loop of the reduced systems, we provide a new and readily detectable method to characterize them compared with the usual Melnikov method when the reduced system forms a generalized rotated vector field. To determine the limit cycles bifurcating from the families of periodic orbits of the layer systems, we apply the averaging *** also provide two four-dimensional singularly perturbed differential systems, which have either heteroclinic or homoclinic orbits located on the slow manifolds and also three limit cycles bifurcating from the periodic orbits of the layer system.

关键词： singular perturbation homoclinic and heteroclinic orbits limit cycle rotating vector fields averaging method

Homoclinic and heteroclinic solutions of cubic strongly nonlinear autonomous oscillators by hyperbolic Lindstedt-Poincaré method

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Science China(Technological Sciences) 2010年第3期53卷 692-702页

作者： SZE Kam Yim Department of Mechanical Engineering The University of Hong Kong

The hyperbolic Lindstedt-Poincaré method is applied to determine the homoclinic and heteroclinic solutions of cubic strongly nonlinear oscillators of the form x + c1 x + c3 x 3= ε f (μ,x,x).In the method,the hyperbolic functions are employed instead of the periodic functions in the Lindstedt-Poincaré *** value of parameter μ under which there exists homoclinic or heteroclinic orbit can be determined by the perturbation *** applications are studied in *** illustrate the accuracy of the present method,its predictions are compared with those of Runge-Kutta method.

关键词： Lindstedt-Poincaré Method nonlinear autonomous oscillator homoclinic orbit heteroclinic orbit

Canard explosion,homoclinic and heteroclinic orbits in singularly perturbed generalist predator-prey systems

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International Journal of Biomathematics 2021年第1期14卷 151-174页

作者： Ali Atabaigi Department of Mathematics Razi UniversityKermanshahIran

This paper studies the dynamics of the generalist predator–prey systems modeled in[***,*** and ***,Bistability and limit cycles in generalist predator–prey dynamics,***.14(2013)48–55].When prey reproduces much faster than predator,by combining the normal form theory of slow-fast systems,the geometric singular perturbation theory and the results near non-hyperbolic points developed by Krupa and Szmolyan[Relaxation oscillation and canard explosion,*** Equations174(2)(2001)312–368;Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions,SIAM ***.33(2)(2001)286–314],we provide a detailed mathematical analysis to show the existence of homoclinic orbits,heteroclinic orbits and canard limit cycles and relaxation oscillations bifurcating from the singular homoclinic ***,on global stability of the unique positive equilibrium,we provide some new *** simulations are also carried out to support the theoretical results.

关键词： Canard cycle relaxation oscillation generalist predator prey singular perturbation homoclinic heteroclinic

Exponential dichotomies and heteroclinic bifurcations in degenerate cases

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Science China Mathematics 1995年第6期38卷 653-661页

作者：曾唯尧井竹君 Hunan Light Industrial College Changsha 410007 China Institute of Mathematics Chinese Academy of Sciences Beijing 100080 China

By making use of theory of exponential dichotomies, the theory of heteroclinic bifurcations in higher degenerate cases is investigated. A Melnikov-type vector is given by which the existence of transversal heteroclinic orbits in degenerate cases can be detected. A functional analytical method of proving the transver-sality of heteroclinic orbits in degenerate cases is also provided.

关键词： exponential dichotomies heteroclinic orbits degenerate case Melnikov vector.

Non-periodic perturbations and transversal heteroclinic orbits

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Chinese Science Bulletin 1995年第9期40卷 715-718页

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

We consider an n-dimensional system x=f(x)+εg(t,x,μ), (1)where f,g∈C_b^2 for x∈ΩR^n,t∈R,0<ε<<1,μ∈R^m,and Ω is some open set.

关键词： heteroclinic orbits transversality exponential dichotomy integral manifold.

heteroclinic and Homoclinic Connections between the Sun-Earth Triangular Points and Quasi-Satellite Orbits for Solar Observations

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Journal of Earth Science and Engineering 2013年第8期3卷 515-526页

作者： Pedro J. Llanos Gerald R. Hintz Martin W. Lo James K. Miller Flight Mechanics Section GMV Space and Defence S. A. Tres Cantos Madrid 28760 Spain Astronautical Engineering Department University of Southern California Los Angeles 90089 USA Jet Propulsion Laboratory California Institute of Technology Pasadena 91109 USA Retired Chief Scientist Space Navigation and Flight Dynamics Section KinetX Inc. Simi Valley California 93065 USA

Investigation of new orbit geometries exhibits a very attractive behavior for a spacecraft to monitor space weather coming from the Sun. Several orbit transfer mechanisms are analyzed as potential alternatives to monitor solar activity such as a sub-solar orbit or quasi-satellite orbit and short and long heteroclinic and homoclinic connections between the triangular points L4 and L5 and the collinear point L3 of the CRTBP （circular restricted three-body problem） in the Sun-Earth system. These trajectories could serve as channels through where material can be transported from L5 to L3 by performing small maneuvers at the departure of the Trojan orbit. The size of these maneuvers at L5 is between 299 m/s and 730 m/s depending on the transfer time of the trajectory and does not need any deterministic maneuvers at L3. Our results suggest that material may also be transported from the Trojan orbits to quasi-satellite orbits or even displaced quasi-satellite orbits.

关键词： Quasi-satellite orbits heteroclinic homoclinic Sun-Earth triangular points invariant manifolds solar observations.

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.