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PERSISTENCE OF RESONANT invariant tori WITH NON-HAMILTONIAN PERTURBATION

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Acta Mathematica Scientia 2005年第3期25卷 481-491页

作者：白玉真朱德明 Department of Mathematics Qufu Normal University Department of Mathematics East China Normal University

A persistence theorem for resonant invariant tori with non-Hamiltonian perturbation is proved. The method is a combination of the theory of normally hyperbolic invariant manifolds and an appropriate continuation metho... 详细信息

关键词： Normally hyperbolic invariant manifolds invariant tori persistence continuation method

Existence of invariant tori for Certain Weakly Reversible Mappings

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Acta Mathematica Sinica,English Series 2002年第4期18卷 701-718页

作者： Bao She WEI Department of Mathematics, Nanjing University. Nanjing 210093. P. R. China Department of Mathematics Nanjing University Nanjing P. R. China

In this paper, we consider certain mappings. M, sufficiently close to an integrable one, which is weakly reversible with respect to the mappings G sufficiently close to an involution of type (m.n). where m, n∈Z_+ are arbitrary. Under some weak non-degeneracy condition, we construct a uniform KAM iteration for proving the existence of a Cantor family of m-tori invariant under the reversible mappings M and the reversing mapping G.

关键词： Weakly reversible mappings Weakly undegenerate condition invariant tori

The Persistence of invariant tori in Nearly Small Twist Mappings with Intersection Property

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Northeastern Mathematical Journal 2004年第2期20卷 175-190页

作者：祝文壮黄庆道刘柏枫 School of Mathematics Jilin University Changchun 130012 School of Mathematics Jilin University Changchun 130012n this paper we investigate the nearly small twist mappings with intersection property. With a certain non-degenerate condition we proved that the most of invariant tori of the original small twist mappings will survive afer small perturtations. The persisted invariant tori are close to the unperturbed ones when the perturbation are small. The orbits reduced by those mappings are quasi-periodic in the invariant tori with the frequences closing to the original ones.

In this paper we investigate the nearly small twist mappings with intersection property. With a certain non-degenerate condition, we proved that the most of invariant tori of the original small twist mappings will survive afer small perturtations. The persisted invariant tori are close to the unperturbed ones when the perturbation are small. The orbits reduced by those mappings are quasi-periodic in the invariant tori with the frequences closing to the original ones.

关键词： small twist mapping intersection property invariant tori

Nonlinear dynamics analysis of cluster-shaped conservative flows generated from a generalized thermostatted system

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Chinese Physics B 2022年第1期31卷 170-178页

作者：李月陈增强王增会仓诗建 College of Artificial Intelligence Nankai UniversityTianjin 300350China Department of Electrical and Mining Engineering University of South AfricaFlorida 1710South Africa Department of Product Design Tianjin University of Science and TechnologyTianjin 300222China

The thermostatted system is a conservative system different from Hamiltonian systems,and has attracted much attention because of its rich and different nonlinear dynamics.We report and analyze the multiple equilibria and curve axes of the cluster-shaped conservative flows generated from a generalized thermostatted system.It is found that the cluster-shaped structure is reflected in the geometry of the Hamiltonian,such as isosurfaces and local centers,and the shapes of cluster-shaped chaotic flows and invariant tori rely on the isosurfaces determined by initial conditions,while the numbers of clusters are subject to the local centers solved by the Hessian matrix of the Hamiltonian.Moreover,the study shows that the cluster-shaped chaotic flows and invariant tori are chained together by curve axes,which are the segments of equilibrium curves of the generalized thermostatted system.Furthermore,the interesting results are vividly demonstrated by the numerical simulations.

关键词： multiple equilibria curve axes invariant tori cluster-shaped conservative chaos

Energy behavior of Boris algorithm

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Chinese Physics B 2021年第5期30卷 492-498页

作者： Abdullah Zafar Majid Khan Department of Engineering and Applied Physics University of Science and Technology of ChinaHefei 230026China Department of Physics Quaid-i-Azam UniversityIslamabad 45320Pakistan

Boris numerical scheme due to its long-time stability,accuracy and conservative properties has been widely applied in many studies of magnetized plasmas.Such algorithms conserve the phase space volume and hence provide accurate charge particle orbits.However,this algorithm does not conserve the energy in some special electromagnetic configurations,particularly for long simulation times.Here,we empirically analyze the energy behavior of Boris algorithm by applying it to a 2D autonomous Hamiltonian.The energy behavior of the Boris method is found to be strongly related to the integrability of our Hamiltonian system.We find that if the invariant tori is preserved under Boris discretization,the energy error can be bounded for an exponentially long time,otherwise the said error will show a linear growth.On the contrary,for a non-integrable Hamiltonian system,a random walk pattern has been observed in the energy error.

关键词： Boris algorithm invariant tori chaotic system Hénon-Heiles potential

Dynamical behaviors for generalized pendulum type equations with p-Laplacian

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Frontiers of Mathematics in China 2020年第5期15卷 959-984页

作者： Yanmin NIU Xiong LI School of Mathematical Sciences Ocean University of ChinaQingdao 266100China Laboratory of Mathematics and Complex Systems(Ministry of Education) School of Mathematical SciencesBeijing Normal UniversityBeijing 100875China

We consider a pendulum type equation with p-Laplacian(φp(x'))'+G'x(t,x)=p(t),where φp(u)=|u|^p-2u,p>1,G(t,x)and p(t)are 1-periodic about every variable.The solutions of this equation present two interesting behaviors.On the one hand,by applying Moser's twist theorem,we find infinitely many invariant tori whenever ∫0^1 p(t)dt=0,which yields the boundedness of all solutions and the existence of quasi-periodic solutions starting at t=0 on the invariant tori.On the other hand,if p(t)=0 and G'x(t,x)has some specific forms,we find a full symbolic dynamical system made by solutions which oscillate between any two different trivial solutions of the equation.Such chaotic solutions stay close to the trivial solutions in some fixed intervals,according to any prescribed coin-tossing sequence.

关键词： p-Laplacian invariant tori quasi-periodic solutions boundedness complex dynamics

Quasiperiodic solutions of higher dimensional Duffing's equations via the KAM theorem

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Science China Mathematics 2001年第5期44卷 631-644页

作者：张福保 Department of Mathematics Nanjing University Nanjing China

In this paper, we consider the higher dimensional second order differential equations of the form x +　 F( x,t) = 0, x ∈Rn, with a class of weakly coupled potentials F(x,t), periodically depending on t. We prove the ... 详细信息

关键词： quasiperiodic solutions invariant tori symplectic mapping KAM theorem.

A KAM-type Theorem for Generalized Hamiltonian Systems

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Communications in Mathematical Research 2009年第1期25卷 37-52页

作者： Liu BAI-FENG ZHU WEN-ZHUANG XU LE-SHUN Department of Mathematics and Information Science Shandong Institute of Business and Technology Yantai Shandong 264005 College of Mathematics Nankai University Tianjin 300071 Department of Applied Mathematics Northwest A ＆ F University Yangling Shanxi 712100

In this paper we mainly concern the persistence of invariant tori in generalized Hamiltonian systems. Here the generalized Hamiltonian systems refer to the systems which may admit a distinct number of action and angle variables. In particular, system under consideration can be odd dimensional. Under the Riissmann type non-degenerate condition, we proved that the majority of the lower-dimension invariant tori of the integrable systems in generalized Hamiltonian system are persistent under small perturbation. The surviving lower-dimensional tori might be elliptic, hyperbolic, or of mixed type.

关键词： KAM theory invariant tori generalized Hamiltonian system

KAM-type Theorem for Nearly Integrable Hamiltonian with a Quasiperiodic Perturbation

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Northeastern Mathematical Journal 2003年第3期19卷 273-282页

作者：刘柏枫史少云王国铭 School of Mathematics Jilin University 130012 Department of Mathematics Beihua University Jilin 132013 School of Mathematics Jilin University 130012 School of Mathematics Jilin University 130012n this paper we consider the persistence of invariant tori of an integrable Hamiltonian system with a quasiperiodic perturbation. It is proved that if the unperturbed system satisfies the Rtissmann non-degenerate condition and the perturbed system satisfies the co-linked non-resonant condition then the majority of invariant tori is persistent under the perturbation.

In this paper we consider the persistence of invariant tori of an integrable Hamiltonian system with a quasiperiodic perturbation. It is proved that if the unperturbed system satisfies the Rtissmann non-degenerate condition and the perturbed system satisfies the co-linked non-resonant condition, then the majority of invariant tori is persistent under the perturbation.

关键词： Hamiltonian system quasiperiodic perturbation invariant tori