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Existence of invariant curves with prescribed frequency for degenerate area preserving mappings

Frontiers of Mathematics in China 2022年第6期17卷 1133-1155页

作者： Dongfeng ZHANG Hao WU School of Mathematics Southeast UniversityNanjing210096China

We consider small perturbations of analytic non-twist area preserving mappings,and prove the existence of invariant curves with prescribed frequency by KAM *** speaking,the frequency of invariant curve may undergo some drift,if the twist condition is not *** in this paper,we deal with a degenerate situation where the unperturbed rotation angle function r→w+r^(2n+1)is odd order degenerate at r=0,and prove the existence of invariant curve without any drift in its ***,we give a more general theorem on the existence of invariant curves with prescribed frequency for non-twist area preserving mappings and discuss the case of degeneracy with various orders.

关键词： invariant curve area preserving mapping non-twist condition KAM iteration

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Existence of invariant curves for area-preserving mappings under weaker non-degeneracy conditions

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Frontiers of Mathematics in China 2020年第3期15卷 571-591页

作者： Kun WANG Junxiang XU School of Mathematics Southeast UniversityNanjing 211189China

We consider a class of analytic area-preserving mappings Cm-smoothly depending on a *** imposing on any non-degeneracy assumption,we prove a formal KAM theorem for the mappings,which implies many previous KAM-type results under some non-degeneracy ***,by this formal KAM theorem,we can also obtain some new interesting results under some weaker non-degeneracy ***,the formal KAM theorem can be regarded as a general KAM theorem for areapreserving mappings.

关键词： Area-preserving mapping invariant curve KAM iteration nondegeneracy condition

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BIFURCATIONS OF invariant curveS OF A DIFFERENCE EQUATION

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Applied Mathematics and Mechanics(English Edition) 2001年第9期22卷 1096-1104页

作者： HE Tian-lan(贺天兰) Institute of Science Kunming University of Science and Technology Kunming 650093 P R China

Bifurcation of the invariant curves of a difference equation is studied. The system defined by the difference equation is integrable, so the study of the invariant curves of the difference system can become the study of topological classification of the planar phase portraits defined by a planar Hamiltonian system. By strict qualitative analysis, the classification of the invariant curves in parameter space can be obtained.

关键词： difference equation Hamiltonian system invariant curve

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Boundedness of Solutions of a Quasi-periodic Sublinear Duffing Equation

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Chinese Annals of Mathematics,Series B 2021年第1期42卷 85-104页

作者： Yaqun PENG Xinli ZHANG Daxiong PIAO School of Mathematics(Zhuhai) Sun Yat-Sen UniversityZhuhai 519082GuangdongChina School of Mathematics and Physics Qingdao University of Science and TechnologyQingdao 266061ShandongChina School of Mathematical Sciences Ocean University of ChinaQingdao 266100ShandongChina

The authors study the Lagrangian stability for the sublinear Duffing equationsx+e(t)|x|^(α-1) x=p(t)with 0<α<1,where e and p are real analytic quasi-periodic functions with frequencyω.It is proved that if the mean value of e is positive and the frequencyωsatisfies Diophantine condition,then every solution of the equation is bounded.

关键词： Hamiltonian system Sublinear Duffing equation Boundedness Quasi-periodic solution invariant curve

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Boundedness of Solutions of Quasi-periodic p-Laplacian Equations with Jumping Nonlinearity

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Acta Mathematica Sinica,English Series 2023年第1期39卷 176-192页

作者： Xin Li ZHANG Ya Qun PENG Da Xiong PIAO School of Mathematics and Physics Qingdao University of Science and TechnologyQingdao 266061P.R.China School of Mathematics(Zhuhai) Sun Yat-sen UniversityZhuhai 519082P.R.China School of Mathematical Sciences Ocean University of ChinaQingdao 266100P.R.China

In this article,we prove the existence of quasi-periodic solutions and the boundedness of all solutions of the p-Laplacian equation(φ_(p)(x’))’+aφ_(p)(x+)-bφ_(p)(x-)=g(x,t)+f(t),where g(x,t)and f(t)are quasi-periodic in t with Diophantine frequency.A new method is presented to obtain the generating function to construct canonical transformation by solving a quasi-periodic homological equation.

关键词： Quasi-periodic twist mapping invariant curve boundedness p-Laplacian

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