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Complete Study on a Bi-Center Problem for the Z2-Equivariant Cubic Vector Fields

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Acta Mathematica Sinica,English Series 2011年第7期27卷 1379-1394页

作者： yi rong liu Ji Bin LI Department ofMathematics Zhejiang Normal University Jinhua 321004 P. R. China Department of Mathematics Zhejiang Normal University Jinhua 321004 P. R. China School of Science Kunming University of Science and Technology Kunming 650093 P. R. China

For the planar Z2-equivariant cubic systems having twoelementary focuses, the characterization of a bi-center problem and shortened expressions of the first six Liapunov constants are completely discussed. The necessary and sufficient conditions for the existence of the bi-center are obtained. All possible first integrals are given. Under small Z2-equivariant cubic perturbations, the conclusion that there exist at most 12 small-amplitude limit cycles with the scheme （6 II 6） is proved.

关键词： Center problem Liapunov constant focal value integral factor invariant integral cubic polynomial system

Z_2-Equivariant Cubic System Which yields 13 Limit Cycles

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Acta Mathematicae Applicatae Sinica 2014年第3期30卷 781-800页

作者： yi-rong liu Ji-bin LI School of Mathematics Central South University Department of Mathematics Zhejiang Normal University School of Science Kunming University of Science and Technology

For the planar Z2-equivariant cubic systems having two elementary focuses, the characterization of a bi-center problem and shortened expressions of the first six Lyapunov constants are completely solved. The necessary and sufficient conditions for the existence of the bi-center are obtained. On the basis of this work, in this paper, we show that under small Z2-equivariant cubic perturbations, this cubic system has at least 13 limit cycles with the scheme 1 6 ∪ 6.

关键词： planar dynamical system limit cycles bifurcations Lyapnov constant weak focus

Generalized Isochronous Centers for Complex Systems

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Acta Mathematica Sinica,English Series 2010年第9期26卷 1779-1792页

作者： Qin Long WANG yi rong liu School of Information and Mathematics Yangtze University Jingzhou 434023 P. R. China Department of Mathematics Zhejiang Normal University Jinhua 321004 P. R. China

In this paper, the definition of generalized isochronous center is given in order to study unitedly real isochronous center and linearizability of polynomial differential systems. An algorithm to compute generalized period constants is obtained, which is a good method to find the necessary conditions of generalized isochronous center for any rational resonance ratio. Its two linear recursive formulas are symbolic and easy to realize with computer algebraic system. The function of time-angle difference is introduced to prove the sufficient conditions. As the application, a class of real cubic Kolmogorov system is investigated and the generalized isochronous center conditions of the origin are obtained.

关键词： Generalized isochronous center generalized period constant time-angle difference complex polynomial system

Center and Isochronous Center Problems for Quasi Analytic Systems

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Acta Mathematica Sinica,English Series 2008年第9期24卷 1569-1582页

作者： yi rong liu Ji Bin Li Department of Mathematics Zhejiang Normal University Jinhua 321004 P. R. China School of Science Kunming University of Science and Technology Kunming 650093 P. R. China

This paper considers the problems of determining center or focus and isochronous centers for the planar quasi-analytic systems. Two recursive formulas to determine the focal values and period constants are given. The convergence of first integral near the center is proved. Using the general results to quasi-quadratic systems, the problem of the isochronous center of the origin is completely solved.

关键词： generalized focal value center integral periodic constant isochronous center quasi-analytic planar differential system

Bifurcations of Limit Circles and Center Conditions for a Class of Non-analytic Cubic Z_2 Polynomial Differential Systems

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Acta Mathematica Sinica,English Series 2012年第11期28卷 2275-2288页

作者： Feng LI yi rong liu yin Lai JIN School of Science Linyi University School of Mathematical Science and Computing Technology Central South University

In this paper, bifurcations of limit cycles at three fine focuses for a class of Z2-equivariant non-analytic cubic planar differential systems axe studied. By a transformation, we first transform non- analytic systems into analytic systems. Then sufficient and necessary conditions for critical points of the systems being centers are obtained. The fact that there exist 12 small amplitude limit cycles created from the critical points is also proved. Henceforth we give a lower bound of cyclicity of Z2-equivaxiant non-analytic cubic differential systems.

关键词： Non-analytic center-focus problem Lyapunov constant limit cycles bifurcation

Multiple Limit Cycles Bifurcation From the Degenerate Singularity for a Class of Three-dimensional Systems

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Acta Mathematicae Applicatae Sinica 2016年第1期32卷 73-80页

作者： Qin-long WANG Wen-tao HUANG yi-rong liu School of Science Hezhou UniversityHezhou542800China Mathematics School of Central South University Changsha 410075China

In this paper,bifurcation of small amplitude limit cycles from the degenerate equilibrium of a three-dimensional system is ***,the method to calculate the focal values at nilpotent critical point on center manifold is *** an example is studied,by computing the quasi-Lyapunov constants,the existence of at least 4 limit cycles on the center manifold is *** terms of degenerate singularity in high-dimensional systems,our work is new.

关键词： Quasi-Lyapunov constant degenerate singularity limit cycles bifurcation three-dimensional system

Center, Limit Cycles and Isochronous Center of a Z_4-equivariant Quintic System

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Acta Mathematica Sinica,English Series 2010年第6期26卷 1183-1196页

作者： Chao Xiong DU Hei Long MI yi rong liu Department of Hu'nan Shaoyang University Shaoyang 422000 P. R. China and School of Mathematics Central South University Changsha 410083 P. R. China Department of Information Science Hu'nan Commercial University Changsha 410083 P. R. China School of Mathematics Central South University Changsha 410083 P. R. China

In this paper, we study the limit cycles bifurcations of four fine focuses in Z4-equivariant vector fields and the problems that its four singular points can be centers and isochronous centers at the same time. By computing the Liapunov constants and periodic constants carefully, we show that for a certain Z4-equivariant quintic systems, there are four fine focuses of five order and five limit cycles can bifurcate from each, we also find conditions of center and isochronous center for this system. The process of proof is algebraic and symbolic by using common computer algebra soft such as Mathematica, the expressions after being simplified in this paper are simple relatively. Moreover, what is worth mentioning is that the result of 20 small limit cycles bifurcating from several fine focuses is good for Z4-equivariant quintic system and the results where multiple singular points become isochronous centers at the same time are less in published references.

关键词： Z4-equivariant focal value center limit cycles isochronous center

Sequential Observation of Alkali-halide Gas Phase Clusters in High Resolution TOF-MS and Prediction of Their Structures

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Chinese Journal of Chemical Physics 2013年第6期26卷 729-738,I0004页

作者： Hui Wen yi-rong liu Kang-ming Xu Teng Huang Chang-jin Hu Wei-jun Zhang Wei Huang Laboratory of Atmospheric Physico-Chemistry Anhui Institute of Optics and Fine MechanicsChinese Academy of Sciences Hefei 230031 China School of Environmental Science and Optoelectronic Technology University of Science and Technology of China Hefei 230026 China

Alkali halide clusters are interesting model systems that can provide information about how crystal properties evolve. To study these properties, a high-resolution atmospheric pressure inlet time-of-flight mass spectrometry （APi-TOF-MS） study of the sequential sodium halides series, C1-（NaC1）n and Br-（NaBr）m, has been reported, and the viability of the APi-TOF- MS equipped with an electrospray ionization source in determining cluster compositions has been demonstrated. The isotopic patterns were well resolved, as n=4 and 7 were determined to be the magic numbers for C1-（NaC1）n clusters, which were particularly abundant in the mass spectra. A global minimum search based on density functional theory enabled basin hopping yield the most stable structures for the mentioned series. The structures exhibit several distinct motifs which can be roughly categorized as linear chain, rock salt, and hexag- onal ring. This work provides an effective way to discover and elucidate the nonstoichiometry sodium halide clusters. These clusters possess very high vertical detachment energies and are generally called as superhalogens, which play important roles in chemistry because they are widely used in the synthesis of new classes of charge-transfer salts.

关键词： Atmospheric pressure interface time-of-flight mass spectrometry Density functional theory Sodium halide cluster

Global Search Algorithm of Atomic and Molecular Clusters

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Global Search Algorithm of Atomic and Molecular Clusters