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504 766 Td (Final Report: 0708331) Tj
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39 -742 Td (Page 1 of 8) Tj
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-525 714 Td (Final Report for Period:) Tj
/Serif 10 Tf
114 0 Td (06/2009 - 05/2010) Tj
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236 0 Td (Submitted on:) Tj
/Serif 10 Tf
64 0 Td (06/05/2010) Tj
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-414 -15 Td (Principal Investigator:) Tj
/Serif 10 Tf
100 0 Td (Yi, Yingfei .) Tj
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250 0 Td (Award ID: ) Tj
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50 0 Td (0708331) Tj
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-400 -15 Td (Organization:) Tj
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64 0 Td (GA Tech Res Corp - GIT ) Tj
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-64 -15 Td (Submitted By: ) Tj
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0 -14 Td (Yi, Yingfei - Principal Investigator) Tj
/SerifB 10 Tf
0 -16 Td (Title:) Tj
/Serif 10 Tf
0 -14 Td (Multi-frequency oscillations in biological, electrical, and mechanical systems) Tj
/SerifB 10 Tf
250 -28 Td (Project Participants) Tj
-250 -15 Td (Senior Personnel) Tj
0 -30 Td (Post-doc) Tj
0 -30 Td (Graduate Student) Tj
0 -30 Td (Undergraduate Student) Tj
0 -30 Td (Technician, Programmer) Tj
0 -30 Td (Other Participant) Tj
0 -30 Td (Research Experience for Undergraduates) Tj
250 -30 Td (Organizational Partners) Tj
-20 -30 Td (Other Collaborators or Contacts) Tj
20 -30 Td (Activities and Findings) Tj
-250 -30 Td (Research and Education Activities:) Tj
/Serif 10 Tf
0 -14 Td (My NSF-supported work focused on multi-frequency phenomena arising in biological, electrical, and mechanical systems, with particular) Tj
0 -12 Td (attention paid to quasi-periodic motions, almost automorphic intermittency, and bio-chemical oscillations. The results obtained under the) Tj
0 -12 Td (support of this grant are listed in the Publications section.\r) Tj
/SerifB 10 Tf
0 -29 Td (Findings:) Tj
/Serif 10 Tf
0 -14 Td (Below is a brief summary of my NSF-supported work on quasi-periodic motions, almost automorphic phenomena, bio-chemical oscillations,) Tj
0 -12 Td (and their related complexity, concerning Hamiltonian systems, forced circle flows, and closed chemical reaction systems. Numbers in brackets) Tj
0 -12 Td (refer to the Publications section in this report.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (1. Hamiltonian systems\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* KAM stability for properly degenerate Hamiltonian systems.\r) Tj
0 -12 Td (A properly degenerate Hamiltonian system is a highly degenerate one whose integrable part involves several time scales and at each time scale) Tj
0 -12 Td (the corresponding Hamiltonian only depends on part of the action variables. Such a Hamiltonian system arises frequently in problems of) Tj
0 -12 Td (celestial mechanics, for instance, in perturbed Kepler problems like the restricted and non-restricted 3-body problems and spatial lunar) Tj
0 -12 Td (problems in which several bodies with very small masses are coupled with two massive bodies.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (In [3], we extended the classical KAM stability result for 2-time-scales to the case of arbitrary high order proper degeneracy, in order to treat a) Tj
0 -12 Td (much broader class of celestial mechanics applications. By developing a quasi-linear KAM scheme, we showed under the weakest higher order) Tj
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504 766 Td (Final Report: 0708331) Tj
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39 -742 Td (Page 2 of 8) Tj
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-525 723 Td (non-degenerate conditions of Bruno-Russmann type that the majority of quasi-periodic, invariant tori associated with the integrable part will) Tj
0 -12 Td (persist after the non-integrable perturbation. This actually concludes the KAM metric stability for such a properly degenerate Hamiltonian) Tj
0 -12 Td (system.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* Quasi-periodic breathers in Hamiltonian networks. \r) Tj
0 -12 Td (Quasi-periodic breathers \(solutions that are quasi-periodic in time and exponentially localized in space\) are among the most important) Tj
0 -12 Td (coherent structures in Hamiltonian networks, which have been largely found via numerics and experiments in solid state physics and) Tj
0 -12 Td (cell-biology. Although breathers are well-known in Hamiltonian PDEs like sine-Gordon equation and cubic NLS equation, they appear to be) Tj
0 -12 Td (rare and non-robust objects in Hamiltonian PDEs. To the contrary, numerical studies of many physical models find that the existence of) Tj
0 -12 Td (breathers and quasi-periodic breathers is a general phenomenon in Hamiltonian networks, suggesting that the localization property is due to the) Tj
0 -12 Td (discreteness and the nature of nonlinearities rather than disorder.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (In [1], we considered a class of Hamiltonian networks of weakly, nonlinear, and long-range coupled anharmonic oscillators arising in biological) Tj
0 -12 Td (and physical models with either exponential or power-law coupling potential. We showed that, for any given N-admissible sites of the lattice,) Tj
0 -12 Td (there is a positive measure set of quasi-periodic breathers having N-frequencies which are only slightly deformed from the on-site frequencies.) Tj
0 -12 Td (This work gives a theoretical justification to the numerical findings in the case of long-range coupling.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* Viscous stability of KAM tori.\r) Tj
0 -12 Td (For a convex Hamiltonian system, viscosity solutions of the corresponding Hamilton-Jaccobi equation form basic dynamical components. They) Tj
0 -12 Td (include KAM tori and Mather sets, and are known to coincide with weak KAM solutions. Restricting to a nearly integrable Hamiltonian) Tj
0 -12 Td (system, variational regularity of these solutions is of great importance to the nature of Mather sets, related stability issues, and the mechanism) Tj
0 -12 Td (of Arnold diffusion.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (In [7], we exam stability of KAM tori among the class of viscosity solutions for a nearly integrable Hamiltonian system by introducing a new) Tj
0 -12 Td (notion of viscosity stability and stability. We showed that a KAM torus associated to any rotation vector is viscously stable with the stability) Tj
0 -12 Td (index less or equal to one-half, and is precisely one-half if the rotation vector is Diophantine. Sharpness of such stability index is shown along) Tj
0 -12 Td (with regularity analysis. This result also gives a uniform description about the Diophantine KAM tori and their general KAM tori nearby.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* Quasi-periodic motions in a periodically forced oscillator with singular potential.\r) Tj
0 -12 Td (In [10], we considered a periodically forced, singular oscillator in which the potential has subquadratic growth at the infinity and admits a) Tj
0 -12 Td (singularity. This type of oscillator often arises in quantum fields theory. The existence of many quasi-periodic solutions is shown. This implies) Tj
0 -12 Td (the boundedness of all solutions as well as the stability of the system. Aubry-Mather solutions are also discussed.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (2. Forced Circle Flows and Cocycles\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* Irregular and complex dynamics in forced circle flows.\r) Tj
0 -12 Td (Continuous almost periodically forced circle flows are among the simplest non-monotone, multi-frequency dynamical systems. They can be) Tj
0 -12 Td (generated from almost periodically forced nonlinear oscillators through integral manifolds reduction in the damped cases and through Mather) Tj
0 -12 Td (theory in the damping-free cases. They also naturally arise in 2D almost periodic Floquet theory as well as in climate models. Discrete almost) Tj
0 -12 Td (periodically forced circle flows arise in the discretization of nonlinear oscillators and discrete counterparts of linear Schrodinger equations with) Tj
0 -12 Td (almost periodic potentials. They have been widely used as models for studying strange, non-chaotic attractors and intermittency phenomena) Tj
0 -12 Td (during the transition from order to chaos. Hence the study of these flows is of fundamental importance to the understanding of) Tj
0 -12 Td (multi-frequency-driven dynamical irregularities and complexities in non-monotone dynamical systems.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (In [2], we made a general study on skew-product circle flows in both continuous and discrete settings, with particular attentions paying to) Tj
0 -12 Td (almost periodically forced circle flows. When a circle flow is either discrete in time and unforced \(i.e., a circle map\) or continuous in time but) Tj
0 -12 Td (periodically forced, behaviors of minimal sets are completely characterized by classical theory. The general case involving almost periodic) Tj
0 -12 Td (forcing is much more complicated due to the presence of multiple forcing frequencies, the topological complexity of the forcing space, and the) Tj
0 -12 Td (possible loss of mean motion property. On one hand, we showed that to some extent behaviors of minimal sets in an almost periodically forced) Tj
0 -12 Td (circle flow resemble those of Denjoy sets of circle maps in the sense that they can be almost automorphic, Cantorian, and everywhere) Tj
0 -12 Td (non-locally connected. But on the other hand, we showed that almost periodic forcing can lead to significant topological and dynamical) Tj
0 -12 Td (complexities on minimal sets which are not covered by the contents of Denjoy theory. For instance, an almost periodically forced circle flow) Tj
0 -12 Td (can be positively transitive and its minimal sets can be Li-Yorke chaotic and non-almost automorphic.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* Non-uniform hyperbolicity and related dynamics issues for cocycles.\r) Tj
0 -12 Td (Cocyles are naturally generated from non-autonomous linear systems in both continuous and discrete setting. Their study closely ties up with) Tj
0 -12 Td (spectral theory and various dynamics issues arising in quantum field theory such as reducibility, stability, and hyperbolicity.\r) Tj
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504 766 Td (Final Report: 0708331) Tj
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-525 723 Td (\r) Tj
0 -12 Td (In [2], we gave a complete classification of minimal sets for the projective flow associated with an almost periodic, sl\(2,R\)-valued cocycle.) Tj
0 -12 Td (This particularly includes cases when the cocycle is non-uniformly hyperbolic.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (In [5], we paid a particular attention to family of quasi-periodic Schrodinger cocyles parametrized by both energy and amplitude of potentials.) Tj
0 -12 Td (In the case that the potential is analytic and the forcing frequency is Diophantine, Lyapunov exponents are known to have a positive lower) Tj
0 -12 Td (bound independent of energy. We showed that this is generically not true for a fixed irrational frequency within the class of continuous) Tj
0 -12 Td (potentials.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (3. Closed Chemical Reaction systems\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* Reversible Lotka-Volterra model.\r) Tj
0 -12 Td (Many new discoveries in cell biology have indicated the importance of bio-chemical oscillations and it is well-believed that these oscillations) Tj
0 -12 Td (can emerge as the collective dynamic behavior of interacting components in the cell. But little understanding is known about the mechanisms) Tj
0 -12 Td (and underlying principles of such oscillations, for which modeling and analysis especially using closed system formalisms are essential.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (In [11], we developed a reversible Lotka-Volterra model to model the oscillatory reaction dynamics in a closed isothermal chemical system.) Tj
0 -12 Td (This is a three-dimensional, dissipative, singular perturbation to the conservative Lotka-Volterra model, with the free energy serving as a global) Tj
0 -12 Td (Lyapunov function. The Second Law of Thermodynamics dictates that the system ultimately reach an equilibrium. Quasi-stationary oscillations) Tj
0 -12 Td (are analyzed while free energy of the system serves as a global Lyapunov function of the dissipative dynamics. A natural distinction between) Tj
0 -12 Td (regions near and far from equilibrium in terms of the free energy is established. We found that the dynamics is analogous to a mechanical) Tj
0 -12 Td (system with time-dependent increasing damping. Near equilibrium, no oscillation is possible as dictated by Onsager reciprocal symmetry) Tj
0 -12 Td (relation. We also observed that while free energy decreases in the closed system dynamics, it does not follow the steepest descending path.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* Bio-chemical oscillations and multiscale dynamics.\r) Tj
0 -12 Td (In [9], we investigate the detail oscillatory chemical dynamics in the reversible Lotka-Volterra model. We showed that there is a natural) Tj
0 -12 Td (distinction between oscillatory and non-oscillatory regions in the phase space, that is, while orbits ultimately reach the equilibrium in a) Tj
0 -12 Td (non-oscillatory fashion, they exhibit damped, oscillatory behaviors as interesting intermediate dynamics. Besides finding of new mechanisms) Tj
0 -12 Td (of biological oscillations, this work has several theoretical novelties: a\) As time evolves, a single orbit in such a system can behave) Tj
0 -12 Td (conservatively, dissipatively, and monotonically; b\) While the long term dynamics of such a system are simple, complexity can occur in finite) Tj
0 -12 Td (time intervals. These are phenomena which have not been previously discovered in dynamical systems.) Tj
/SerifB 10 Tf
0 -39 Td (Training and Development:) Tj
/Serif 10 Tf
0 -14 Td (* Mentoring and Career Development\r) Tj
0 -12 Td (Three postdocs: Zhenguo Liang, Zhenxin Liu, Shuguan Ji, and four graduate students: Yongfeng Li, Jorge Viveros, Yao Li, Xun Huang, were) Tj
0 -12 Td (partially supported by this grant. As a result, Liang made a substantial contribution to weak KAM solutions of Hamiltonian systems and he is) Tj
0 -12 Td (now an assistant professor at Fudan University; Liu worked on stochastic chemical reaction models and he is now an assistant professor at) Tj
0 -12 Td (Jilin University; and Ji worked on periodic solutions of nonlinear wave equations and he is now an associate professor at Jilin University. Y.-F.) Tj
0 -12 Td (Li obtained his Ph.D in Fall 2008 on topics of biochemical oscillations and he is now a postdoctoral fellow at IMA; and Viveros obtained his) Tj
0 -12 Td (Ph.D in Winter 2007 on topics of quasi-periodic breathers and he is now an assistant professor at Universidad Autonoma del Estado de) Tj
0 -12 Td (Hidalgo. Y. Li began to work on his Ph.D thesis on topics of bio-complexity; and Huang obtained his M.S. degree in Summer 2010.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (My other postdocs/graduate students supervision during the NSF support period included two young female mathematicians: Qian Wang, a) Tj
0 -12 Td (postdocs under the Canadian NSERC Fellowship and Qiuxia Liu, an exchange Ph.D student under a Chinese scholarship.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (I also supervised REU projects for two senior students at Georgia Tech: Ahmet Eser from ISYE and Rachel Lunde from CME, who worked) Tj
0 -12 Td (with me on projects concerning modeling of production lines and chemical rate equations respectively. The projects generated notable interests) Tj
0 -12 Td (among these students towards the study of dynamics. In particular, Lunde has expressed a clear interest in pursuing her graduate study in the) Tj
0 -12 Td (School of Math at Georgia Tech in the area of applied dynamics.) Tj
/SerifB 10 Tf
0 -36 Td (Outreach Activities:) Tj
/Serif 10 Tf
0 -14 Td (As for my other educational activities during the NSF support period, I gave a 12-hour mini-course at the Shanghai Summer School on) Tj
0 -12 Td (Dynamical Systems in 2007, a series of lectures at the 13th Chinese National Summer School in Mathematics in 2008, and a frontier lecture at) Tj
0 -12 Td (the the Northeast College Math Teacher's Training Camp in summer 2007. I was the main organizer of the 2nd Summer School on Dynamical) Tj
0 -12 Td (Systems held in Changchun, China in 2008, and I served on the international scientific committees of the VII Americas School in Differential ) Tj
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504 766 Td (Final Report: 0708331) Tj
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39 -742 Td (Page 4 of 8) Tj
/Serif 10 Tf
-525 723 Td (Equations and Nonlinear Analysis, held in Cartagena, Colombia in 2007 and PASI2009 on Differential Equations and Nonlinear Analysis,) Tj
0 -12 Td (held in Mexico City, Mexico.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (In addition, I served as a chair of thesis committees at USTC, China \(1 Masters thesis in 2009, and 2 Ph.D and 1 Masters theses in 2007\) and) Tj
0 -12 Td (Jilin U., China \(14 Masters theses in 2009\), as well as member of thesis committees at Nanjing U. China \(2 Ph.D theses in 2008\) and Jilin U.,) Tj
0 -12 Td (China \(6 Ph.D theses in 2008 and 2 Ph.D theses in 2007\).) Tj
/SerifB 10 Tf
250 -27 Td (Journal Publications) Tj
/Serif 10 Tf
-250 -29 Td ([1] J. Geng, J. Viveros, Y. Yi, "Quasi-periodic breathers in Hamiltonian networks of long-range coupling", Physica D, p. 2866-2892, vol. 237,) Tj
0 -12 Td (\(2008\). Published, ) Tj
0 -28 Td ([2] W. Huang, Y. Yi, "Almost periodically forced circle flows", J. Funct Analysis, p. 832-902, vol. 257, \(2009\). Published, ) Tj
0 -27 Td ([3] Y. Han, Y. Li, Y. Yi, "Invariant tori for Hamiltonian systems with high order proper degeneracy", Ann. Henri Poincare, p. 1419-1436, vol.) Tj
0 -12 Td (8, \(2010\). Published, ) Tj
0 -28 Td ([4] Z. Liu, W. Huang, Y. Yi, "Random chemical reactions, measure stability, and Fokker-Planck equation", in preparation, p. , vol. , \(2010\). in) Tj
0 -12 Td (preparation, ) Tj
0 -28 Td ([5] D. Dou, W. Huang, Y. Yi, "Quasi-periodic and weakly mixing dynamics in Hamiltonian systems", Preprint, p. , vol. , \(2010\). preprint, ) Tj
0 -27 Td ([6] W. Huang, Y. Yi, "On Lyapunov exponents of continuous Schrodinger cocycles over irrational rotations", J. Dynam. Differential Equations,) Tj
0 -12 Td (p. , vol. , \(2010\). Submitted, ) Tj
0 -28 Td ([7] Z. Liang, J. Yan, Y. Yi, "Viscous stability of Lagrangian tori", Invent. Math., p. , vol. , \(2010\). Submitted, ) Tj
0 -27 Td ([8] D.-W. Huang, H. Wang, Y. Yi, "Bifurcations in a stochastic cycle model", Int. J. Bifur. Chaos, p. , vol. , \(2010\). Accepted, ) Tj
0 -27 Td ([9] Y.-F. Li, H. Qian, Y. Yi, "Nonlinear oscillations and multiscale dynamics in a closed chemical reaction system", J. Dynam. Differential) Tj
0 -12 Td (Equations, p. , vol. , \(2010\). Accepted, ) Tj
0 -28 Td ([10] X. Li, Y. Yi, "Littlewood problem for a singular subquadratic potential", Fields Inst. Commun., p. , vol. , \(2010\). Accepted, ) Tj
0 -27 Td ([11] Y.-F. Li, H. Qian, y. Yi, "Oscillations and multiscale dynamics in a closed chemical reaction system: Second law of thermodynamics and) Tj
0 -12 Td (temporal complexity", J. Chem. Phys., p. 154505, vol. 129, \(2008\). Published, ) Tj
/SerifB 10 Tf
205 -44 Td (Books or Other One-time Publications) Tj
45 -45 Td (Web/Internet Site) Tj
-250 -30 Td (URL\(s\):) Tj
/Serif 10 Tf
0 -14 Td (www.math.gatech.edu/~yi) Tj
/SerifB 10 Tf
0 -13 Td (Description:) Tj
/Serif 10 Tf
0 -14 Td (The papers that are listed in this report are posted at the URL and all the papers acknowledge NSF support. In addition, many of the papers are) Tj
0 -12 Td (posted on other preprint servers, such as the ArXiv, further enhancing the dissemination of research supported by this grant.) Tj
/SerifB 10 Tf
240 -29 Td (Other Specific Products) Tj
10 -30 Td (Contributions) Tj
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504 766 Td (Final Report: 0708331) Tj
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39 -742 Td (Page 5 of 8) Tj
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-525 722 Td (Contributions within Discipline: ) Tj
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0 -14 Td (1. Conferences\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (The following is a list of my invited lectures given at international conferences during the support period, under partial support of this grant.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (\(a\) Plenary Lectures/Mini-Courses on Conferences:\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* Plenary lecture, 2010 ICMC Summer Meeting in Diff. Eqns, Sao Carlos, Brazil \(45 min\) Feb 2010\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* Plenary lecture, Int. Conf. on Nonl. & Stochastic Dynamics, Chengdu, China \(45 min\) Jun 2009\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* Plenary lecture, CRM Conf. on Stability & Instability in Mech. Sys., Barcelona, Spain \(1 hr\) Sep 2008\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* Frontier lectures, The 13th National Summer School in Mathematics, Hefei, China \(2 hrs\) Jul 2008\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* Plenary lecture, Int. Conf. Diff. Equations Appl. Math. Biology, Le Havre, France \(1 hr\) Jun 2008\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* Plenary lecture, 2008 ICMC Summer Meeting in Diff. Eqns, Sao Carlos, Brazil \(45 min\) Jan 2008 \r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* Plenary lecture, Int. Conf. on Dynam. & Math. Modeling, Valladolid, Spain \(45 min\) Sep 2007\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* Plenary lecture, Symposium on Dynamical Systems, Beijing, China \(1 hr\) Aug 2007\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* Invited address, Northeast College Math Teacher's Training Camp, Changchun, China \(1 hr\) Jul 2007\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* Plenary lecture, VII Americas School: DENA, Cartagena, Colombia \(45 min\) Jul 2007\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* Mini-course, Shanghai Summer School on Dynam. Sys., Shanghai, China \(12 hrs\) Jun 2007\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (\(b\) Other Invited Lectures on Conferences:\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* The 3rd Int. Conf. on Recent Advance in App. Dynam. Sys., Guangzhou, China \(40 min\) Dec 2009\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* LCDS Workshop on Lattice Diff. Equations, Providence, USA \(1 hr\) Nov 2009\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* Americas VIII Conf. on Diff. Equations, Veracruz, Mexico \(30 min\) Oct 2009\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* Int. Workshop on Global Dynam. Beyond Uniform Hyperbolicity, Beijing, China \(45 min\) Aug 2009\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* Int. Workshop on Reaction-Diffusion Models & Math. Biology, Harbin, China \(45 min\) Jun 2009\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* Int. Conf. on Asymptotic Anal. & Infinite dim. Dynam. Sys., Hong Kong \(40 min.\) Jun 2009\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* Workshop on Meas. & Top. Dynam. Sys., Suwon, Korea \(30 min\) Jun 2009\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* CAS Workshop on Dynamical Systems, Beijing, China \(1 hr\) Dec 2008\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* Lorentz Center Workshop on KAM Theory & Appl., Leiden, The Nethelands \(45 min\) Dec 2008\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (* Special Session Dyn. Appl. Diff. Eq., AMS Sectional Meeting, Huntsville, US \(30 min\) Oct 2008\r) Tj
0 -12 Td (\r) Tj
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0 -12 Td (\r) Tj
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0 -12 Td (\r) Tj
0 -12 Td (* Dynamical Systems Seminar, University of Science and Technology of China, May 2008\r) Tj
0 -12 Td (\r) Tj
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0 -12 Td (\r) Tj
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