H. Dumas

H. Scott Dumas

Professor Emeritus

French Hall


Professional Summary

Research Interests:  Applied mathematics and dynamical systems, with emphasis on both theory and applications of averaging methods for ordinary differential equations, ergodic theory, Hamiltonian perturbation theory, and kinetic theory.  Applied work extends into these areas of physics: condensed matter-particle interactions, beam stability and spin-orbit interactions in particle accelerators, gas dynamics, and celestial mechanics.

Positions and Work Experience

1982 -2020 More than twenty visiting positions at universities, laboratories and research institutes in the U.S., France, Germany, Brazil, and China.,

Research Support

1986 -1987 American Fulbright Advanced Student Award (Host institution: ENS-Paris)

1989 -1990 IMA Postdoctoral Membership, University of Minnesota-Twin Cities

1992 -1994 NSF Principal Investigatorship (DMS/Applied Mathematics)

1995 -1998 NSF Principal Investigatorship (DMS/Geometric Analysis)

1993 -1999 Ohio Supercomputer Center Award

1991 -2013 University of Cincinnati C.P. Taft Travel, Research, and Publication Grants

Abbreviated Publications

Peer Reviewed Publications

Dumas, H.S. & Ellison, J.A.  (1986).  Particle channeling and the method of averaging, in Local and Global Methods of Nonlinear Dynamics (R. Cawley, A.W. Sáenz, and W.W. Zachary, Eds.) Lecture Notes in Physics No. 252: 200–230, Springer-Verlag, New York.

Dumas, H.S.  (1989).  Nekhoroshev's theorem and particle channeling in crystals, in Integrable Systems and Applications (M. Balabane, P. Lochak, and C. Sulem, Eds.) Lecture Notes in Physics No. 342: 87–94, Springer-Verlag, New York.

Dumas, H.S. & Kimball, J. (1990).  Symmetry in motion, relation to chaos.  Physics Letters A 144:  201–205.

Ellison, J.A., Sáenz, A.W. & Dumas, H.S. (1990).  Improved Nth order averaging for periodic systems.  Journal of Differential Equations 84: 383–403.

Dumas, H.S.  (1991).  Existence and stability of particle channeling in crystals on timescales beyond all orders, in Asymptotics Beyond All Orders (H. Segur, S. Tanveer, and H. Levine, Eds.)  NATO ASI Series B: Physics 284: 267–273, Plenum Press, New York.

Dumas, H.S., Ellison, J.A. & Sáenz, A.W. (1991).  Axial channeling in perfect crystals, the continuum model, and the method of averaging.  Annals of Physics (NY) 209: 97–123.

Dumas, H.S.  (1991).  Ergodization rates for linear flow on the torus.  Journal of Dynamics and Differential Equations 3 (4): 593–610.

Dumas, H.S. & Ellison, J.A.  (1991).  Nekhoroshev's theorem, ergodicity, and the motion of energetic charged particles in crystals, in Essays on Quantum and Classical Dynamics (J.A. Ellison and H. Überall, Eds.): 17–56, Gordon and Breach, Philadelphia.

Dumas, H.S. (1993).  A Nekhoroshev-like theory of classical particle channeling in perfect crystals.  Dynamics Reported (New Series) 2: 69–115.

Dumas, H.S. & Laskar, J.  (1993).  Global dynamics and long-time stability in Hamiltonian systems via numerical frequency analysis.  Physical Review Letters 70 (20): 2975–2979.

Dumas, H.S., Golse, F. & Lochak, P.  (1994).  Multiphase averaging for generalized flows on manifolds.  Ergodic Theory and Dynamical Systems 14: 53–68.

Dumas, H.S.  (1995).  A new proof of Anosov`s averaging theorem, in Hamiltonian Dynamical Systems:  History, Theory, and Applications  (H.S. Dumas, K.R. Meyer, D. Schmidt, Eds.),  IMA Volumes in Mathematics and its Applications 63: 127–136, Springer-Verlag, New York.

Dumas, H.S., Dumas, L. & Golse, F.  (1996).  On the mean free path for a periodic array of spherical obstacles.  Journal of Statistical Physics 82 (5/6): 1385–1407.

Dumas, H.S. & Golse, F. (1997).  The averaging method for perturbations of mixing flows.  Ergodic Theory and Dynamical Systems 17: 1339–1358.

Dumas, H.S., Dumas, L. & Golse, F. (1997).  Remarks on the notion of mean free path for a periodic array of spherical obstacles.  Journal of Statistical Physics 87 (3/4): 943–950.

Dumas, H.S. (1998).  Mathematical aspects of high-energy particle motion in periodic structures, in Nonlinear and Stochastic Beam Dynamics in Accelerators – A Challenge to Theoretical and Computational Physics (A. Bazzani, J.A. Ellison, H. Mais, G. Turchetti, Eds.), 140–160, DESY, Hamburg.

Dumas, H.S. (1999).  Filling rates for linear flow on the torus: recent progress and applications, in Hamiltonian Systems with Three or More Degrees of Freedom (C. Simó, Ed.), 335–339, Kluwer Academic Publishers, Dordrecht.

Dumas, H.S., Ellison, J.A., & Golse, F. (2000). A mathematical theory of planar particle channeling in crystals. Physica D 146 (1–4), 341–366.

Dumas, H.S., Ellison, J.A., & Hoffstaetter, G. (2002).  Adiabatic invariants for spin-orbit motion. in Proceedings of the Eighth European Particle Accelerator Conference (Paris, France, June 3–7, 2002) EPS-IGA & CERN, Geneva, 2002, pp. 332–334.

Dumas, H.S., Ellison, J.A., Salas, M., Sen, T., Sobol, A., & Vogt, M. (2003). Weak-strong beam-beam: averaging and tune diagrams. Beam Halo Dynamics, Diagnostics, and Collimation: 29th ICFA Advanced Beam Dynamics Workshop on Beam-Halo Dynamics and the Beam-Beam Interaction, pp. 282–286.

Dumas, H.S., Ellison, J.A., & Vogt, M. (2004).  First-order averaging principles for maps with applications to accelerator beam dynamics. SIAM Journal on Applied Dynamical Systems, (4), 409–432.

Dumas, H.S., & Ellison, J.A. (2005). Averaging for Quasiperiodic Systems. in Equadiff 2003: Proceedings of the International Conference on Differential Equations, Hasselt, Belgium (F. Dumortier, H.W. Broer, J. Mawhin, A. Vanderbauwhede, & S. Verduyn Lunel, Eds.), World Scientific, Singapore, 2005, pp. 726–731.

Dumas, H.S. (2005).  Mathematical theories of classical particle channeling in perfect crystals. NIM Physics Research Section B: Beam Interactions with Materials and Atoms 234 (1–2), 3–13.

Dumas, H.S., Ellison, J.A., & Hoffstaetter, G. (2006). Adiabatic invariance of spin-orbit motion in accelerators. Physical Review ST - Accelerators and Beams, 9 (1): 014001 [13 pp].

Dumas, H.S., Meyer, K.R., Palacián, J., & Yanguas, P. (2007).  Qualitative features of Hamiltonian systems through averaging and reduction, in Actas del XX Congreso de Ecuaciones Diferenciales y Aplicaciones, X Congreso de Matemática Aplicada (held in Sevilla, Spain, September 24–28, 2007), 8 pp., Sociedad Española de Matemática Aplicada, Universidad de Sevilla.

Yanguas, P., Palacián, J., Meyer, K.R., & Dumas, H.S. (2008).  Periodic solutions in Hamiltonian systems, averaging, and the lunar problem. SIAM Journal on Applied Dynamical Systems 7 (2): 311–340.

Graça, D.S., Zhong, N., & Dumas, H.S. (2012). The connection between computability of a nonlinear problem and its linearization: the Hartman-Grobman theorem revisited.  Theoretical Computer Science 457, 101–110.

Dumas, H.S. (2013).  A specialized multiphase averaging method for quasiperiodic systems of ODEs.  Qualitative Theory of Dynamical Systems 12 (1) 225–242. 

Dumas, H.S., Meyer, K.R., Palacian, J.F., & Yanguas, P.  (2017).  Asymptotic stability estimates near an equilibrium point.  Journal of Differential Equations  263, 1125–1139.



Lochak, P. & Meunier, C. [Translated by H.S. Dumas].  Multiphase Averaging for Classical Systems with Applications to Adiabatic Theorems (Applied Mathematical Sciences Vol. 72), Springer-Verlag, New York, 1988.

Dumas, H.S., Meyer, K.R. & Schmidt, D.S., Eds.  Hamiltonian Dynamical Systems: History, Theory, and Applications (IMA Volumes in Mathematics and its Applications, Vol. 63), Springer-Verlag, New York, 1995.

Dumas, H.S.  The KAM Story (A friendly introduction to the content, history, and significance of classical Kolmogorov-Arnold-Moser theory), World Scientific Publishers, Singapore, 2014
[Chinese translation in progress; Russian translation see below].

Dumas, H.S. [Translated by V.V. Shulikovskaya, with Scientific Editors I.V. Bizyaev and A.V. Borisov].  Teoriya KAM: kak eto bylo (Kratkoe znakomstvo s soderzhaniem, istoriey i znacheniem klassicheskoy teorii Kolmogorova-Arnolda-Mozera), Regulyarnaya i haoticheskaya dinamika, Institut kompyuternyh issledovaniy, Moskva, 2017
[Russian translation of The KAM Story, above].

See also H.S. Dumas’ Amazon Author Page for more information about these books, as well as other books containing some of the articles listed above:


Merle, F.  [Translated by H.S. Dumas] (1988).  Existence of stationary states for nonlinear Dirac equations.  Journal of Differential Equations 74: 50–68.

Albouy, A. [Translated by H.S. Dumas] (1993).  Integral manifolds of the N-body problem.  Inventiones Mathematicae 114: 463--488.

Albouy, A. [Translated by H.S. Dumas] (2002).  Lectures on the two-body problem, in Classical and Celestial Mechanics: The Recife Lectures (H. Cabral and F. Diacu, Eds.) 63–116, Princeton University Press, Princeton, NJ.

Other Publication

Dumas, H.S.  (1992).  The role of low-dimensional symplectic maps in the dynamics of particle accelerators.  Imagine That!  (The Geometry Center Quarterly, University of Minnesota) 1 (2): 6–7.

Dumas, H.S. (1996).  Review of the book Classical Dynamics with Applications to Beam Dynamics, by L. Michelotti, Wiley Interscience, New York, 1994, in SIAM Review 38 (4): 685–687.

Dumas, H.S. (2004).  More than 30 signed reviews of articles in Mathematical Reviews/MathSciNet, 1999–2004.

Dumas, H.S. (2008).  Oscillations near the eye of the long-tailed fish, Electronic Journal of Differential Equations, Differential Equations Problems Page, Problem 2007–1; Solution 2008.

Dumas, H.S. (2011).  Hamiltonian dynamics and celestial mechanics, a conference in honor of Ken Meyer in his 75th year.  Dynamical Systems Magazine (July, 2011, section: Dynamical Systems Today, 3 pp., electronic)

Dumas, H.S. (2011).  A brief biographical sketch of Ken Meyer. Dynamical Systems Magazine (July, 2011, section: In the Spotlight, 5 pp., electronic)

Dumas, H.S. & Golse, F.  (2013).  The Lorentz gas: periodic or not? Notes of the Canadian Mathematical Society (research notes section) 45 (1) 16–17.

Palacián, J.F., Yanguas, P. & Dumas, H.S. (2013).  Preface to Special Issue on Hamiltonian Systems and Celestial Mechanics, Qualitative Theory of Dynamical Systems  12 (1) : 1–3 (DOI  10.1007/s12346-013-0100-x).

Dumas, H.S. (2014).  KAM Theory (book blurb), in The Right Angle (Newsletter of the University of Cincinnati Department of Mathematical Sciences) Vol. 21, Fall 2014, page 2.



(03-22-2000. ) Séminaire systèmes dynamiques .Université de Lille I, France.

(05-18-2000. ) Audition, Laboratoire de mathématiques .Université d’Orléans, France.

(04-16-2001. ) PACM Dynamical Systems/Nonlinear Science Seminar .Princeton University, NJ.

(09-14-2001. ) Applied Mathematics Seminar .University of New Mexico, Albuquerque, NM.

(01-17-2002. ) Physics Colloquium .University of Cincinnati, OH.

(03-19-2002. ) Seminar in Perturbation Theory .University of New Mexico, Albuquerque, NM.

(04-25-2002. ) Mathematics Colloquium .University of Cincinnati, OH.

(07-22-2003. ) Invited talk, Equadiff 2003 .Limburg University, Hasselt, Belgium.

(07-29-2003. ) Oberseminar, IRAM .Rheinisch-Westfaelische Technische Hochschule, Aachen, Germany.

(03-24-2004. ) Invited talk .Workshop on Relativistic Channeling, Frascati, Italy.

(02-2005. ) Invited lectures, Matemairacorana .Universidade Federal de Pernambuco, Recife, Brazil.

(09-15-2005. ) Theoretical Physics Seminar .University of Warwick, Coventry, England.

(10-11-2005. ) Joint DESY/University of Hamburg Physics Seminar, DESY-MPY .Hamburg, Germany.

(02-23-2006. ) Seminar in Harmonic Analysis and PDE .University of New Mexico, Albuquerque, NM.

(06-07-2006. ) Invited talk .International Conference on Nonlinear and Stochastic Dynamics, Chengdu, China.

(04-15-2010. ) Seminar in KAM theory 1 .Jiangsu University, Zhenjiang, China.

(04-16-2010. ) Seminar in KAM theory 2 .Jiangsu University, Zhenjiang, China.

(03-16-2015. ) Applied Mathematics Seminar .University of New Mexico, Albuquerque, NM.

(11-15-2017. ) Keynote speaker .Opening Ceremony of 7th Sino-Foreign Postgraduate Academic Forum, Jiangsu University, Zhenjiang, China.

(12-10-2019. ) Applied Mathematics Seminar .University of New Mexico, Albuquerque, NM.

(01-02-2020. ) Applied Mathematics Seminar .National Sun Yat-sen University, Kaohsiung, Taiwan.

(01-07-2020. ) Applied Mathematics Seminar .Institute of Mathematics, Academia Sinica, Taipei, Taiwan.

(01-09-2020. ) Mathematics Colloquium .Institute of Mathematics, Academia Sinica, Taipei, Taiwan.

Event Organized

03-1992 International Conference on Hamiltonian Dynamical Systems, University of Cincinnati

05-1995 Minisymposium, 3rd SIAM Conference on Applications of Dynamical Systems, Snowbird, Utah

09-2000 30th Anniversary Midwest Dynamical Systems Conference, University of Cincinnati

09-2002 Midwest Dynamical Systems Seminar, University of Cincinnati

12-2006 Workshop on Theory & Applications of Fluid Mechanics, University of Cincinnati

05-2011 Hamiltonian Dynamics and Celestial Mechanics, a Conference in Honor of Ken Meyer in his 75th year, Castro Urdiales, Spain

Honors and Awards

1991 Tom L. Popejoy Dissertation Prize (Best PhD Thesis in the Sciences, Mathematics, or Engineering during the period 1987–90), University of New Mexico

1994 Qualification aux fonctions de professeur des universités, French Ministry of Higher Education and Research

2008 Mrs. A.B. (Dolly) Cohen Award for Outstanding Contributions and Excellence in Teaching, University of Cincinnati

2009 Edith C. Alexander Award for Distinguished Teaching, McMicken College of Arts and Sciences, University of Cincinnati

2009 Representative of Excellence in Teaching, Greater Cincinnati Consortium of Colleges & Universities