Applied Mathematics Group 

Degrees 

Professional Appointments 

Professional Activities 

Honors 

Lectures 

Courses Taught 

Selected Publications 

Click here to view Dr. Jones' vita in PDF or HTML format.

Degrees 

• B.A. (First Class) Mathematics and Philosophy, University of Bristol, 1974.
• Ph.D., Mathematics, University of Wisconsin-Madison (Advisor: C. Conley), 1979.

Dynamical systems as a tool for solving problems which originate in applications is the main thrust of my research. I have been interested in the dynamics of optical systems, resonance phenomena in celestial mechanics and pattern formation. A particular focus has been the use of dynamical systems methods in the study of nonlinear wave motion. I have developed techniques for determining the stability of traveling and standing waves with applications in nerve impulse and optical beam propagation. I have recently worked on the problem of stable (multiple) pulse propagation in optical fibers with phase-sensitive amplifiers. I have also been involved in a group effort to understand transport and mixing in mesoscale ocean structures. I have focussed on the use of geometric methods, particularly invariant manifolds in phase space. A major collaborative mathematical study has been on the use of geometric methods in singularly perturbed systems.

Professional Appointments 

• Visiting Professor, University of British Columbia.
• Visiting Professor, University of Minnesota.
• Visiting Professor, Universitaet Stuttgart.
• 1999 - present: Director, Lefschetz Center for Dynamical Systems, Brown University.
• 1990 - present: Brown University: Professor.
• 1987 - 1990: Associate Professor, University of Maryland-College Park.
• 1985 - 1987: Assistant Professor, University of Maryland-College Park.
• 1981 - 1985: Assistant Professor, University of Arizona.
• 1980 - 1981: Visiting Assistant Professor, University of Minnesota-Minneapolis.
• 1980: Postdoctoral Fellow, University of British Columbia: Postdoctoral Fellow.

Professional Activities 

• Member, Board of Trustees, SIAM, 1998 - 2000.
• President, SIAM Activity Group on Dynamical Systems, 1993 - 1995.
• Advisory Board, SIAM Activity Group on Dynamical Systems, 1995 - 1997.
• Coordinating Editor, Physica D.
• Member, Editorial Board, Dynamics and Differential Equations; Methods and Applications of Analysis.
• Co-organizer, SIAM Meeting on Applications of Dynamical Systems, Snowbird, Utah, October, 1992.
• Organizer, ONR Workshop on Dynamical Systems and Oceanography, Little Compton, RI, May 1993.
• Member, Organizing committee, SIAM Annual Meeting, Kansas City, July 1996.
• Co-organizer, Workshop on "Dynamical Systems and Nonlinear Optics," May, 1997, Rhode Island.
• Chair, Organizing committee, Workshop on "Multiple Time Scale Dynamical Systems" Oct. 1997, IMA, Minnesota.
• Organizer, Workshop on "Dynamical Systems Methods for Lagrangian Transport in the Ocean," May, 1998, IMA, Minnesota.
• Co-organizer, Workshop on "Recent Advances in the Stability Theory of Nonlinear Waves," May, 1998, Rhode Island.

Honors 

• Graduate Research Board Award, University of Maryland, 1989.
• Humboldt Award for Senior U.S. Scientist, 1993-1994.
• C.N.R. Visiting Professor and C.I.M.E. Lecturer, Italy, June 1994.
• Plenary Lecturer, SIAM Activity Group on Dynamical Systems Conference, Snowbird, 1997.

Lectures 

• Nonlinear Optics Meeting Tucson, Arizona, Sept. 24-26, 1997.
• Tutorial on Singular Perturbation Theory IMA, Minneapolis, MN Oct. 23-24, 1997.
• Student Workshop on Nonlinear Optics Oberwolfach, Germany, Nov. 7-8, 1997.
• Mathematics Colloquium University of Stuttgart, Nov. 10, 1997.

Courses Taught 

• Fall 1996, Fall 1997, Fall 1999: AM0219 - Nonlinear Dynamical Systems: Theory and Applications.
• Spring 1997, Spring 1998, Spring 2000: AM0220 - Nonlinear Dynamical Systems: Theory and Applications.
• Spring 1999: AM0221 - Topics in Dynamical Systems.
• Spring 1999: AM0282 - Seminar in Applied Mathematics.

Selected Publications 

• Do Invariant Manifolds Hold Water? (with S. Winkler), to appear,
  Handbook of Dynamical Systems III: Towards Applications, B. Fiedler, G. Iooss, & N. Koppell, Eds.
  Download:     compressed postscript.
• Quantifying transport in numerically-generated velocity fields (with P. D. Miller, A.M. Rogerson and L.J. Pratt), Physica D, in revision 1997.
• Viscous perturbations of vorticity conserving flows and splitting of separatrices (with B. Sandstede and S.Balasuriya), submitted 1997.
• Bifurcations and edge oscillations in the semiconductor Fabry-P'erot interferometer (with J. Rubin), Optics Communications, 140 1997 pp. 93-98.
• Chaotic mixing in a barotropic meandering jet (with A.M. Rogerson, P. D. Miller, L.J. Pratt and J. Biello), Journal of Physics Oceanograrphy, in revision 1997.
• Chaotic mixing across oceanic jets (with P. D. Miller, G. Haller and L. Pratt), Chaotic, fractal, and nonlinear signal processing, R. Katz, ed., AIP conference proceedings 375, AIP Press, 1996, pp. 591-604
• Settling and chaotic mixing in oceanic flows: the role of heteroclinic orbits (with P.D. Miller, G. Haller and J. Rubin), Naval Res., Rev. XLVII, 1995, pp. 17-28
• Existence of standing pulse solutions to an inhomogeneous reaction-diffusion equation (with J. Rubin), JDDE, 1997.
• Phase field models for hypercooled solidification (with P. Fife, P. Bates and R. Gardner), Physica D, 1041997, pp. 1-31.
• Existence and stability of N-pulses on optical fibers with phase-sensitive amplifiers (with B. Sandstede and J. Alexander), Physica D, 1061997 pp. 167-206.
• Stability of pulses on optical fibers with phase-sensitive amplifiers (with J.C. Alexander, M.G. Grillakis and B. Sandstede), ZAMP, 48 1997 pp. 175-192.
• The existence of travelling wave solutions of a generalized phase-field model (with P. Fife, P. Bates and R. Gardner), SIAM Journal of Mathematical Analysis, 28, pp. 60-93, 1997.
• Geometric Singular Perturbation Theory in CIME Lectures in Dynamical Systems, Lecture Notes in Mathematics, Springer-Verlag 1994.
• Settling and asymptotic motion of aerosol particles in a cellular flow field (with J. Rubin and M. Maxey), Journal of Nonlinear Science, 5, pp. 337-358, 1995.
• Stability of the in-phase travelling wave solution in a pair of coupld nerve fibers (with A. Bose), Indiana Univ. Math. Journal, 44, 189-220 1995.
• Invariant Manifolds and Singularly Perturbed Boundary Value Problems (with S.-K. Tin and N. Kopell), SIAM Journal of Numerical Analysis, 31, pp. 1558-1576, 1994.
• Tracking invariant manifolds up to exponentially small errors (with T.Kaper and N.Kopell), SIAM Journal of Mathematical Analysis, 27, pp. 558-577 1996.
• A mechanism for capture into resonance (with T. Burns), Physica D, 69, pp. 85-106 (1993).
• Instability of quasiperiodic solutions of the Ginsburg-Landau equations (with A. Doelman and R. Gardner), Proc. Royal Society of Edinburgh, 125A, pp.501-517 1995.