Research of James Meiss

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Differential equations are the basis for models of any physical systems that exhibit smooth change. This book combines much of the material found in a traditional course on ordinary differential equations with an introduction to the more modern theory of dynamical systems. Applications of this theory to physics, biology, chemistry, and engineering are shown through examples in such areas as population modeling, fluid dynamics, electronics, and mechanics.

Differential Dynamical Systems begins with coverage of linear systems, including matrix algebra; the focus then shifts to foundational material on nonlinear differential equations, making heavy use of the contraction-mapping theorem. Subsequent chapters deal specifically with dynamical systems concepts-flow, stability, invariant manifolds, the phase plane, bifurcation, chaos, and Hamiltonian dynamics.

Throughout the book, the author includes exercises to help students develop an analytical and geometrical understanding of dynamics. Many of the exercises and examples are based on applications and some involve computation; an appendix offers simple codes written in Maple®, Mathematica®, and MATLAB® software to give students practice with computation applied to dynamical systems problems.

This textbook is intended for senior undergraduates and first-year graduate students in pure and applied mathematics, engineering, and the physical sciences. Readers should be comfortable with elementary differential equations and linear algebra and should have had exposure to advanced calculus.

My Vita (pdf file)

My Erdös number is at most 4.

Books

• MacKay, R. S. and J.D. Meiss, Eds. (1987). Hamiltonian Dynamical Systems: a reprint selection. London, Adam-Hilgar Press, 784pp., ISBN 0-85274-205-3. (Buy from Amazon)
• Hazeltine, R. D. and J.D. Meiss (1991). Plasma Confinement. Redwood City, CA, Addison-Wesley, 394 pp., ISBN 0201-53353-5.
• R.D. Hazeltine and J.D. Meiss, Plasma Confinement, (2003) 2nd Edition, Dover Press, 480 pp., ISBN 0486432424. (Buy from Amazon)
• J.D. Meiss, Differential Dynamical Systems, (2007) SIAM, Philadelphia 412 pp., ISBN 978-0-899816-35-1.

Pedagogical Articles

• J.D. Meiss, Symplectic Maps , "Variational Principles, and Transport," Reviews of Modern Physics 64 795-848 (1992) (reprint)
• J.D. Meiss, Hamiltonian Systems, Symplectic Maps, and The Standard Map, articles in the Encyclopedia of Nonlinear Science, ed. Alwyn Scott. (New York, Routledge) (2005). ISBN: 1-57958-385-7
• J.D. Meiss, Dynamical systems, Scholarpedia 2(2):1629 (2007).
• J.D. Meiss, Hamiltonian systems, Scholarpedia 2(8):1943 (2007).
• J.D. Meiss, "Visual Explorations of Dynamics: the Standard Mapping," Pramana, Indian Academy of Sciences, 70 965-988(2008) (arXiv preprint).

Physics

• A. Y. Aydemir, R.D. Hazeltine, M. Kotschenreuther, J.D. Meiss, P.J. Morrison, D.W Ross, F. L. Waelbroeck, J.C. Wiley, "Nonlinear MHD Studies in Toroidal Geometry," Plasma Physics and Controlled Nuclear Fusion Research 1988, Lausanne, Switzerland (International Atomic Energy Agency, Vienna, 1989), 131-143.
• J.D. Meiss, "Comment on Microwave Ionization of H-atoms: breakdown of classical dynamics for high frequencies," Physical Review Letters 62, 1576 (1989).
• J.D. Meiss and R.D. Hazeltine, "Canonical Coordinates for Guiding Center Particles," Physics of Fluids, B2 2563-2567 (1990).
• Hayashi, T., T. Sato, H.J. Gardner and J.D. Meiss, "Evolution of Magnetic Islands in a Heliac," Physics of Plasmas 2 752-759 (1994).
• J.L. Tennyson J.D. Meiss and P.J. Morrison, "Self-Consistent Chaos in the Beam-Plasma Instability," Physica D 71 1-17 (1994). (PDF preprint no figures).
• J.E. Howard and J.D. Meiss "Straight Line Orbits in Hamiltonian Flows," Celestial Mech. and Dyn. Astronomy 105(4) 337-352 (2009). (arXiv Preprint).

Area-Preserving Maps

• J.D. Meiss and R.L. Dewar, "Minimizing Flux," Proceedings of the Centre for Mathematical Analysis, Australian National University, Mini-conference on CHAOS & ORDER, 1-3 February 1990, Canberra Australia, Nalini Joshi and Robert L. Dewar (eds.), (World Scientific, Singapore, 1991) pp. 97-103.
• J.D. Meiss, "Phenomenology of Area Preserving Twist Maps," in Nonlinear Dynamics and Chaos, R. L. Dewar and B. I. Henry (eds.), (World Scientific Press, 1992), pp. 15-40.
• J.D. Meiss, "Regular Orbits for the Stadium Billiard," in Quantum Chaos-Quantum Measurement, P. Cvitanovic, I. Percival and A. Wirzga (eds.) (Kluwer Academic, Dordrecht, 1991), NATO ASI Series C Vol 358, pp. 145-166.
• R.L. Dewar and J.D. Meiss, "Flux-Minimizing Curves for Reversible Area-Preserving Maps," Physica D 57 476-506 (1992) (A laTeX version of this paper).
• J.D. Meiss, "Cantori for the Stadium Billiard," Chaos 2 267-272 (1992).
• J.D. Meiss, "Regular Orbits for the Stadium Billiard," In Quantum Chaos-Quantum Measurement P. Cvitanovic, I. C. Percival and A. Wirzba. (Dordrecht, Kluwer Academic) 145-166 (1992).
• J.D. Meiss, "Transient Measures for the Standard Map," Physica D 74 254-267 (1994). (PS Preprint).
• E. Bollt and J.D. Meiss, "Targeting Chaotic Orbits to the Moon," Phys. Lett. A 204, 373-378 (1995). (PS preprint figures: 1, 2, 3.)
• H. E. Lomelí and J.D. Meiss "Heteroclinic Orbits and Transport in a Perturbed, Integrable Standard Map". Phys. Lett A 269 (5/6) 309-318 (1999).. (arXiv Preprint)
• H.R. Dullin, D. Sterling and J.D. Meiss "Self-Rotation Number using the Turning Angle," Physica D 145(1-2) 25-46 (2000).
• J.D. Meiss, "Visual Explorations of Dynamics: the Standard Mapping," Pramana, Indian Academy of Sciences, 70 965-988(2008) (arXiv preprint).
• M. Gidea, J.D. Meiss, I. Ugarcovici, H. Weiss, "Applications of KAM Theory to Population Dynamics," submitted to Biological Dynamics July 2009.

Symplectic Maps

• H.T. Kook and J.D. Meiss, "Periodic Orbits for Reversible, Symplectic Mappings," Physica D 35, 65-86 (1989).
• H.T. Kook and J.D. Meiss, "Application of Newton's Method to Lagrangian Dynamical Systems," Physica D 36, 317-326 (1989).
• R.S. MacKay, J.D. Meiss, and J. Stark, "Converse KAM Theory for Symplectic Twist Maps," Nonlinearity 2, 555-570 (1989).
• H.T. Kook and J.D. Meiss, "Diffusion in Symplectic Maps," Physical Review A 41 4143-4150 (1990).
• J.D. Meiss, Symplectic Maps , "Variational Principles, and Transport," Reviews of Modern Physics 64 795-848 (1992)
• E.Bollt and J.D. Meiss, "Breakup of Invariant Tori for the Four Dimensional Semi-Standard Map," Physica D 66 282-297 (1993). ( A laTeX version of this paper).
• R.W. Easton, J.D. Meiss and S. Carver, "Exit Times and Transport for Symplectic Twist Maps," Chaos 3 153-165 (1993).
• E. Bollt and J.D. Meiss, "Controlling Transport Through Recurrences," Physica D, 81 280-294 (1994).
• MacKay, R. S., J.D. Meiss and J. Stark, "An Approximate Renormalization for the Break-up of Invariant Tori with Three Frequencies," Phys. Lett. A 190 417 (1994). ( PS Preprint).
• J.D. Meiss, "Towards an Understanding of the Break-up of Invariant Tori," in Proceedings of the International Conference on Dynamical Systems and Chaos, Y. Aizawa, S. Saito and K. Shiraiwa (eds.), (World Scientific,Singapore), 385-394 (1995). (PS Preprint)
• J.D. Meiss, "On the Break-up of Invariant Tori with Three Frequencies," In Hamiltonian Systems with Three or More Degrees of Freedom (Ed, Simo, C.) Kluwer, Sagaro, Spain, pp. 494-498 (1999). (PS Preprint)
• H.R. Dullin and J.D. Meiss, "Stability of Minimal Periodic Orbits," Phys. Lett. A 247 227-234 (1998). (PDF preprint), (PS preprint)

Computational Topology

• V. Robins, J.D. Meiss and E. Bradley, "Computing Connectedness: an exercise in computational topology," Nonlinearity 11 913-922 (1997). (PS Preprint).
• V. Robins, J.D. Meiss and L. Bradley, "Computing Connectedness: Disconnectedness and Discreteness," Phys. D 139 276-300 (1999). (PDF reprint),

Anti-Integrability

• Q. Chen, R.S. MacKay, and J.D. Meiss, "Cantori for Symplectic Maps," Journal of Physics A 23 L1093-L1100 (1990).
• R.S. MacKay and J.D. Meiss, "Cantori for Symplectic Maps near the Anti-integrable Limit," Nonlinearity 5 149-160 (1992).
• D. Sterling and J.D. Meiss, "Computing Periodic Orbits using the Anti-Integrable Limit," Phys. Lett. A 241(1/2) 46-52 (1998). (Preprint)
• D. Sterling, H. R. Dullin and J.D. Meiss, "Homoclinic Bifurcations for the Hénon Map," Physica D 134, 153-184 (1999). (Preprint).
• R. W. Easton, J.D. Meiss, G. Roberts, "Drift by Coupling to an Anti-Integrable Limit," Physica D 156 201-218 (2001). PDF preprint
• H.R. Dullin, J.D. Meiss, and D. Sterling, "Symbolic Codes for Rotational Orbits," SIAM J.Appl. Dyn. Sys 4 515-562 (2005). (arXiv Preprint)

Polynomial Maps

• H.R. Dullin and J.D. Meiss, "Generalized Hénon Maps: the Cubic Polynomial Diffeomorphisms of the Plane," Physica D 143(1-4) 265-292 (2000) .
• A. Gómez and J.D.Meiss, "Reversible Polynomial Automorphisms of the Plane: the Involutory Case," Physics Letters A 312 49-58 (2003). (PDF preprint)
• A. Gómez and J.D. Meiss, "Reversible Polynomial Automorphisms in the Plane," Nonlinearity 17 975-1000 (2003). (PDF Preprint)
• Gonchenko, S.V., J.D. Meiss and I.I. Ovsyannikov, "Chaotic Dynamics of Three-Dimensional Hénon Maps That Originate from a Homoclinic Bifurcation," Reg. and Chaotic Dynamics 11 191-212 (2006) (arXiv preprint)

Twistless Bifurcations

• H. R. Dullin, J.D. Meiss and D. Sterling, "Generic Twistless Bifurcations," Nonlinearity 13 203-224 (2000). (Preprint)
• H.R. Dullin and J.D. Meiss, "Twist Singularities for Symplectic Maps," Chaos 13 1-16 (2003). (PDF preprint)
• H.R. Dullin, A.V. Ivanov and J.D. Meiss, "Normal Forms for 4D Symplectic Maps with Twist Singularities," Phyisca D 215 175-190(2006). (arXiv Preprint)

Volume Preserving Maps

• J.D. Meiss, "Average Exit Times in Volume Preserving Maps," Chaos 7,139-147 (1997). (PS preprint Color Figures: 1, 2.)
• H.E. Lomelí and J.D. Meiss, "Quadratic Volume Preserving Maps," Nonlinearity 11 557-574 (1998).
• K. E. Lenz, H. E. Lomelí; and J.D. Meiss, "Quadratic Volume Preserving Maps: an Extension of a Result of Moser," Regular and Chaotic Dynamics 3, 122-130 (1999). (PDF preprint).
• H. E. Lomelí and J.D. Meiss, "Heteroclinic Primary Intersections and Codimension one Melnikov Method for Volume Preserving Maps," Chaos 10(1) 109-121 (2000). (PDF Preprint), (PS Preprint)
• A. Gómez and J.D.Meiss, "Volume-Preserving Maps with an Invariant", Chaos 12 289-299 (2002). (PDF reprint)
• H.E. Lomelí and J.D. Meiss, "Heteroclinic intersections between Invariant Circles of Volume-Preserving Maps," Nonlinearity 16 1573-1595 (2003). (PDF preprint)
• P. Mullowney, K. Julian and J.D. Meiss, "Blinking rolls: chaotic advection in a 3D flow with an Invariant," SIAM J. Appl. Dyn. Sys. 4 159-186 (2005).
• D.B. Wysham and J.D. Meiss, "Numerical Computation of the Stable Manifolds of Tori," Chaos 16 023129 (2006). (arXiv Preprint)
• J.D. Meiss "Dynamics of Volume Preserving Maps", Lecture at MSRI, Jan 2007
• H.R. Dullin and J.D. Meiss, "Nilpotent Normal form for Divergence Free Vector Fields and Volume-Preserving Maps," Physica D 237(2): 156-166 (2008) (arXiv preprint).
• H.E. Lomelí, J.D. Meiss, and R. Ramírez-Ros, "Canonical Melnikov Theory for Diffeomorphisms," Nonlinearity 21 485-508 (2008) (arXiv preprint).
• P. Mullowney, K. Julien, and J.D. Meiss, "Chaotic Advection in the Küppers-Lortz State," Chaos 18 033104 (2008). (arXiv preprint)
• H.E. Lomelí and J.D. Meiss, "Generating Forms for Exact Volume-Preserving Maps," Discrete and Continuous Dynamical Systems 2 361-377 (2009) (arXiv preprint)
• H.R. Dullin and J.D. Meiss, "Quadratic Volume-Preserving Maps: Invariant Circles and Bifurcations" SIAM J. Appl. Dyn. Sys. 8 76-128 (2009) (arXiv preprint)
• H.E. Lomelí and J.D. Meiss, "Resonance Zones and Lobe Volumes for Volume-Preserving Maps," Nonlinearity, 22 1761-1789 (2009) (arXiv preprint)

Piecewise Smooth Bifurcations

• D.J.W. Simpson and J.D. Meiss, "Andronov-Hopf Bifurcations in Planar, Piecewise-Smooth, Continuous Flows", Phys. Lett. A 371(3) 213-220 (2007) (arXiv preprint).
• D.J.W. Simpson and J.D. Meiss, "Neimark-Sacker Bifurcations in Planar, Piecewise Smooth, Continuous Maps", SIAM J. Appl. Dyn. Sys. 7(3) 795-824 (2008)
• D.J.W. Simpson and J.D. Meiss, "Unfolding a Codimension-Two, Discontinuous, Andronov-Hopf Bifurcation", Chaos 18 033125 (2008) (arXiv preprint)
• D.J.W. Simpson, D.S. Kompala, and J.D. Meiss, "Discontinuity Induced Bifurcations in a Model of Saccharomyces cerevisiae", Math. Biosciences 218 40-49(2009) (pdf preprint) (arXiv preprint)
• D.J.W. Simpson and J.D. Meiss, "Shrinking Point Bifurcations of Resonance Tongues for Piecewise-Smooth, Continuous Maps", Nonlinearity 22 1123-1144 (2009) (arXiv preprint)
• D.J.W. Simpson and J.D. Meiss, "Simultaneous Border-Collision and Period-Doubling Bifurcations", Chaos 19 033146 (2009) (arXiv preprint)