Research of James Meiss

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Acceleration of Particles by trapping in a moving well

A "transitory'' dynamical system is one whose time-dependence is confined to a compact interval. In this paper we show how to quantify transport between Lagrangian coherent structures for the Hamiltonian case. This quantification requires knowing only the relevant heteroclinic orbits on the intersection of invariant manifolds of ``forward" and ``backward" hyperbolic orbits. These manifolds can be easily computed by leveraging the autonomous nature of the vector fields on either side of the time-dependent transition. As examples we consider a fluid flow in a rotating double-gyre configuration and a simple model of a resonant particle accelerator. We compare our results to those obtained using finite-time Lyapunov exponents and to adiabatic theory, discussing the benefits and limitations of each method.

The figure shows the lobes constructed from the images of the unstable manifold of a resonance in the past, with the stable manifolds of a resonance at the current time for a simple Hamilitonian model of an accelerating potential well. More details are in the paper Transport in Transitory Dynamical Systems.

This research, and much of the research listed below has received support from the National Science Foundation, most recently under grants DMS-0202032 and DMS-0707659.

My Vita (pdf file)

My Erdös number is at most 4.

Some Papers since 1987

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), (Corrected Reprint).

Physics

A. Aydemir, R.D. Hazeltine, J.D. Meiss, and M. Kotschenreuther, "Destabilization of Alfvén-Resonant Modes by Resistivity and Diamagnetic Drifts," Physics of Fluids 30 4-6 (1987).
J.D. Meiss, "Transport Near the Onset of Chaos," Physics Today, Physics News of 1986, January (1987).
R.S. MacKay and J.D. Meiss, "The Relation between Quantum and Classical Thresholds for Multi-photon Ionization of Excited Atoms," Physical Review A 37 4702-4706 (1988).
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

R. S. MacKay, J.D. Meiss, and I. C. Percival, "Resonances in Area Preserving Maps," Physica D 27 1-20 (1987).
Q. Chen and J.D. Meiss, "Flux, Resonances and the Devil's Staircase for the Sawtooth Map," Nonlinearity 2 347-356 (1988).
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). (Corrected Reprint).
M. Gidea, J.D. Meiss, I. Ugarcovici, H. Weiss, "Applications of KAM Theory to Population Dynamics," J. Biological Dynamics 5:1,44-63 (2011).

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),
Z. Alexander, E. Bradley, J. Garland and J.D. Meiss, "Iterated Function System Models in Data Analysis: Detection and Separation," submitted to Chaos, Oct 2011. (Preprint).

Anti-Integrability

Q. Chen, R.S. MacKay, and J.D. Meiss, "Cantori for Symplectic Maps," J 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) 262-289 (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) (arXiv 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)
H.R. Dullin and J.D. Meiss, "Resonances and Twist in Volume-Preserving Maps," (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). (arXiv preprint)
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) (arXiv 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)
H.R. Dullin and J.D. Meiss, "Resonances and Twist in Volume-Preserving Maps," submitted to Discrete and Continuous Dynamical Systems, (2010) (arXiv Preprint)
J.D. Meiss, "The Destruction of Tori in Volume-Preserving Maps," Comm. in Nonlinear Science and Numerical Simulation (2011) in press (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)
D.J.W. Simpson and J.D. Meiss, "Resonance near Border-Collision Bifurcations in Piecewise-Smooth, Continuous Maps", Nonlinearity 23 3091-3118 (2010) (arXiv preprint)
D.J.W. Simpson and J.D. Meiss, "Aspects of Bifurcation Theory for Piecewise-Smooth, Continuous Systems", submitted to Physica D, (2010) (arXiv preprint)

Transitory Dynamics

B. Mosovsky and J.D. Meiss, "Transport in Transitory Dynamical Systems", SIAM J. Dyn. Syst. 10 35-65 (2011) (arXiv preprint) ( PDF Reprint)

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revised Oct 27, 2011