Research

Research of James Meiss

Summaries

Subjects

Much of the research listed below has received support from the National Science Foundation, most recently under grants DMS-0707659, DMS-1211350, CMMI-1447440, CMMI-1553297, DMS-1812481 and AGS-2001670. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the NSF. Support from the Simons Foundation, grant #601972, is also gratefully acknowledged.

Triplet Interactions

Newton formulated the theory of graviation as what turned out to be a Hamiltonian system with interactions between pairs of masses. For the point mass case the system has a potential energy that is a function of pairwise distances between the particles. Inspired by the many recent network studies that look at syncrhonizaton for interactions on hypergraphs, in we study a system of particles that interact in triplets.

We postulate a potential energy that depends on the distances between the particles, but that cannot be written as a sum of pair interactions. Similar interactions do arise in applications. For example, polarizable molecules have a three-body force that was first studied in 1943 by Axilrod and Teller (and contemporaneously by Muto in Japan). Similarly colloids an nucleon interactions can give rise to such forces. In these applications the three-body force is a correction (usually in a power-series sense) to a more familiar two-body interaction. In our case, we assume there is only a triplet potential.

It seems natural to study forces that depend on the geometry of the triangle. Here we study the perimetric and areal cases: the potential energy is a function of the perimeter or area of the triangle. Even for the three-body case, the dynamics can be complex, since at its most reduced (rotating, center-of-mass) form, this system has three degrees of freedom. An example is shown in the movie (right). In this case the potential is U(P) = P^2/2.

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Weighted Birkhoff Averages and Detecting Chaos

The dynamics of an integrable Hamiltonian or volume-preserving system consists of periodic and quasi-periodic motion on invariant tori. When such a system is smoothly perturbed, Kolmogorov-Arnold-Moser (KAM) theory implies that some of these tori persist and some are replaced by isolated periodic orbits, islands, or chaotic regions. On each KAM torus, the dynamics is conjugate to a rigid rotation with some fixed frequency vector. Typically, as the perturbation grows the proportion of chaotic orbits increases and more of the tori are destroyed.

In a paper with Evelyn Sander, we explore an alternative technique, based on windowed Birkhoff averages, to distinguish between chaotic, resonant, and quasiperiodic dynamics for area preserving maps. We applied this technique, in to find two-dimensional tori for 3D volume-preserving maps. An important question in both these situations is: how can one distinguish between ``irrational" and ``rational" numbers numerically. We show how an answer to question can be computed if it is reformulated: what is the smallest period rational within a given tolerance. This leads, in the invariant circle case, to a method based on the Farey tree expansion. In higher dimensions, a similar method can be applied to find commensurabilities

In 2021, Nathan Duignan and I, applied these methods to flows. We show how the super-convergence of the weighted Birkhoff average also applies to the case of a smooth flow when the rotation vector is Diophantine, generalizing earlier work of Das, Sander & Yorke— for the map case.

We applied these methods to distinguish regular and chaotic regions for one-and-a-half degree of freedom Hamiltonian systems, using the two-wave model (that we also studied usng converse KAM methods), and a simple model for magnetic field line flow. We also show that it can distinguish chaotic orbits in a "strange-nonchaotic-attractor" (SNC) first studied by Grebogi, Ott, Pelikan and Yorke. The interesting aspect of these orbits is that they lie on geometrically strange attractors, but have zero maximal Lyapunov exponents.

p along a line x = 0.0). Orbits on the Poincare section that are chaotic (digits less than 5) are blue.

-->The picture (right) shows the detection of ``weak chaos'' or strange nonchaotic attractors for a quasiperioidically forced Arnold circle map as a function of the coupling amplitude and the intrinsic rotation number. The color scale represents the Lyapunov exponent. Regions with negative Lyapunov exponent (grey) are nevertheless weakly chaotic since the WBA converges slowly.

Presymplectic Formulation of Field Line Flow

Though the flow of an incompressible vector field in 3 space, such as a magnetic field, is often thought of as a one-and-a-half degree-of-freedom Hamiltonian system, i.e., H(q,p,t) with periodic time-dependence, we argue, in a paper with Josh Burby and Nathan Duignan, that it is more appropriate to think of it as a presymplectic system. There is a two-form that generates the dynamics, but it cannot be non-degenerate on a three-dimensional manifold. We use this idea to reformulate the problem of magneto-hydro-static (MHS) equilibria, and to show that there exist normal form coordinates near a magnetic axis (a non-degenerate closed loop) that are analogous to Hamada and Boozer coordinates. The second invariant in this system corresponds to the current vector field (the diamagnetic current) and it is generated by the "Hamiltonian" given by the pressure.

Moving away from the integrable case, Nathan Duignan and I use this reformulation to compute asymptotic normal forms analogous to the Gustavson-Birkhoff form, near a magnetic axis both then the local rotational transform is irrational and rational. As shown in the figure at the right, this can be used in the near-resonant case to give a good approximation to the field lines, though the normal form necessarily breaks-down near the separatrix where it becomes chaotic.

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Anti-Integrable Limits for Three-Dimensional Maps

The concept of anti-integrability was introduced by Aubry and Abramovicci in 1983 for the standard map, viewed as a linear chain of particles connected by springs in a periodic potential. They reasoned that the integrable limit corresponded to vanishing potential energy, so that the springs dominated giving equal spacing at equilibrium. By contrast, anti-integrability corresponds to vanishing kinetic energy, so that particles sit at critical points of the potential. What is most interesting about this limit is that it is relatively easy, using a contraction mapping style argument, to show that AI states persist, and this gives conjugacy to a shift on a symbolic dynamics.

In the paper, Amanda Hampton and I generalize these ideas to the family of quadratic three-dimensional diffeomorphisms that were obtained in Lomelí & Meiss . We write the map as a third difference equation, and scale to isolate the nonlinear terms. A unique feature of this study is that the AI limit corresponds to a quadratic correspondence---a quadratic curve that corresponds to a one-dimensional dynamical system. We show that there are a number of parameter values for which a full shift on two-symbols exists at the AI limit and that these orbits can be continued away from the limit.

The figure at the right shows an orbit of a 3D quadratic map. continued away from the AI limit. At the limit, the orbit falls on the intersection of the two elliptic cylinders. As we move away from this limit, the orbit maintains some of this structure.

More recently, Hampton and I studied bifurcations that create strange attractors for a special case of this family that can be thought of as a 3D version of Henon's map.

Birkhoff Averages and Rotational Invariant Tori

The dynamics of an integrable Hamiltonian or volume-preserving system consists of periodic and quasi-periodic motion on invariant tori. When such a system is smoothly perturbed, Kolmogorov-Arnold-Moser (KAM) theory implies that some of these tori persist and some are replaced by isolated periodic orbits, islands, or chaotic regions. On each KAM torus, the dynamics is conjugate to a rigid rotation with some fixed frequency vector. Typically, as the perturbation grows the proportion of chaotic orbits increases and more of the tori are destroyed.

In a paper with Evelyn Sander, we explore an alternative technique, based on windowed Birkhoff averages, to distinguish between chaotic, resonant, and quasiperiodic dynamics. We applied a smoothed version of time averaging—based on earlier work of Das, Sander & Yorke—to accurately determine whether an orbit of an area-preserving map is chaotic or not, and when it is regular to compute its rotation number. The picture (right) shows results for the standard map: the color indicates the number of digits computed in the average (up to 18-red) as a function of initial condition (varying y along a line x = 0.321), and of the parameter k. Orbits that are chaotic (digits less than 5) are black. We use this method to construct the so-called critical function: the maximum value of k for which there is an invariant circle of a given rotation number

Invariant tori can also be found numerically by taking limits of periodic orbits and by iterative methods based on the conjugacy to rotation. In these methods, one fixes a frequency vector and attempts to find invariant sets on which the dynamics has this frequency. In the current paper we do not fix the rotation vector in advance, so this method permits one to accurately compute the rotation vector for each initial condition that lies on a regular orbit. As such the method is analogous to Laskar's frequency analysis, which uses a windowed Fourier transform to compute rotation numbers.

More recently we have extended this method to 3D Volume preserving maps, to compute two-tori.

--> In 1994 Moser generalized Hénon's famous quadratic map to the four dimensional case. Moser's quadratic sympletic map has at most four fixed points, and they are organized by a codimension three bifurcation that creates four fixed points at a single point in phase space. In a paper with Arnd Bäcker we study this quadfurcation, and show that it also occurs a when an accelerator mode is created in a four dimensional Froeschlé map.

The figure shows a three-dimsional slice through the 4D phase space. For this case there are four fixed points, two are doubly elliptic (red spheres) and two are elliptic-hyperbolic (green spheres). Invariant two-tori typically intersect the slice in a pair of rings. One such torus is shown projected from 4D, with the fourth dimension indicated by the color scale shown.

In a second paper we discuss the differences between the generic case and that of a weakly coupled pair of Hénon maps. Transport in Chirikov's area-preserving Standard Map appears to be ``quasilinear", that is described by a random walk, when the parameter k is much larger than one: the action of an ensemble of initial conditions diffuses. However, this normal diffusion fails dramatically when upon certain saddle-center bifurcations that lead to accelerator modes. These new orbits create sticky, stable islands that accelerate, and drag chaotic orbits along, leading to super-diffusive behavior.

In a recent paper with Narcis Miguel, Carles Simo, and Arturo Vieiro, we studied the generic form of such accelerator modes in three-dimensional volume-preserving maps. We consider the case of a map with two angle variables, and one action. We show that the local form of a bubble can be described by a quadratic VP map, a special case of that derived in a paper with Lomelí. We discuss the trapping statistics for orbits near the bubble of stabilty, showing that there is a power-law decay similar to that seen in the area-preserving case, and to that we saw in a map with Mullowney and Julien.

The picture shows one example of orbits trapped near 3D bubble, outlining a family of invariant two-tori. --> Transport in Symplectic maps has been studied extensively in the area-preserving case, and is especially well-understood using the ideas of flux through cantori developed by MacKay, Meiss and Percival. The corresponding picture for higher-dimensional symlectic maps is still, largely, open. Nevertheless, we know a number of things. For nearly integrable maps there are typically many invariant tori, by the KAM theorem, and transport, due to Arnold's mechanism, is very slow, according to Nekhoroshev's theorem.

How do these restrictions apply to the volume-prerserving case? There is still a version of KAM theory that applies, so nearly integrable VP maps have many tori. But does Nekhoroshev's theorem apply? In a paper with Guillery, we show that it does not: even for maps with a positive definite twist, there can be rapid transport along resonance channels. The figure shows a 2D action-slice through a phase space of a four dimensional VP map. The grayscale is the FLI: dark gray regions correspond to small Lyapunov exponent, and many invariant tori. The white regions are chaotic due to resonances. The red dots show an orbit that drifts rapidly along a resonance channel, switching from one to another upon intersection. --> Optimal Mixing for a sequence of Harper Maps The mixing of a passive scalar in a fluid is a familiar process~~-you see it in action whenever you stir milk into tea, for example. Mixing requires two things: effective stirring and diffusive spreading. Diffusion is only effective when the scales are very small, thus to design an effective mixer, one must first create a stirring process that stretches and folds the fluids. This process has applications to the remediation of contaminated groundwater, as discussed in a paper with R. Neupauer and D. Mays. When the two fluids being mixed can react, the resulting striations can cause localized enhancement of the reaction rates, as discussed in a paper with K. Pratt and J. Crimaldi.

Recently, Rebecca Mitchell (PhD, 2017) and I studied the problem of how to design an effective mixer taking into account that the device acts over a finite time (so infinite time considerations of Lyapunov exponents and entropy are not really appropriate) and that there are constraints on its design~~-for example the energy of the mixer is limited and the shape of the stirring elements is constrained. For a simple model consisting of sequential shears (essentially Harper maps), we find that one can use a step-by-step method to choose the next stirring action and obtain a near optimal result. The sequence of images at the right shows an optimal protocol for a Gaussian density profile. --> The extremely complicated intermixture of regular and chaotic orbits, seen so often in numerical simulations has not been rigorously verified for generic, smooth symplectic maps. It is known, of course, by KAM theory, that nearly integrable systems have many invariant circles. This same structure holds in the neigbhorhood of an elliptic periodic orbit. At the other extreme, the ergodicity and hyperbolicity properties of Anosov diffeomorphisms are well-understood. This extreme of uniform hyperbolicity can be thought of as a complementary limit to integrability: the study of perturbations from "anti-integrability"; however, this do not lead to proofs of a positive measure of chaotic orbits.

In a recent paper with Lev Lerman, we study a family of area-preserving maps that is homotopic to Arnold's famous cat map, but that has a pair of fixed points, one hyperbolic and one parabolic. We show that the unstable and stable manifolds of these two points define a channel that seems to confine the nonhyperbolic behavior of the system. In particular, the channel contains elliptic orbits, and its complement appears to have positive Lyapunov exponent almost everywhere. The area of the channel is, for small perturbations, strictly less than one. The figure to the right shows a computation of the channel, see the paper for the details. --> The phase space of a typical Hamiltonian system \(H(q,p)\) contains both chaotic and regular orbits, mixed in a complex, fractal pattern. One oft-studied phenomenon is the algebraic decay of correlations and recurrence time distributions. For area-preserving maps, this has been attributed to the stickiness of boundary circles, which separate chaotic and regular components. Though such dynamics has been extensively studied, a full understanding depends on many fine details that typically are beyond experimental and numerical resolution. In a recent paper with Or Alus and Shmuel Fishman we study the statistics of boundary circle winding numbers and island periods. Since phase space transport is of great interest for dynamics, we compute the distributions of fluxes through island chains.

The figure at the right shows an island hierarchy for the Hénon quadratic area-preserving map near the tripling bifurcation of its stable fixed point. --> Finite-time transport between distinct flow regions is of great relevance to many scientific applications, yet quantitative studies remain scarce to date. The primary obstacle is computing the evolution of material volumes, which is often infeasible due to extreme interfacial stretching. In a recent PRL, Brock Mosovsky, Michel Speetjens and I present a framework for describing and computing finite-time transport in n-dimensional (chaotic) volume-preserving flows that relies on the reduced dynamics of an (n-2)-dimensional “minimal set” of fundamental trajectories. This approach has essential advantages over existing methods: the regions between which transport is investigated can be arbitrarily specified; no knowledge of the flow outside the finite transport interval is needed; and computational effort is substantially reduced. We demonstrate our framework in 2D for an industrial mixing device, the RAM mixer, as shown in the figure. Here we compute the transport from an inner disk (the grey region) to the outer annulus, for six different pipe lengths, or flow times. --> Nearly integrable volume-preserving maps with two angles and one action have robust invariant tori with Diophantine rotation vectors (under smoothness and twist conditions); this is a consequence of Jeff Xia’s KAM theory. How long do these tori persist? Are they replaced by “cantori” when they are destroyed? In Fox and Meiss we study a volume-preserving family introduced by Dullin and Meiss that represents (just like Chirikov’s standard map) a generic form near resonance. We show that a generalization of John Greene’s residue criterion for the standard map holds: tori exist when nearby periodic orbits are stable, i.e., their residues are small, and are destroyed when these residues blow-up. We study tori with “spiral tails” in their generalized Farey tree expansions, and look for the most robust torus, in an attempt to discover a higher-dimensional version of the noble tori of the area-preserving case.

-->Differential Dynamical Systems James D. Meiss (SIAM, 2007)

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.

The phase space of an integrable volume-preserving map with one action is foliated by a one-parameter family of invariant tori. Perturbations lead to chaotic dynamics with interesting transport properties. In Resonances and Twist in Volume-Preserving Maps we show that near a rank-one resonant torus the mapping can be reduced to a volume-preserving standard map. This map is a twist map only when the frequency map crosses the resonance curve transversely. We show that these maps can be reduced using averaging theory to the usual area-preserving twist or nontwist standard maps. In the volume-preserving setting the twist condition is shown to be distinct from the nondegeneracy condition of used in KAM theory.

The figure shows some "reconnecting" orbits of the normal form near a tangency of the frequency map with the (1,0,1) resonance. More details are in the preprint Resonances and Twist in Volume-Preserving Maps. 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. -->

Bibliographic Information

  1. My Vita (pdf file)
  2. My Erdös number is at most 4

News Items

Books, Pedagogy and Reviews

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. ()
  • Hazeltine, R.D. and J.D. Meiss (1991). Plasma Confinement. Redwood City, CA, Addison-Wesley, 394 pp., ISBN 0201-53353-5.
  • Hazeltine, R.D. and J.D. Meiss, , (2003) 2nd Edition, Dover Press, 480 pp., ISBN 0486432424. ()
  • Meiss, J.D., Differential Dynamical Systems, (2007) SIAM, Philadelphia 412 pp., ISBN 978-0-899816-35-1.
  • Meiss, J.D., . (2017) SIAM, Philadelphia, 392 pp., ISBN 978-1-61197-463-8.

Pedagogical Articles

  • Meiss, J.D., "Symplectic Maps, Variational Principles, and Transport", (reprint).
  • Meiss, J.D., Hamiltonian Systems, Symplectic Maps, and The Standard Map, articles in the , ed. Alwyn Scott. (New York, Routledge) (2005). ISBN: 1-57958-385-7
  • Meiss, J.D., "Dynamical systems"", .
  • Meiss, J.D., "Hamiltonian systems"", .
  • Meiss, J.D., "Visual Explorations of Dynamics: the Standard Mapping", (), (Corrected reprint).
  • Meiss, J.D., "Thirty Years of Turnstiles and Transport", ().
  • Meiss, J.D., "Ordinary Differential Equations",
  • Meiss, J.D., "Integrability, Anti-Integrability and Volume-Preserving Maps",

Fields of Research

Computational Topology

  • Robins, V., J.D. Meiss, and E. Bradley, "Computing Connectedness: an exercise in computational topology", .
  • Robins, V., J.D. Meiss, and L. Bradley, "Computing Connectedness: Disconnectedness and Discreteness", . (PDF reprint),
  • Alexander, V., J.D. Meiss, E. Bradley, and J. Garland, "Iterated Function System Models in Data Analysis: Detection and Separation", . ().
  • Alexander, V., E. Bradley, J.D. Meiss, and N. Sanderson, "Simplicial Multivalued Maps and the Witness Complex for Dynamical Analysis of Time Series", . ().
  • Garland, J., E. Bradley, and J.D. Meiss, "Exploring the Topology of Dynamical Reconstructions", ().
  • Sanderson, N., E. Shugerman, S Molnar, J.D. Meiss, and E. Bradley, "Computational Topology Techniques for Characterizing Time-Series Data", , ().
  • Deshmukh, V., T.E. Berger, E. Bradley, and J.D. Meiss, "Leveraging the Mathematics of Shape for Solar Magnetic Eruption Prediction", . ()
  • Deshmukh, V., V., E. Bradley, J. Garland, J.D. Meiss, "Using Curvature to Select the Time Lag for Delay Reconstruction", . ()
  • Deshmukh, V., T. Berger, J.D. Meiss and E. Bradley "Shape-based Feature Engineering for Solar Flare Prediction", . ()
  • Deshmukh, V., E. Bradley, J. Garland, J.D. Meiss "Towards automated extraction and characterization of scaling regions in dynamical systems", . ()
  • Deshmukh,V. R. Meikle, E. Bradley, J.D. Meiss, and J. Garland "Using scaling-region distributions to select embedding parameters", ()
  • Deshmukh, V., S. Baskar, T.E. Berger, E. Bradley, and J.D. Meiss, "Comparing Feature Sets and Machine Learning Models for Prediction of Solar Flares:Topology, Physics, and Model Complexity", ()

Fluid Dynamics

  • Meiss, J.D. and K.M. Watson (1978). "Discussion of Some Weakly Nonlinear Systems in Continuum Mechanics", AIP Conf. Proceedings 46: 296-323.
  • Meiss, J.D. (1979) "Integrability of Multiple Three-Wave Interactions", .
  • Meiss, J.D., N. Pomphrey and K.M. Watson, "Numerical Analysis of Weakly Nonlinear Wave Turbulence", .
  • Meiss, J.D. (1980), "Internal Wave Interactions in the Induced Diffusion Approximation", Geophysical Fluid Dynamics, Woods Hole, WHOI (1980).
  • Meiss, J.D. and W.R. Young (1980) "A Forcer Burgers Equation" Geophysical Fluid Dynnamics, Woods Hole, WHOI (1980).
  • Pomphrey, N., J.D. Meiss and K.M. Watson, "Description of Nonlinear Internal Wave Interactions Using Langevin Methods", .
  • Meiss, J.D., "Numerical Computation of Relaxation Rates for the Test Wave Model", .
  • Mullowney, P., K. Julian and J.D. Meiss, "Blinking rolls: chaotic advection in a 3D flow with an Invariant", . (PDF reprint).
  • Mullowney, P., K. Julien, and J.D. Meiss, "Chaotic Advection in the Kuppers-Lortz State", . ().
  • Mosovsky, B.A., M.F.M. Speetjens, and J.D. Meiss, "Finite-Time Transport in Volume-Preserving Flows",
  • Neupauer, R.M., J.D. Meiss, and D.C. Mays "Chaotic advection and reaction during engineered injection and extraction in heterogeneous porous media", .
  • Pratt, K.R., J.D. Meiss, and J.P. Crimaldi, "Reaction Enhancement of Initially Distant Scalars by Lagrangian Coherent Structures", . ().
  • Mitchell, R.A., and J.D. Meiss, "Designing a Finite-Time Mixer: Optimizing Stirring for Two-Dimensional Maps." . ().
  • Dullin, H.R., J.D. Meiss, and J. Worthington, "Poisson Structure of the Three-Dimensional Euler Equations in Fourier Space." . ().

Hamiltonian Dynamics

  • MacKay, R.S.and J.D. Meiss, "Flux and Differences in Action for Continuous Time Hamiltonian Systems." .
  • Meiss, J.D.. "Transport Near the Onset of Stochasticity." J. Part. Accel. 19: 9-24. (1986)
  • Meiss, J.D. and E. Ott, "Markov Tree Model of Transport in Area-Preserving Maps." .
  • Chen, Q., J.D. Meiss, and I.C. Percival, "Orbit Extension Method for Finding Unstable Orbits", .
  • Meiss, J.D. ", Physics Today, Physics News of 1986, January (1987).
  • MacKay, R.S. and J.D. Meiss, "The Relation between Quantum and Classical Thresholds for Multi-photon Ionization of Excited Atoms", .
  • Meiss, J.D., "Comment on Microwave Ionization of H-atoms: breakdown of classical dynamics for high frequencies", .
  • Bollt E. and J.D. Meiss, "Targeting Chaotic Orbits to the Moon", . (PDF reprint).
  • Howard, J.E. and J.D. Meiss "Straight Line Orbits in Hamiltonian Flows", . ().
  • Restrepo, J.G. and J.D. Meiss, "Onset of Synchronization in the Disordered Hamiltonian Mean Field Model", . ().
  • Virkar, Y.S., J.G. Restrepo and J.D. Meiss, "The Hamiltonian Mean Field model: effect of network structure on synchronization dynamics", . ().
  • Duignan, N. and J.D. Meiss, "Nonexistence of Invariant Tori Transverse to Foliations: An Application of Converse KAM Theory", . ()
  • Duignan, N. and J.D. Meiss "Distinguishing between Regular and Chaotic orbits of Flows by the Weighted Birkhoff Average", ()
  • J.D. Meiss "Hamiltonian Triplet Interactions: Areal and Perimetric Forces", accepted for SIAM J. Dyn. Systems (2025). ()

Plasma Physics

  • Meiss, J.D. and C. W. Horton, "Drift-Wave Turbulence from a Soliton Gas." .
  • Meiss, J.D. and C. W. Horton, "Fluctuation Spectra of a Drift Wave Soliton Gas." .
  • Meiss, J.D. and C. W. Horton (1983). "Solitary Drift Waves in the Presence of Magnetic Shear." .
  • Meiss, J.D. and P. J. Morrison, "Nonlinear Electron Landau Damping of Ion Acoustic Solitons." .
  • Horton, W., J. Liu, J.D. Meiss and J. E. Sedlak, "Solitary Vortices in a Rotating Plasma." .
  • Mirnov, V.V., J.D. Meiss and J. L. Tennyson (1986). "Relaxation to the Steady State in Neutral-Beam Injected Mirrors", .
  • Aydemir, A., R.D. Hazeltine, J.D. Meiss, and M. Kotschenreuther, "Destabilization of Alfven-Resonant Modes by Resistivity and Diamagnetic Drifts", .
  • Meiss, J.D., "Transport Near the Onset of Chaos", Physics Today, Physics News of 1986, January (1987).
  • Aydemir, A. Y., 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.
  • Meiss, J.D., "Comment on Microwave Ionization of H-atoms: breakdown of classical dynamics for high frequencies", .
  • Meiss, J.D. and R.D. Hazeltine, "Canonical Coordinates for Guiding Center Particles", .
  • Hayashi, T., T. Sato, H.J. Gardner and J.D. Meiss, "Evolution of Magnetic Islands in a Heliac", .
  • Tennyson, J.L., J.D. Meiss and P.J. Morrison, "Self-Consistent Chaos in the Beam-Plasma Instability", . (PDF reprint).
  • Burby, J., N. Duignan and J.D. Meiss, "Integrability, Normal Forms, and Magnetic Axis Coordinates", . ()
  • Duignan, N. and J.D. Meiss, "Normal Forms and Near-Axis Expansions for Beltrami Magnetic Fields", ()
  • Burby, J., N. Duignan and J.D. Meiss, "Minimizing Separatrix Crossings through Isoprominence", ()

Classes of Dynamical Systems

Area-Preserving Maps

  • Cary, J. R. and J.D. Meiss, "Rigorously Diffusive Deterministic Map", .
  • Cary, J. R., J.D. Meiss and A. Bhattacharjee, "Statistical Characterization of Periodic Area Preserving Mappings." .
  • Meiss, J.D., "Class Renormalization: Islands around Islands", .
  • Meiss, J.D. and E. Ott, "Markov Tree Model of Transport in Area-Preserving Maps." .
  • Chen, Q., J.D. Meiss, and I.C. Percival, "Orbit Extension Method for Finding Unstable Orbits", .
  • MacKay, R.S., J.D. Meiss, and I.C. Percival, "Resonances in Area Preserving Maps", .
  • Chen, Q. and J.D. Meiss, "Flux, Resonances and the Devil's Staircase for the Sawtooth Map", .
  • Meiss, J.D. 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.
  • Meiss, J.D., "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.
  • Meiss, J.D., "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.
  • Dewar, R.L. and J.D. Meiss, "Flux-Minimizing Curves for Reversible Area-Preserving Maps", (PDF reprint).
  • Meiss, J.D., "Cantori for the Stadium Billiard", .
  • Meiss, J.D., "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).
  • Meiss, J.D., "Transient Measures for the Standard Map", . (PDF reprint).
  • Lomelí, H.E. and J.D. Meiss "Heteroclinic Orbits and Transport in a Perturbed, Integrable Standard Map". . ().
  • Dullin, H.R., D. Sterling and J.D. Meiss "Self-Rotation Number using the Turning Angle", .
  • Meiss, J.D., "Visual Explorations of Dynamics: the Standard Mapping", ()., (Corrected reprint).
  • Gidea, M., J.D. Meiss, I. Ugarcovici, H. Weiss, "Applications of KAM Theory to Population Dynamics", .
  • Fox, A.M., and J.D. Meiss, "Critical Invariant Circles in Asymmetric and Multiharmonic Generalized Standard Maps", . ().
  • Alus, O., S. Fishman, and J.D. Meiss, "Statistics of the Island-Around-Island Hierarchy in Hamiltonian Phase Space", . ().
  • Meiss, J.D., "Thirty Years of Turnstiles and Transport", ().
  • Lerman, L.M. and J.D. Meiss, "Mixed Dynamics in a Parabolic Standard Map", . ().
  • Alus, O., S. Fishman, and J.D. Meiss, "Probing the statistics of transport in the Henon Map", () ().
  • Mitchell, R.A. and J.D. Meiss, "Designing a Finite-Time Mixer: Optimizing Stirring for Two-Dimensional Maps", . ().
  • Alus, O., S. Fishman, and J.D. Meiss, "Universal exponent for transport in mixed Hamiltonian dynamics", ().
  • Sander, E. and J.D. Meiss, "Birkhoff Averages and Rotational Invariant Circles for Area-Preserving Maps", ().
  • Lomelí, H.E. and J.D. Meiss, "Symmetry Reduction and Rotation Numbers for Poncelet maps", submitted to Nonlinearity (2023) ().

Symplectic Maps

  • Kook, H.T. and J.D. Meiss, "Periodic Orbits for Reversible, Symplectic Mappings", .
  • Kook, H.T. and J.D. Meiss, "Application of Newton's Method to Lagrangian Dynamical Systems", .
  • MacKay, R.S., J.D. Meiss, and J. Stark, "Converse KAM Theory for Symplectic Twist Maps", .
  • Kook, H.T. and J.D. Meiss, "Diffusion in Symplectic Maps", .
  • Meiss, J.D., "Symplectic Maps, Variational Principles, and Transport",
  • Bollt, E. and J.D. Meiss, "Breakup of Invariant Tori for the Four Dimensional Semi-Standard Map", . (PDF reprint).
  • Easton, R.W., J.D. Meiss and S. Carver, "Exit Times and Transport for Symplectic Twist Maps", .
  • Bollt, E. and J.D. Meiss, "Controlling Transport Through Recurrences", .
  • MacKay, R. S., J.D. Meiss and J. Stark, "An Approximate Renormalization for the Break-up of Invariant Tori with Three Frequencies", . (reprint).
  • Meiss, J.D., "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). (PDF Preprint).
  • Meiss, J.D., "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). (PDF Preprint).
  • Dullin, H.R. and J.D. Meiss, "Stability of Minimal Periodic Orbits", . (PDF reprint).
  • Bäcker, A. and J.D. Meiss, "Moser's Quadratic, Symplectic Map", . ().
  • Bäcker, A. and J.D. Meiss, (2018).
  • Bäcker, A. and J.D. Meiss, "The Quadfurcation", "Elliptic Bubbles in Moser's 4D Quadratic Map: the Quadfurcation", , ().

Three-Dimensional Maps

  • Meiss, J.D., "Average Exit Times in Volume-Preserving Maps", . (PDF reprint).
  • Lomelí, H.E. and J.D. Meiss, "Quadratic Volume-Preserving Maps", . () (PDF preprint).
  • Lenz, K.E., H.E. Lomelí and J.D. Meiss, "Quadratic Volume Preserving Maps: an Extension of a Result of Moser", . (PDF preprint).
  • Lomelí, H.E. and J.D. Meiss, "Heteroclinic Primary Intersections and Codimension one Melnikov Method for Volume-Preserving Maps", (PDF reprint),
  • Gomez, A. and J.D. Meiss, "Volume-Preserving Maps with an Invariant", . (PDF reprint)
  • Lomelí, H.E. and J.D. Meiss, "Heteroclinic intersections between Invariant Circles of Volume-Preserving Maps", () (PDF preprint)
  • Mullowney, P., K. Julian and J.D. Meiss, "Blinking rolls: chaotic advection in a 3D flow with an Invariant", . (PDF reprint)
  • Wysham, D.B. and J.D. Meiss, "Numerical Computation of the Stable Manifolds of Tori", . ()
  • Gonchenko, S.V., J.D. Meiss and I.I. Ovsyannikov, "Chaotic Dynamics of Three-Dimensional Henon Maps That Originate from a Homoclinic Bifurcation", ().
  • Meiss, J.D. "Dynamics of Volume-Preserving Maps",
  • Dullin, H.R. and J.D. Meiss, "Nilpotent Normal form for Divergence Free Vector Fields and Volume-Preserving Maps", ().
  • Lomelí, H.E., J.D. Meiss, and R. Ramirez-Ros, "Canonical Melnikov Theory for Diffeomorphisms", ().
  • Mullowney, P., K. Julien, and J.D. Meiss, "Chaotic Advection in the Kuppers-Lortz State", . ().
  • Lomelí, H.E. and J.D. Meiss, "Generating Forms for Exact Volume-Preserving Maps", ().
  • Dullin, H.R. and J.D. Meiss, "Quadratic Volume-Preserving Maps: Invariant Circles and Bifurcations", ().
  • Lomelí, H.E. and J.D. Meiss, "Resonance Zones and Lobe Volumes for Volume-Preserving Maps", ().
  • Dullin, H.R. and J.D. Meiss, "Resonances and Twist in Volume-Preserving Maps", ().
  • Meiss, J.D., "The Destruction of Tori in Volume-Preserving Maps", ().
  • Dullin, H. R., H. Lomelí and J.D. Meiss, "Symmetry Reduction by Lifting for Maps", ().
  • Fox, A.M. and J.D. Meiss, "Greene's Residue Criterion for the Breakup of Invariant Tori of Volume-Preserving Maps", . ().
  • Fox, A. M. and J.D. Meiss, "Computing the Conjugacy of Invariant Tori for Volume-Preserving Maps", SIAM J. Appl. Dyn. Sys. 15(1): 557-579 (2016) . ().
  • Guillery, N. and J.D. Meiss, "Diffusion and Drift in Volume-Preserving Maps",. ().
  • Meiss, J.D., N. Miguel, C. Simo, A. Vieiro, "Accelerator modes and anomalous diffusion in 3D volume-preserving maps",. ()
  • Meiss, J.D. and E. Sander, "Birkhoff Averages and the Breakdown of Invariant Tori in Volume-Preserving Maps", , ()
  • Hampton, A.E. and J.D. Meiss, "Anti-Integrability for Three-Dimensional Quadratic Maps", , ()
  • Hampton, A.E. and J.D. Meiss, “The three-dimensional generalized Hénon map: Bifurcations and attractors”, ()
  • Hampton, A.E. and J.D. Meiss, “Connecting Anti-integrability to Attractors for Three-Dimensional, Quadratic Diffeomorphisms” ()

Phenomena and Methods

Anti-Integrability

  • Chen, Q., R.S. MacKay, and J.D. Meiss, "Cantori for Symplectic Maps", .
  • MacKay, R.S. and J.D. Meiss, "Cantori for Symplectic Maps near the Anti-integrable Limit", .
  • Sterling, D. and J.D. Meiss, "Computing Periodic Orbits using the Anti-Integrable Limit", . ().
  • Sterling, D., H. R. Dullin and J.D. Meiss, "Homoclinic Bifurcations for the Henon Map", . ().
  • Easton, R.W., J.D. Meiss, G. Roberts, "Drift by Coupling to an Anti-Integrable Limit", . (PDF reprint).
  • Dullin, H.R., J.D. Meiss, and D. Sterling, "Symbolic Codes for Rotational Orbits", SIAM J.Appl. Dyn. Sys. 4 515-562 (2005). (), (PDF reprint).
  • Meiss, J.D. "Integrability, Anti-Integrability and Volume-Preserving Maps",
  • Hampton, A.E. and J.D. Meiss "Anti-Integrability for Three-Dimensional Quadratic Maps",, ()
  • Hampton, A.E. and J.D. Meiss, “The three-dimensional generalized Hénon map: Bifurcations and attractors”, ()
  • Hampton, A.E. and J.D. Meiss, “Connecting Anti-integrability to Attractors for Three-Dimensional, Quadratic Diffeomorphisms” ()

Converse KAM Theory

  • MacKay, R.S., J.D. Meiss, and J. Stark, "Converse KAM Theory for Symplectic Twist Maps", .
  • Duignan, N. and J.D. Meiss "Nonexistence of Invariant Tori Transverse to Foliations: An Application of Converse KAM Theory", . ()

Invariant Tori

  • Bollt, E. and J.D. Meiss, "Breakup of Invariant Tori for the Four Dimensional Semi-Standard Map", . (PDF reprint).
  • MacKay, R. S., J.D. Meiss and J. Stark, "An Approximate Renormalization for the Break-up of Invariant Tori with Three Frequencies", . (PDF reprint).
  • Meiss, J.D., "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). (PDF Preprint).
  • Meiss, J.D., "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). (PDF Preprint).
  • Wysham, D.B. and J.D. Meiss, "Numerical Computation of the Stable Manifolds of Tori", . ().
  • Dullin, H.R. and J.D. Meiss, "Quadratic Volume-Preserving Maps: Invariant Circles and Bifurcations", ().
  • Meiss, J.D., "The Destruction of Tori in Volume-Preserving Maps", ().
  • Fox, A.M., and J.D. Meiss, "Greene's Residue Criterion for the Breakup of Invariant Tori of Volume-Preserving Maps", . ().
  • Fox, A.M., and J.D. Meiss, "Critical Invariant Circles in Asymmetric and Multiharmonic Generalized Standard Maps", ().
  • Fox, A. M. and J.D. Meiss, "Computing the Conjugacy of Invariant Tori for Volume-Preserving Maps." SIAM J. Appl. Dyn. Sys. 15(1): 557-579 (2016) . ().
  • Sander, E. and J.D. Meiss, "Birkhoff Averages and Rotational Invariant Circles for Area-Preserving Maps", (arXiv Preprint ).
  • Duignan, N. and J.D. Meiss "Nonexistence of Invariant Tori Transverse to Foliations: An Application of Converse KAM Theory", . ()
  • Meiss, J.D. and E. Sander "Birkhoff Averages and the Breakdown of Invariant Tori in Volume-Preserving Maps", , ()
  • Duignan, N. and J.D. Meiss "Distinguishing between Regular and Chaotic orbits of Flows by the Weighted Birkhoff Average", ()

Piecewise Smooth Bifurcations

  • Simpson, D.J.W. and J.D. Meiss, "Andronov-Hopf Bifurcations in Planar, Piecewise-Smooth, Continuous Flows", ().
  • Simpson, D.J.W. and J.D. Meiss, "Neimark-Sacker Bifurcations in Planar, Piecewise Smooth, Continuous Maps",
  • Simpson, D.J.W. and J.D. Meiss, "Unfolding a Codimension-Two, Discontinuous, Andronov-Hopf Bifurcation", ().
  • Simpson, D.J.W., D.S. Kompala, and J.D. Meiss, "Discontinuity Induced Bifurcations in a Model of Saccharomyces cerevisiae", () (PDF reprint).
  • Simpson, D.J.W. and J.D. Meiss, "Shrinking Point Bifurcations of Resonance Tongues for Piecewise-Smooth, Continuous Maps", ().
  • Simpson, D.J.W. and J.D. Meiss, "Simultaneous Border-Collision and Period-Doubling Bifurcations", ().
  • Simpson, D.J.W. and J.D. Meiss, "Resonance near Border-Collision Bifurcations in Piecewise-Smooth, Continuous Maps", ().
  • Simpson, D.J.W. and J.D. Meiss, "Aspects of Bifurcation Theory for Piecewise-Smooth, Continuous Systems", ().

Polynomial Maps

  • Dullin, H.R. and J.D. Meiss, "Generalized Hénon Maps: the Cubic Polynomial Diffeomorphisms of the Plane", .
  • Gomez, A. and J.D. Meiss, "Reversible Polynomial Automorphisms of the Plane: the Involutory Case", . (PDF reprint).
  • Gomez, A. and J.D. Meiss, "Reversible Polynomial Automorphisms in the Plane", . () (PDF reprint).
  • Gonchenko, S.V., J.D. Meiss and I.I. Ovsyannikov, "Chaotic Dynamics of Three-Dimensional Henon Maps That Originate from a Homoclinic Bifurcation", ().
  • Bäcker, A. and J.D. Meiss, "Moser's Quadratic, Symplectic Map", . ().
  • Bäcker, A. and J.D. Meiss, "The Quadfurcation", (2018).
  • Bäcker, A. and J.D. Meiss, Elliptic Bubbles in Moser's 4D Quadratic Map: the Quadfurcation", , (). ()
  • Hampton, A.E. and J.D. Meiss, "Anti-Integrability for Three-Dimensional Quadratic Maps", , ()
  • Hampton, A.E. and J.D. Meiss, “The three-dimensional generalized Hénon map: Bifurcations and attractors”, ()
  • Hampton, A.E. and J.D. Meiss “Connecting Anti-integrability to Attractors for Three-Dimensional, Quadratic Diffeomorphisms” ()

Semantics and Textual Dynamics

  • Doxas, I., J. Meiss, S. Bottone, T. Strelich, A. Plummer, A. Breland, S. Dennis, K. Garvin-Doxas, and M. Klymkowsky "Narrative as a Dynamical System", ()
  • Doxas, I., J. Meiss, S. Bottone, T. Strelich, A. Plummer, A. Breland, S. Dennis, K. Garvin-Doxas, and M. Klymkowsky "The Dynamical Principles of Storytelling" ()

Solitons

  • Meiss, J.D. and N. R. Pereira, "Internal Wave Solitons", .
  • Meiss, J.D. and C. W. Horton, "Drift-Wave Turbulence from a Soliton Gas." .
  • Meiss, J.D. and C. W. Horton, "Fluctuation Spectra of a Drift Wave Soliton Gas." .
  • Meiss, J.D. and C. W. Horton (1983). "Solitary Drift Waves in the Presence of Magnetic Shear." .
  • Meiss, J.D. and P. J. Morrison, "Nonlinear Electron Landau Damping of Ion Acoustic Solitons." .
  • Morrison, P. J., J.D. Meiss and J. R. Cary, "Scattering of Regularized-Long-Wave Solitary Waves." .
  • Horton, W., J. Liu, J.D. Meiss and J. E. Sedlak, "Solitary Vortices in a Rotating Plasma." .

Synchronization

  • Restrepo, J.G. and J.D. Meiss, "Onset of Synchronization in the Disordered Hamiltonian Mean Field Model", . ().
  • Virkar, Y.S., J.G. Restrepo and J.D. Meiss, "The Hamiltonian Mean Field model: effect of network structure on synchronization dynamics", . ().

Transitory Dynamics

Transport

  • MacKay,R.S.and J.D. Meiss, "Flux and Differences in Action for Continuous Time Hamiltonian Systems." .
  • Meiss, J.D.. "Transport Near the Onset of Stochasticity." J. Part. Accel. 19: 9-24. (1986)
  • Meiss, J.D., "Class Renormalization: Islands around Islands", .
  • MacKay, R.S. and Meiss, J.D. and Percival, I.C., "Transport in Hamiltonian Systems', .
  • Meiss, J.D. and E. Ott, "Markov Tree Model of Transport in Area-Preserving Maps." .
  • Meiss, J.D., "Transport Near the Onset of Chaos", Physics Today, Physics News of 1986, January (1987).
  • Meiss, J.D., "Symplectic Maps, Variational Principles, and Transport", (reprint).
  • Easton, R.W., J.D. Meiss and S. Carver, "Exit Times and Transport for Symplectic Twist Maps", .
  • Bollt, E. and J.D. Meiss, "Controlling Transport Through Recurrences", .
  • Lomelí, H.E. and J.D. Meiss "Heteroclinic Orbits and Flux in a Perturbed, Integrable Standard Map", . ()
  • Lomelí, H.E. ; and J.D. Meiss, "Heteroclinic Primary Intersections and Codimension one Melnikov Method for Volume-Preserving Maps", (PDF reprint).
  • Mosovsky, B.A. and J.D. Meiss, "Transport in Transitory Dynamical Systems", () (PDF reprint).
  • Mosovsky, B.A. and J.D. Meiss, "Transport in Transitory, Three-Dimensional, Liouville Flows", () (PDF reprint).
  • Mosovsky, B.A., M.F.M. Speetjens, and J.D. Meiss, "Finite-Time Transport in Volume-Preserving Flows",
  • Alus, O., S. Fishman, and J.D. Meiss, "Statistics of the Island-Around-Island Hierarchy in Hamiltonian Phase Space", . ().
  • Pratt, K.R., J.D. Meiss, and J.P. Crimaldi, "Reaction Enhancement of Initially Distant Scalars by Lagrangian Coherent Structures", . ().
  • Meiss, J.D., "Thirty Years of Turnstiles and Transport", ().
  • Lerman, L.M. and J.D. Meiss, "Mixed Dynamics in a Parabolic Standard Map", . ().
  • Alus, O., S. Fishman, and J.D. Meiss, "Probing the statistics of transport in the Henon Map", ().
  • Mitchell, R.A., and J.D. Meiss, "Designing a Finite-Time Mixer: Optimizing Stirring for Two-Dimensional Maps." . ().
  • Alus, O., S. Fishman, and J.D. Meiss, Universal exponent for transport in mixed Hamiltonian dynamics, ().
  • Guillery, N. and J.D. Meiss, "Diffusion and Drift in Volume-Preserving Maps", ().
  • Meiss, J.D., N. Miguel, C. Simo, A. Vieiro, Accelerator modes and anomalous diffusion in 3D volume-preserving maps", . ()
  • Homan, J.R., and J.D. Meiss, "Noise-enhanced Stickiness in the Harper Map." ()

Twistless Bifurcations

  • Dullin, H.R., J.D. Meiss and D. Sterling, "Generic Twistless Bifurcations", . (
  • Dullin, H.R. and J.D. Meiss, "Twist Singularities for Symplectic Maps", (PDF reprint).
  • Dullin, H.R., A.V. Ivanov and J.D. Meiss, "Normal Forms for 4D Symplectic Maps with Twist Singularities", . ().
  • Dullin, H.R. and J.D. Meiss, "Resonances and Twist in Volume-Preserving Maps", ().

Weighted Birkhoff Averages

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