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, "Hidden Symmetries and Fusion Energy" 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.

Play 

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.

 

-->

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
  3. ORCID
  4. Researcher ID
  5. Google Scholar
  6. zbMath
  7. Research.com Ranking

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. (Buy from Amazon)
  • 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, Plasma Confinement, (2003) 2nd Edition, Dover Press, 480 pp., ISBN 0486432424. (Buy from Amazon)
  • Meiss, J.D., Differential Dynamical Systems, (2007) SIAM, Philadelphia 412 pp., ISBN 978-0-899816-35-1.
  • Meiss, J.D., Differential Dynamical Systems: Revised Edition. (2017) SIAM, Philadelphia, 392 pp., ISBN 978-1-61197-463-8.

Pedagogical Articles

Fields of Research

Computational Topology

Fluid Dynamics

Hamiltonian Dynamics

Plasma Physics

Classes of Dynamical Systems

Area-Preserving Maps

Symplectic Maps

Three-Dimensional Maps

Phenomena and Methods

Anti-Integrability

Converse KAM Theory

Invariant Tori

Piecewise Smooth Bifurcations

Polynomial Maps

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", (arXiv Preprint)
  • 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" (arXiv Preprint)

Solitons

Synchronization

Transitory Dynamics

Transport

Twistless Bifurcations

Weighted Birkhoff Averages

Return to my home pagerevised March 16, 2026