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James D. Meiss

Publications and source records attributed to James D. Meiss.

At least 19 recordsLinked to original sources

All field strengths are possible locally in Boozer coordinates

The magnetic field strength is a central design object in the theory of magnetic confinement of plasma. Desirable confinement properties such as quasi-symmetry and omnigenity are characterized by special forms of the field strength when it is given in the particular coordinate system called ``Boozer coordinates." We ask a basic realizability question: which functions of Boozer coordinates can arise as the magnitude of a magnetic field that is written in Boozer coordinates? Using exterior differential systems and the Cartan-Kähler theorem, we show that every positive analytic function is locally realizable. Thus there is no local analytic obstruction to prescribing the field strength in Boozer coordinates; though further restrictions might arise from global geometric or additional physical constraints.

physics.plasm-ph

Distinguishing between Regular and Chaotic orbits of Flows by the Weighted Birkhoff Average

This paper investigates the utility of the weighted Birkhoff average (WBA) for distinguishing between regular and chaotic orbits of flows, extending previous results that applied the WBA to maps. It is shown that the WBA can be super-convergent for flows when the dynamics and phase space function are smooth, and the dynamics is conjugate to a rigid rotation with Diophantine rotation vector. The dependence of the accuracy of the average on orbit length and width of the weight function width are investigated. In practice, the average achieves machine precision of the rotation frequency of quasiperiodic orbits for an integration time of O(10^3) periods. The contrasting, relatively slow convergence for chaotic trajectories allows an efficient discrimination criterion. Three example systems are studied: a two-wave Hamiltonian system, a quasiperiodically forced, dissipative system that has a strange attractor with no positive Lyapunov exponents, and a model for magnetic field line flow.

math.DS

Using scaling-region distributions to select embedding parameters

Reconstructing state-space dynamics from scalar data using time-delay embedding requires choosing values for the delay $τ$ and the dimension $m$. Both parameters are critical to the success of the procedure and neither is easy to formally validate. While embedding theorems do offer formal guidance for these choices, in practice one has to resort to heuristics, such as the average mutual information (AMI) method of Fraser & Swinney for $τ$ or the false near neighbor (FNN) method of Kennel et al. for $m$. Best practice suggests an iterative approach: one of these heuristics is used to make a good first guess for the corresponding free parameter and then an "asymptotic invariant" approach is then used to firm up its value by, e.g., computing the correlation dimension or Lyapunov exponent for a range of values and looking for convergence. This process can be subjective, as these computations often involve finding, and fitting a line to, a scaling region in a plot: a process that is generally done by eye and is not immune to confirmation bias. Moreover, most of these heuristics do not provide confidence intervals, making it difficult to say what "convergence" is. Here, we propose an approach that automates the first step, removing the subjectivity, and formalizes the second, offering a statistical test for convergence. Our approach rests upon a recently developed method for automated scaling-region selection that includes confidence intervals on the results. We demonstrate this methodology by selecting values for the embedding dimension for several real and simulated dynamical systems. We compare these results to those produced by FNN and validate them against known results -- e.g., of the correlation dimension -- where these are available. We note that this method extends to any free parameter in the theory or practice of delay reconstruction.

physics.data-an

Minimizing Separatrix Crossings through Isoprominence

A simple property of magnetic fields that minimizes bouncing to passing type transitions of guiding center orbits is defined and discussed. This property, called isoprominence, is explored through the framework of a near-axis expansion. It is shown that isoprominent magnetic fields for a toroidal configuration exist to all orders in a formal expansion about a magnetic axis. Some key geometric features of these fields are described.

physics.plasm-ph

Machine Learning Approaches to Solar-Flare Forecasting: Is Complex Better?

Recently, there has been growing interest in the use of machine-learning methods for predicting solar flares. Initial efforts along these lines employed comparatively simple models, correlating features extracted from observations of sunspot active regions with known instances of flaring. Typically, these models have used physics-inspired features that have been carefully chosen by experts in order to capture the salient features of such magnetic field structures. Over time, the sophistication and complexity of the models involved has grown. However, there has been little evolution in the choice of feature sets, nor any systematic study of whether the additional model complexity is truly useful. Our goal is to address these issues. To that end, we compare the relative prediction performance of machine-learning-based, flare-forecasting models with varying degrees of complexity. We also revisit the feature set design, using topological data analysis to extract shape-based features from magnetic field images of the active regions. Using hyperparameter training for fair comparison of different machine-learning models across different feature sets, we show that simpler models with fewer free parameters \textit{generally perform better than more-complicated models}, ie., powerful machinery does not necessarily guarantee better prediction performance. Secondly, we find that \textit{abstract, shape-based features contain just as much useful information}, for the purposes of flare prediction, as the set of hand-crafted features developed by the solar-physics community over the years. Finally, we study the effects of dimensionality reduction, using principal component analysis, to show that streamlined feature sets, overall, perform just as well as the corresponding full-dimensional versions.

astro-ph.SR

Towards automated extraction and characterization of scaling regions in dynamical systems

Scaling regions -- intervals on a graph where the dependent variable depends linearly on the independent variable -- abound in dynamical systems, notably in calculations of invariants like the correlation dimension or a Lyapunov exponent. In these applications, scaling regions are generally selected by hand, a process that is subjective and often challenging due to noise, algorithmic effects, and confirmation bias. In this paper, we propose an automated technique for extracting and characterizing such regions. Starting with a two-dimensional plot -- e.g., the values of the correlation integral, calculated using the Grassberger-Procaccia algorithm over a range of scales -- we create an ensemble of intervals by considering all possible combinations of endpoints, generating a distribution of slopes from least-squares fits weighted by the length of the fitting line and the inverse square of the fit error. The mode of this distribution gives an estimate of the slope of the scaling region (if it exists). The endpoints of the intervals that correspond to the mode provide an estimate for the extent of that region. When there is no scaling region, the distributions will be wide and the resulting error estimates for the slope will be large. We demonstrate this method for computations of dimension and Lyapunov exponent for several dynamical systems, and show that it can be useful in selecting values for the parameters in time-delay reconstructions.

physics.data-an

Normal Forms and Near-Axis Expansions for Beltrami Magnetic Fields

A formal series transformation to Birkhoff-Gustavson normal form is obtained for toroidal magnetic field configurations in the neighborhood of a magnetic axis. Bishop's rotation-minimizing coordinates are used to obtain a local orthogonal frame near the axis in which the metric is diagonal, even if the curvature has zeros. We treat the cases of vacuum and force-free (Beltrami) fields in a unified way, noting that the vector potential is essentially the Poincaré-Liouville one-form of Hamiltonian dynamics, and the resulting magnetic field corresponds to the canonical two-form of a nonautonomous one-degree-of-freedom system. Canonical coordinates are obtained and Floquet theory is used to transform to a frame in which the lowest-order Hamiltonian is autonomous. The resulting magnetic axis can be elliptic or hyperbolic, and resonant elliptic cases are treated. The resulting expansion for the field is shown to be well-defined to all orders, and is explicitly computed to degree four. An example is given for an axis with constant torsion near a 1:3 resonance.

math-ph

Integrability, Normal Forms, and Magnetic Axis Coordinates

Integrable or near-integrable magnetic fields are prominent in the design of plasma confinement devices. Such a field is characterized by the existence of a singular foliation consisting entirely of invariant submanifolds. A regular leaf, known as a flux surface,of this foliation must be diffeomorphic to the two-torus. In a neighborhood of a flux surface, it is known that the magnetic field admits several exact, smooth normal forms in which the field lines are straight. However, these normal forms break down near singular leaves including elliptic and hyperbolic magnetic axes. In this paper, the existence of exact, smooth normal forms for integrable magnetic fields near elliptic and hyperbolic magnetic axes is established. In the elliptic case, smooth near-axis Hamada and Boozer coordinates are defined and constructed. Ultimately, these results establish previously conjectured smoothness properties for smooth solutions of the magnetohydrodynamic equilibrium equations. The key arguments are a consequence of a geometric reframing of integrability and magnetic fields; that they are presymplectic systems.

math-ph

Nonexistence of Invariant Tori Transverse to Foliations: An Application of Converse KAM Theory

Invariant manifolds are of fundamental importance to the qualitative understanding of dynamical systems. In this work, we explore and extend MacKay's converse KAM condition to obtain a sufficient condition for the nonexistence of invariant surfaces that are transverse to a chosen 1D foliation. We show how useful foliations can be constructed from approximate integrals of the system. This theory is implemented numerically for two models, a particle in a two-wave potential and a Beltrami flow studied by Zaslavsky (Q-flows). These are both 3D volume-preserving flows, and they exemplify the dynamics seen in time-dependent Hamiltonian systems and incompressible fluids, respectively. Through both numerical and theoretical considerations, it is revealed how to choose foliations that capture the nonexistence of invariant tori with varying homologies.

math.DS

Using Curvature to Select the Time Lag for Delay Reconstruction

We propose a curvature-based approach for choosing good values for the time-delay parameter $τ$ in delay reconstructions. The idea is based on the effects of the delay on the geometry of the reconstructions. If the delay is chosen too small, the reconstructed dynamics are flattened along the main diagonal of the embedding space; too-large delays, on the other hand, can overfold the dynamics. Calculating the curvature of a two-dimensional delay reconstruction is an effective way to identify these extremes and to find a middle ground between them: both the sharp reversals at the ends of an insufficiently unfolded reconstruction and the folds in an overfolded one create spikes in the curvature. We operationalize this observation by computing the mean over the Menger curvature of 2D reconstructions for different time delays. We show that the mean of these values gives an effective heuristic for choosing the time delay. In addition, we show that this curvature-based heuristic is useful even in cases where the customary approach, which uses average mutual information, fails --- e.g., noisy or filtered data.

physics.data-an

Poisson Structure of the Three-Dimensional Euler Equations in Fourier Space

We derive a simple Poisson structure in the space of Fourier modes for the vorticity formulation of the Euler equations on a three-dimensional periodic domain. This allows us to analyse the structure of the Euler equations using a Hamiltonian framework. The Poisson structure is valid on the divergence free subspace only, and we show that using a projection operator it can be extended to be valid in the full space. We then restrict the simple Poisson structure to the divergence-free subspace on which the dynamics of the Euler equations take place, reducing the size of the system of ODEs by a third. The projected and the restricted Poisson structures are shown to have the helicity as a Casimir invariant. We conclude by showing that periodic shear flows in three dimensions are equilibria that correspond to singular points of the projected Poisson structure, and hence that the usual approach to study their nonlinear stability through the Energy-Casimir method fails.

math-ph

Computational Topology Techniques for Characterizing Time-Series Data

Topological data analysis (TDA), while abstract, allows a characterization of time-series data obtained from nonlinear and complex dynamical systems. Though it is surprising that such an abstract measure of structure - counting pieces and holes - could be useful for real-world data, TDA lets us compare different systems, and even do membership testing or change-point detection. However, TDA is computationally expensive and involves a number of free parameters. This complexity can be obviated by coarse-graining, using a construct called the witness complex. The parametric dependence gives rise to the concept of persistent homology: how shape changes with scale. Its results allow us to distinguish time-series data from different systems - e.g., the same note played on different musical instruments.

cs.CG

Moser's quadratic, symplectic map

In 1994, Jürgen Moser generalized Hénon's area-preserving quadratic map to obtain a normal form for the family of four-dimensional, quadratic, symplectic maps. This map has at most four isolated fixed points. We show that the bounded dynamics of Moser's six parameter family is organized by a codimension-three bifurcation, which we call a quadfurcation, that can create all four fixed points from none. The bounded dynamics is typically associated with Cantor families of invariant tori around fixed points that are doubly elliptic. For Moser's map there can be two such fixed points: this structure is not what one would expect from dynamics near the cross product of a pair of uncoupled Hénon maps, where there is at most one doubly elliptic point. We visualize the dynamics by escape time plots on 2D planes through the phase space and by 3D slices through the tori.

nlin.CD

Elliptic Bubbles in Moser's 4D Quadratic Map: the Quadfurcation

Moser derived a normal form for the family of four-dimensional, quadratic, symplectic maps in 1994. This six-parameter family generalizes Hénon's ubiquitous 2D map and provides a local approximation for the dynamics of more general 4D maps. We show that the bounded dynamics of Moser's family is organized by a codimension-three bifurcation that creates four fixed points---a bifurcation analogous to a doubled, saddle-center---which we call a quadfurcation. In some sectors of parameter space a quadfurcation creates four fixed points from none, and in others it is the collision of a pair of fixed points that re-emerge as two or possibly four. In the simplest case the dynamics is similar to the cross product of a pair of Hénon maps, but more typically the stability of the created fixed points does not have this simple form. Up to two of the fixed points can be doubly-elliptic and be surrounded by bubbles of invariant two-tori; these dominate the set of bounded orbits. The quadfurcation can also create one or two complex-unstable (Krein) fixed points. Special cases of the quadfurcation correspond to a pair of weakly coupled Hénon maps near their saddle-center bifurcations. The quadfurcation also occurs in the creation of accelerator modes in a 4D standard map.

nlin.CD

Accelerator modes and anomalous diffusion in 3D volume-preserving maps

Angle-action maps that are periodic in the action direction can have accelerator modes: orbits that are periodic when projected onto the torus, but that lift to unbounded orbits in an action variable. In this paper we construct a volume-preserving family of maps, with two angles and one action, that have accelerator modes created at Hopf-one (or saddle-center-Hopf) bifurcations. Near such a bifurcation we show that there is often a bubble of invariant tori. Computations of chaotic orbits near such a bubble show that the trapping times have an algebraic decay similar to that seen around stability islands in area-preserving maps. As in the 2D case, this gives rise to anomalous diffusive properties of the action in our 3D map.

nlin.CD

Universal Exponent for Transport in Mixed Hamiltonian Dynamics

We compute universal distributions for the transition probabilities of a Markov model for transport in the mixed phase space of area-preserving maps and verify that the survival probability distribution for trajectories near an infinite island-around-island hierarchy exhibits, on average, a power law decay with exponent $γ= 1.57$. This exponent agrees with that found from simulations of the Hénon and Chirikov-Taylor maps. This provides evidence that the Meiss-Ott Markov tree model describes the transport for mixed systems.

nlin.CD

Boundary Circles of Mixed Phase Space, Hamiltonian Systems

The phase space of an area-preserving map typically contains infinitely many elliptic islands embedded in a chaotic sea. Orbits near the boundary of a chaotic region have been observed to stick for long times, strongly influencing their transport properties. The boundary is composed of invariant circles, called "Boundary circles." We investigate the distribution of rotation numbers of boundary circles for the Henon quadratic map and show that the probability of occurrence of small elements of their continued fraction expansions is larger than would be expected for a number chosen at random. However, large elements occur with probabilities distributed proportionally to the random case. These results have implications for models of transport in mixed phase space.

nlin.CD

Probing the statistics of transport in the Hénon Map

The phase space of an area-preserving map typically contains infinitely many elliptic islands embedded in a chaotic sea. Orbits near the boundary of a chaotic region have been observed to stick for long times, strongly influencing their transport properties. The boundary is composed of invariant "boundary circles". We briefly report recent results of the distribution of rotation numbers of boundary circles for the Hénon quadratic map and show that the probability of occurrence of small elements of their continued fraction expansions is larger than would be expected for a number chosen at random. However, large elements occur with probabilities distributed proportionally to the random case. The probability distributions of ratios of fluxes through island chains is reported as well. These island chains are neighbours in the sense of the Meiss-Ott Markov-tree model. Two distinct universality families are found. The distributions of the ratio between the flux and orbital period are also presented. All of these results have implications for models of transport in mixed phase space.

nlin.CD