publications

2023

1. Existence of Global Symmetries of Divergence-Free Fields with First Integrals

   Perrella, David,  Duignan, Nathan, and Pfefferlé, David

   Journal of Mathematical Physics May 2023

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   The relationship between symmetry fields and first integrals of divergence-free vector fields is explored in three dimensions in light of its relevance to plasma physics and magnetic confinement fusion. A Noether-type theorem is known: for each such symmetry, there corresponds a first integral. The extent to which the converse is true is investigated. In doing so, a reformulation of this Noether-type theorem is found for which the converse holds on what is called the toroidal region. Some consequences of the methods presented are quick proofs of the existence of flux coordinates for magnetic fields in high generality, without needing to assume a symmetry such as in the cases of magneto-hydrostatics or quasi-symmetry.

2. Distinguishing between Regular and Chaotic Orbits of Flows by the Weighted Birkhoff Average

   Duignan, Nathan, and Meiss, James D.

   Physica D: Nonlinear Phenomena Jul 2023

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   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 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.

3. Minimizing Separatrix Crossings through Isoprominence

   Burby, J. W., Duignan, N., and Meiss, J. D.

   Plasma Physics and Controlled Fusion Feb 2023

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   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.

2021

1. Nonexistence of invariant tori transverse to foliations: An application of converse KAM theory

   Duignan, Nathan, and Meiss, James D.

   Chaos: An Interdisciplinary Journal of Nonlinear Science Jan 2021

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   Invariant manifolds are of fundamental importance to the qualitative understanding of dynamical systems. In this work, we explore and extend MacKay’s converse Kolmogorov–Arnol’d–Moser 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.

2. Normal Forms for Manifolds of Normally Hyperbolic Singularities and Asymptotic Properties of Nearby Transitions

   Duignan, Nathan

   Qualitative Theory of Dynamical Systems Jul 2021

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   This paper contains theory on two related topics relevant to manifolds of normally hyperbolic singularities. First, theorems on the formal and \ C^k \ C k normal forms for these objects are proved. Then, the theorems are applied to give asymptotic properties of the transition map between sections transverse to the centre-stable and centre-unstable manifolds of some normally hyperbolic manifolds. A method is given for explicitly computing these so called Dulac maps. The Dulac map is revealed to have similar asymptotic structures as in the case of a saddle singularity in the plane.

3. On the C 8/3-regularisation of simultaneous binary collisions in the planar four-body problem

   Duignan, Nathan, and Dullin, Holger R.

   Nonlinearity Jun 2021

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   The dynamics of the four-body problem allows for two binary collisions to occur simultaneously. It is known that in the collinear four-body problem this simultaneous binary collision (SBC) can be block-regularised, but that the resulting block map is only C 8/3 differentiable. In this paper, it is proved that the C 8/3 differentiability persists for the SBC in the planar four-body problem. The proof uses several geometric tools, namely, blow-up, normal forms, dynamics near normally hyperbolic manifolds of equilibrium points, and Dulac maps.

4. Normal forms and near-axis expansions for Beltrami magnetic fields

   Duignan, Nathan, and Meiss, James D.

   Physics of Plasmas Dec 2021

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   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 non-autonomous 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.

5. Integrability, normal forms, and magnetic axis coordinates

   Burby, Joshua W.,  Duignan, Nathan, and Meiss, James D.

   Journal of Mathematical Physics Dec 2021

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   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 entirely consisting of invariant submanifolds. A compact regular leaf (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: they are presymplectic systems.

2020

1. On the C8/3-regularisation of simultaneous binary collisions in the collinear 4-body problem

   Duignan, Nathan, and Dullin, Holger

   Journal of Differential Equations Nov 2020

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   A conjecture on the nature of the simultaneous binary collision is explored in the collinear 4-body problem. It is known that any attempt to remove the singularity via block regularisation will result in a regularised flow that is no more than C8/3 differentiable with respect to initial conditions. Through a blow-up of the singularity, this loss of differentiability is investigated and a new proof of the C8/3 regularity is provided. In the process, it is revealed that the collision manifold consists of two manifolds of normally hyperbolic saddle singularities which are connected by a manifold of heteroclinics. By utilising recent work on transitions near such objects and their normal forms, an asymptotic series of the transition past the singularity is explicitly computed. It becomes remarkably apparent that the finite differentiability at 8/3 is due to the inability to construct a set of integrals local to the simultaneous binary collision. The finite differentiability is shown to be independent of the value of the masses.

2. Chazy-Type Asymptotics and Hyperbolic Scattering for the n-Body Problem

   Duignan, Nathan, Moeckel, Richard, Montgomery, Richard, and Yu, Guowei

   Archive for Rational Mechanics and Analysis Jun 2020

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   We study solutions of the Newtonian n-body problem which tend to infinity hyperbolically, that is, all mutual distances tend to infinity with nonzero speed as t → +∞ or as t → −∞. In suitable coordinates, such solutions form the stable or unstable manifolds of normally hyperbolic equilibrium points in a boundary manifold “at infinity”. We show that the flow near these manifolds can be analytically linearized and use this to give a new proof of Chazy’s classical asymptotic formulas. We also address the scattering problem, namely: for solutions which are hyperbolic in both forward and backward time, how are the limiting equilibrium points related? After proving some basic theorems about this scattering relation, we use perturbations of our manifold at infinity to study scattering “near infinity”, that is, when the bodies stay far apart and interact only weakly.

2019

1. Regularisation for Planar vector fields

   Duignan, Nathan, and Dullin, Holger R.

   Nonlinearity Nov 2019

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   This paper serves as a first foray on regularisation for planar vector fields. Motivated by singularities in celestial mechanics, the block regularisation of a generic class of degenerate singularities is studied. The paper is concerned with asymptotic properties of the transition map between a section before and after the singularity. Block regularisation is reviewed before topological and explicit conditions for the C 0-regularity of the map are given. Computation of the C 1-regularisation is reduced to summing residues of a rational function. It is shown that the transition map is in general only finitely differentiable and a method of computing the map is conveyed. In particular, a perturbation of an example derived from the four-body problem is shown to be C 4/3. The regularisation of all homogeneous quadratic vector fields is computed.

2. Thesis

   Duignan, N. (2019). On the Regularisation of Simultaneous Binary Collisions.

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   This dissertation contains work on the simultaneous binary collision in the n-body problem. Martínez and Simó have conjectured that removal of the singularity at this collision via block regularisation results in a regularised flow that is no more than C^(8/3) differentiable with respect to initial conditions. Remarkably, the same authors proved the conjecture for the collinear 4-body problem. The conjecture remains open for the planar case or for n > 4 . This thesis explores the loss of differentiability in the collinear and planar 4-body problem. In the collinear problem, a new proof is provided of the C^(8/3)-regularisation. In the planar problem, a proof that the simultaneous binary collisions are at least C^2-regularisable is given. In both cases a remarkable link between the finite differentiability and the inability to construct a set of integrals local to the singularities is established. The theoretical framework for improving the C^2 result in the plane is established. The method of proof in both cases brings together the theory of blow-up, normal forms, hyperbolic transitions, and computation of regular transition maps to explicitly compute an asymptotic expansion of the transition past the singularities. These tools are first explored in novel work on the regularisation of a generic class of degenerate singularities in planar vector fields. In particular, a relatively simple perturbation of an example derived from the 4-body problem is shown to be C^(4/3) However, the study of simultaneous binary collisions requires that each of these tools be extended to higher dimensions, in particular to manifolds of normally hyperbolic fixed points. General theory on normal forms and asymptotic properties of nearby transitions of such manifolds are detailed. The normal forms are studied in the formal and C^k categories. The hyperbolic transitions are shown to have similar properties to the well studied Dulac maps of planar saddles.

2016

1. The diver with a rotor

   Bharadwaj, Sudarsh,  Duignan, Nathan, Dullin, Holger R., Leung, Karen, and Tong, William

   Indagationes Mathematicae Dec 2016

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   We present and analyse a simple model for the twisting somersault. The model consists of a rigid body with a rotor attached that can be switched on and off. This makes it simple enough to devise explicit analytical formulas whilst still maintaining sufficient complexity to preserve the shape-changing dynamics essential for twisting somersaults performed in springboard and platform diving. With “rotor on” and with “rotor off” the corresponding Euler-type equations can be solved and the essential quantities characterising the dynamics, such as the periods and rotation numbers, can be computed in terms of complete elliptic integrals. We arrive at explicit formulas for how to achieve a dive with m somersaults and n twists in a given total time. This can be thought of as a special case of a geometric phase formula due to Cabrera (2007).