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.Comment: 25 pages, 18 figure

The Diver with a Rotor

We present and analyse a simple model for the twisting somersault. The model is a rigid body with a rotor attached which 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 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. Thus 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.Comment: 15 pages, 6 figure

Existence of global symmetries of divergence-free fields with first integrals

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 (MHS) or quasi-symmetry.Comment: 31 pages, 3 figures. This version of the article features an example involving Reeb cylinders whose idea was suggested by Daniel Peralta-Sala

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. &nbsp;</p

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 (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.</p

On the Regularisation of Simultaneous Binary Collisions

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