Mike R Jeffrey

PostDoctoral Researcher in Applied Nonlinear Mathematics

Address:
Room 2.80
Applied Nonlinear Mathematics Group
Department of Engineering Mathematics
University of Bristol
Queen's Building, University Walk
Bristol BS8 1TR, United Kingdom

• Research Interests
• Publications
• Events in Nonsmoothland
• Teaching
• 2007/8 Complex Variable Theory
• 2008/9 Advanced Partial Differential Equations

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Research Interests:

• Geometrical Asymptotics. Although the bulk of the world satisfies nice simple rules laid down from Descartes, to Newton, Hamilton, through to Dirac, the really interesting structures are born from the outer fringes of these theories - sonic booms, rainbows, ripples on ponds, wave-particle or wave-ray dualities. On these fringes the theories break down, but in their wake, beautiful geometry reveals intricate physical phenomena, such as swirling Mach cones in helicopter noise, the speed-of-light caustic in a high energy synchrotron, cusps, whiskers and complex rays in conical diffraction; ...
  
• Grazing Catastrophes. The theory of dynamical systems with discontinuities, such as friction, switching, or impact, is young and growing. At the heart of most theories is the assymption of smoothness, but from the squeal of brakes to the rattle of a fan blade, it is clear that in the real world things rarely run smoothly. These events are discontinuous, they involve sudden changes and loss of reversibility, and occur whenever two systems come into contact. Mathematics was not built to handle discontinuities, so we treat the interaction ad hoc, we look at the "before" and "after" in isolation, then stitch things back together later. But perhaps nature is not so malicious. Perhaps not everything is lost when a system jumps. By studying orbits that "graze" discontinuities - that almost jump but not quite - we hope to get to the heart of discontinuity.
• From nonsmooth to slow-fast - Discontinuity Asymptotics. What does the borderland between a system with a discontinuity, and a system with a sudden (slow-fast) change, look like? Experimental laws of the second type are incredibly complicated. Theoretical equations of the first type are too idealistically simple. Somewhere between them are the bifurcations and the catastrophes that define real world events everywhere from engineering to physiology, where contact between systems takes place much more quickly than changes in the systems themselves, but hugely affects their dynamics. I am interested in a nonsingular understanding of singular perturbation theory: a geometrical relationship between nonsmooth and slow-fast systems.
  
  
  

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  Publications

  See also Group Preprint Server
  
• M. R. Jeffrey, A. R. Champneys, M. di Bernardo, S. W. Shaw (2009) Catastrophic Sliding Bifrucations, & onset of oscillations in a superconducting resonator PRE submitted pdf
It is well established that smooth periodical motion can acquire segments of `sticking' behaviour, by means of sliding bifurcations in nonsmooth systems. It happens because a codimension one surface in phase space, where a discontinuity occurs in the dynamics, can contain superstable regions where the system slides along it. In the opposite case, when the sliding region is unstable, we show that sliding bifurcations can still occur but they are catastrophic - in the sense of a "blue-sky catastrophe", the attractor is destroyed instantaneously and without prior warning. The picture shows the two coexisting fates at the moment of bifurcation in a resonator model - a periodic orbit, or collapse to equilibrium.




• M. R. Jeffrey, S. J. Hogan (2009) The geometry of generic sliding bifurcations. SIAM review submitted pdf
What happens to a dynamical system when it meets a discontinuity in phase space, such as when friction changes direction or a switch is flipped at some threshold? Catastrophes (see pic) in the shape of the threshold give rise to "sliding bifurcations" - where the system may start to slide along the threshold or suddenly lose stability - providing a systematic study and classification of the most fundamental discontinuity induced bifurcations.




• A. Colombo, M. di Bernardo, E. Fossas, M. R. Jeffrey (2009) Teixeira singularities in 3D switched feedback control systems. Systems & Control Letters submitted pdf
The Teixeira singularity should be commonplace, so why has it not been seen? Here's how to find it and how it will look, with examples and simulations in some control systems.




• M. R. Jeffrey (2009) Two-folds in nonsmooth dynamical systems. IFAC Chaos09 proceedings, Queen Mary June 2009 pdf
The two-fold singularity: elusive or illusive? The two-fold is a fundamental and yet troublesome point in dynamical systems that arises at a discontinuity (a piecewise-smooth system). It permits dynamics to leap between periodicity, sliding, and nondeterministic chaos. It should be typical in three (or more) dimensions, from control systems to neurons to mechanics. Why, then, has it not been seen? Perhaps it is hiding in plain sight, even in nongeneric designed systems. Here we summarize the basic dynamics of the singularity, and highlight some simple conditions for its existence.




• M. R. Jeffrey, A. Colombo (2009) The two-fold singularity of discontinuous vector fields. SIAM J. Appl. Dyn. Syst. 8, 624-640 pdf
Unfolding the dynamics around a pivotal point of nonsmooth dynamics: the "Teixeira singularity", a generic point in 3D where a vector field is tangent to both sides of a discontinuity.




• M. Schmidt, M. R. Jeffrey (2007) Peel or coat spheres by convolution, repeatedly. JMP 48, 123507 pdf
The mathematics of peeling onions -- geometric and tensor representations for the convolution of spheres - the "peeling" and "coating" of spherical layers in colloid theory.




• M. V. Berry, M. R. Jeffrey, M. Mansuripur (2005). Orbital and spin angular momentum in conical diffraction. J. Opt.A 7, 685-690. Also featured as a "Highlight of 2006" pdf
Modern surprises from a 176 year old problem: the polarization singularity presented by conical diffraction converts optical spin angular momentum into orbital angular momentum, halving it in the process, and resulting in a torque on the crystal. The picture shows the rays and wave surfaces of conical diffraction, centering around Hamilton's diabolical point.



• J. H. Hannay, M. R. Jeffrey (2005) The electric field of synchrotron radiation. Proc. R. Soc. A 461, 3599-3610 pdf
Surprisingly (perhaps), an explicit formula for the relativistic electric field from a synchrotron is something neglected in the wealth of synchrotron literature. The result is a beautifully simple pattern of electric field lines, which twist around a pair of arched watch-spring like spirals, squashed up with dimensions inversely proportional to powers of the Lorentz factor. The picture shows two field lines spiralling through space in unsquashed coordinates (right), and their mapping into real space. The problem is closely related to similar caustics appearing in problems from x-ray galaxies to helicopter noise.

More from the Conical Diffraction Years:

• M. R. Jeffrey (2007) Conical Diffraction: Complexifying Hamilton's Diabolical Legacy, Ph.D. Thesis, University of Bristol pdf
  In the early 1800's William Rowan Hamilton united wave and ray optics as geometrical optics, and demonstrated his new theory with the discovery of conical diffraction - the birth of singular optics, the first time a mathematical prediction had preceeded physical observation, the proof of Fresnel's diffraction theory, and the start of 175 years of head-scratching. The resolution of conical diffraction requires modern concepts from asymptotics, catastrophe theory, and a delicate interplay of waves with complex rays. The picture shows conical diffraction images on the exit face of anisotropic (left), dichroic (middle), and chiral (right) crystals. Below, the singularity of conical diffraction is at once destroyed and reborn in a chiral crystal: Hamilton's diabolical point (pictured above) is no more, replaced by a ring of inflection points, whence springs Hamilton's fleeting fanfare, a trumpet-horn caustic that awaits experimental detection.
• M. R. Jeffrey (2007) The spun cusp complexified: complex ray focusing in chiral conical diffraction. J. Opt.A 9, 634-641 pdf
• M. R. Jeffrey (2006) Conical diffraction in optically active crystals. Photon06 online conference proceedings, IOP pdf
• M. V. Berry, M. R. Jeffrey (2006) Conical diffraction complexified: dichroism and the transition to double refraction. J. Opt.A 8, 1043-1051 pdf
• M. V. Berry, M. R. Jeffrey, J. G. Lunney (2006). Conical diffraction: observations and theory. Proc. R. Soc. A 462, 1629-1642. pdf
• M. V. Berry, M. R. Jeffrey (2006) Chiral conical diffraction. J. Optics A 8, 363-372. pdf
• M. V. Berry, M. R. Jeffrey (2006) Conical diffraction: Hamilton's diabolical point at the heart of crystal optics. Progress in Optics 50, 13-50. pdf
A downloadable site for the ANM group preprints since 1995

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Papers on this web page are protected by copyright, and are intended as a scientific resource for private use.  They may not be used commercially, or for financial gain.  Requests for permission to make public use of any of the papers, or material therein, should be sought from the original publisher, or from Mike Jeffrey, as appropriate.  




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The design of physical science is .. to learn the language and interpret the oracles of the Universe.

(Sir William Rowan Hamilton, lecture on Astronomy 1831)

Do you really think we'd be ****ing about with crystal optics if we could solve black holes?

(Mark Dennis on the field of Physical Asymptotics, unpublished New Scientist interview, Physics coffee room 2005)