Projects per year
Personal profile
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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Projects
- 2 Finished
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When Worlds Collide: the asymptotics of interacting systems (Career Acceleration Fellowship)
1/08/12 → 1/08/16
Project: Research
Research output
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Novel slow-fast behaviour in an oscillator driven by a frequency-switching force
Bonet, C., Jeffrey, M. R., Martin, P. & Olm, J. M., 1 Apr 2023, In: Communications in Nonlinear Science and Numerical Simulation. 118, 107032, 107032.Research output: Contribution to journal › Article (Academic Journal) › peer-review
Open Access2 Citations (Scopus) -
Uncountably Many Cases of Filippov’s Sewed Focus
Glendinning, P., Hogan, S. J., Homer, M. E., Jeffrey, M. R. & Szalai, R., 5 May 2023, In: Journal of Nonlinear Science. 33, 4, p. 1-14 14 p., 52 (2023).Research output: Contribution to journal › Article (Academic Journal) › peer-review
Open Access -
Catastrophe conditions for vector fields and varieties in n-dimensions
Jeffrey, M. R., 25 Nov 2022, In: Journal of Physics A: Mathematical and Theoretical. 55, 464006.Research output: Contribution to journal › Article (Academic Journal) › peer-review
Open Access
Activities
- 1 Fellowship awarded competitively
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Career Acceleration Fellowship - When Worlds Collide: the asymptotics of interacting systems
Mike R Jeffrey (Recipient)
1 Aug 2012 → 1 Aug 2016Activity: Other activity types › Fellowship awarded competitively
Thesis
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Conical Diffraction: Complexifying Hamilton's Diabolical Legacy
Author: Jeffrey, M. R., 3 Oct 2007Supervisor: Berry, M. (Supervisor)
Student thesis: Doctoral Thesis › Doctor of Philosophy (PhD)
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