# Research Articles

## Publications

** Calculating Fourier Transforms of
Long-Tailed Functions,**
* SIAM Journal on Scientific and Statistical Computing,*
volume 8 (1987) pages 1005-1011,
with J.N. Lyness

** A commentary `On the periodic solutions of
a forced second order equation' by SP Hastings and JB McLeod,**
*Journal of Nonlinear Science,*
volume 1 (1991) pages 247-253,
with S. Wiggins

** Lobe area in adiabatic Hamiltonian systems,**
*PhysicaD, *
volume 51 (1991) pages 205-212,
with S. Wiggins

** On the structure of separatrix-swept regions in
singularly-perturbed
Hamiltonian systems,**
*Differential and
Integral Equations,*
volume 5 (1992) pages 1363-1381,
with S. Wiggins

** An analytical study of transport in Stokes flows exhibiting
large-scale
chaos in the eccentric journal bearing, **
* Journal of Fluid Mechanics,*
volume 253 (1993) pages 211-243,
with S. Wiggins

** A geometric criterion for adiabatic chaos,**
* Journal of Mathematical Physics,*
volume 35 (1994) pages 1202-1218,
with Gregor Kovacic

** A simple model for chaotic advection and scattering, **
* CHAOS, *
volume 5 (1995) pages 671-686,
with Gustavo Stolovitzky and Lawrence Sirovich

** Periodic orbits in singularly-perturbed systems,**
in * Nonlinear dynamics and pattern formation in the natural
environment,*
(A. Doelman and A. van Harten, editors), Pitman Research Notes
in Mathematics,
volume 335 (1995) pages 295-314,
with Cristina Soto-Trevino

** Wave-number transport:
scattering of small-scale internal waves by large-scale
wave packets, **
* Journal of Fluid Mechanics,*
volume 289 (1995) pages 379-405,
with David L. Bruhwiler

** N-th order split operator schemes and nonreversible systems,**
* SIAM Journal on Numerical Analysis, *
volume 33 (1996) pages 349-367,
with Daniel Goldman

** Tracking invariant manifolds up to exponentially
small error,**
* SIAM Journal on Mathematical Analysis,*
volume 27 (1996) pages 558-577,
with Christopher K.R.T. Jones and Nancy Kopell

** Multi-bump orbits homoclinic to resonance bands, **
* Transactions of the American Mathematical
Society, *
volume 348 (1996) pages 3835-3887,
with Gregor Kovacic

** Higher order Melnikov theory
for adiabatic systems,**
* Journal of Mathematical Physics, *
volume 37 (1996) pages 6220-6249,
with Cristina Soto-Trevino

** Global dynamics of a rapidly forced cart and pendulum,**
*Nonlinear Dynamics, *
volume 13 (1997) pages 131-170,
with Stephen Weibel and John Baillieul

**
Pattern formation in the one-dimensional
Gray-Scott model, **
* Nonlinearity,*
volume 10 (1997) pages 523-563,
with Arjen Doelman and Paul Zegeling

**
Stability analysis of singular patterns
in the 1-D Gray-Scott model: A matched
asymptotics approach, **
* Physica D,*
volume 122 (1998) pages 1-36,
with Arjen Doelman and Robert A. Gardner
[ pdf ]

**
On acoustic cavitation of slightly subcritical bubbles, **
* Physics of Fluids,*
volume 11 (1999) pages 274-287,
with Anthony Harkin and Ali Nadim
[ pdf ]

** Stationary periodic
patterns in the 1D Gray-Scott model, **
* Methods and Applications of Analysis,*
volume 7(#1) (2000) 105--150,
with David S. Morgan and Arjen Doelman
[ pdf ]

**
A reaction-diffusion system with periodic front dynamics, **
* SIAM Journal on Applied Mathematics, *
volume 60(#5) (2000) pages 1601-1638,
with Georgiy Medvedev and Nancy Kopell
[ pdf ]

** On axisymmetric traveling waves and radial solutions
of semilinear elliptic equations, **
* Natural Resource Modeling,*
volume 13 (2000) pages 339-388,
with Thomas P. Witelski and Kinya Ono
[ pdf ]

** Alpha-frequency rhythms desynchronize over long cortical distances:
a modeling study,
**
* Journal of Computational Neuroscience *,
volume 9(#3) (2000) 271--291,
with Stephanie R. Jones, David J. Pinto and Nancy Kopell

**
Large stable pulse solutions
in reaction-diffusion equations,
**
Indiana University Math Journal,
volume 50, #1 (2001) 443--507,
with Arjen Doelman and Robert A. Gardner
[ pdf ]

** Critical wave speeds
for a family of scalar reaction diffusion equations, **
* Applied Mathematics Letters*,
volume 14, #1 (2001),
pages 65--73,
with Thomas P. Witelski and Kinya Ono

**
Slowly-modulated two-pulse solutions
in the Gray-Scott model I:
Asymptotic construction and stability, **
* SIAM Journal on Applied Mathematics, *
volume 61, #3 (2001),
pages 1080--1102,
with Arjen Doelman and Wiktor Eckhaus
[ pdf ]

**
Slowly-modulated two-pulse solutions
in the Gray-Scott model II:
Geometric theory, bifurcations and splitting dynamics, **
* SIAM Journal on Applied Mathematics, *
volume 61, #6 (2001),
pages 2036--2062,
with Arjen Doelman and Wiktor Eckhaus
[ pdf ]

**
Spatially periodic and aperiodic multi-pulse patterns
in the one-dimensional
Gierer-Meinhardt equation, **
* Methods and Applications of Analysis, *
volume 8 (2001),
pages 387--414,
with Arjen Doelman and Harmen van der Ploeg
[ pdf ]

**
An unfolding theory approach to bursting
in fast-slow systems, **
in * Global Analysis of Dynamical Systems, *
A Festschrift/Liber Amicorum dedicated to
Floris Takens for his 60-th birthday,
eds. H. Broer, B. Krauskopf, G. Vegter,
IOP Pub., Bristol
(2001), pages 277--308,
with Martin Golubitsky and Kresimir Josic

**
Coupled pulsation and translation of
two gas bubbles in a liquid, **
* Journal of Fluid Mechanics, *
volume 445 (2001),
pages 377--411,
with Anthony Harkin and Ali Nadim
[ pdf ]

** A stability index analysis of 1-D patterns of the Gray-Scott model, **
* Memoirs of the AMS,*
volume 155, number 737, (2002),
ISBN 0-8218-2739-1,
with Arjen Doelman and Robert A. Gardner
[ pdf ]

**
Asymptotic analysis
of two reduction methods
for chemical kinetics, **
* PhysicaD *
volume 165 (2002),
pages 66--93,
with Hans G. Kaper
[ pdf ]

**
Blowup in the nonlinear Schr\"odinger equation
near critical dimension, **
* Journal of Mathematical Analysis and Applications, *
volume 268 (2002),
pages 517--549,
with Vivi Rottsch\"afer
[ pdf ]

**
Geometric theory for multi-bump, self-similar, blowup sloutions
of the cubic nonlinear Schr\"odinger equation, **
* Nonlinearity, *
volume 16 (2003), pages 929--961,
with Vivi Rottsch\"afer
[ pdf ]

**
Semi-strong pulse interactions
in a class
of coupled reaction-diffusion equations, **
* SIAM Journal on Applied Dynamical Systems, *
volume 2 (2003), pages 53--96,
with Arjen Doelman
[ pdf ]

**
Coordination of cellular pattern-generating circuits
that control limb movements: the sources
of stable differences in intersegmental phases, **
* Journal of Neuroscience, *
volume 23(8) (2003), pages 3457--3469,
with Stephanie R. Jones, Brian Mulloney, and Nancy Kopell
[ pdf ]

**
Analysis of state-dependent transitions in frequency and long-distance coordination
in a model oscillatory cortical circuit, **
* Journal of Computational Neuroscience, *
volume 15(2) (2003), pages 283--298,
with David J. Pinto, Stephanie R. Jones, and Nancy Kopell
[ pdf ]

**
Analysis of the computational singular perturbation
reduction method for chemical kinetics, **
* Journal of Nonlinear Science, *
volume 14(1) (2004), pages 59--91,
with Antonios Zagaris and Hans G. Kaper
[ pdf ]

**
Axisymmetric ring solutions
of the 2-D Gray-Scott model
and their destabilization into spots, **
* PhysicaD, *
volume 192 (1-2) (2004), pages 33--62,
with David S. Morgan
[ pdf ]

**
Fast and slow dynamics
for the computational singular perturbation
method, **
* SIAM Multiscale Modelling and Simulations *
volume 2 (4) (2004), pages 613--638,
with Antonios Zagaris and Hans G. Kaper
[ pdf ]

**
Two perspectives on reduction in ordinary differential equations, **
* Mathematische Nachrichten,*
volume 278 (12-13) (2005), pages 1629--1642,
with Antonios Zagaris and Hans G. Kaper
[ pdf ]

**
Projecting to a slow manifold:
Singularly perturbed systems and legacy codes,
**
* SIAM Journal on Applied Dynamical Systems, *
volume 4 (2005)
711--731,
with C. William Gear, Yannis Kevrekidis, and Antonios Zagaris
[ pdf ]

**
Rigorous asymptotic expansions
for critical wave speeds
in a family of reaction-diffusion equations
**
* Journal of Dynamics and Differential Equations, *
volume 18(1) (2006)
103--139,
with Nikola Popovic
[ pdf ]

**
A geometric construction of traveling waves
in a bioremediation model,
**
* Journal of Nonlinear Science, *
volume 16 (2006)
329--349,
with Margaret Beck and Arjen Doelman
[ pdf ]

**
Homoclinic bifurcations at the onset of pulse self-replication,
**
* Journal of Differential Equations, *
volume 231(1) (2006)
359--423,
with Arjen Doelman and Lambertus A. Peletier
[ pdf ]

**
The asymptotic critical wave speed
in a family of scalar reaction-diffusion equations,
**
* Journal of Mathematical Analysis and Applications, *
volume 326 (2007) 1007--1023,
with Freddy Dumortier and Nikola Popovic
[ pdf ]

**
The critical wave speed
for the FKPP equation with cut-off,
**
* Nonlinearity, *
volume 20 (2007) 855--877,
with Freddy Dumortier and Nikola Popovic
[ pdf ]

**
Nonlinear asymptotic stability of the semi-strong pulse dynamics
in a regularized Gierer-Meinhart model,
**
* SIAM Journal on Mathematical Analysis, *
volume 38 (2007) 1760--1787,
with Arjen Doelman and Keith Promislow
[ pdf ]

**
Reduction for Michaelis-Menten-Henri kinetics
in the presence of diffusion,
**
* Electronic Journal of Differential Equations, *
volume 16 (2007) 155--184,
with Leonid Kalachev, Hans G. Kaper, Nikola Popovic, and Antonios Zagaris
[ pdf ]

**
Introduction to Focus Issue:
Mixed-mode oscillations: Experiment, computation, and analysis,
**
* CHAOS, *
volume 18 (2008) article number 015101.
This Focus Issue includes research articles on MMOs and
was edited jointly with Morten Brons and Horacio Rotstein
[ pdf ]

**
Analysis of a renormalization group method
and normal form theory for perturbed ordinary differential equations,
**
* PhysicaD, *
volume 237(8) (2008) pages 1029--1052,
with R.E. Lee DeVille, Anthony Harkin, Matt Holzer, and Kreso Josic
[ pdf ]

**
Pulse dynamics in a three-component system:
existence analysis,
**
* Journal of Dynamics and Differential Equations, *
volume 21(1) (2009) pages 73--115,
DOI: 10.1007/s10884-008-9125-2,
with Peter van Heijster and Arjen Doelman
[ pdf ]

**
Pulse dynamics in a three-component system:
stability and bifurcations,
**
* PhysicaD, *
volume 237(24) (2008) pages 3335--3368,
DOI: 10.1016/j.physd.2008.07.014,
with Peter van Heijster and Arjen Doelman
[ pdf ]

**
Analysis of the accuracy and convergence
of equation-free projection to a slow manifold,
**
* Modelisation Mathematique et Analyse Numerique,
Mathematical Modeling and Numerical Analysis, M2AN,*
volume 43 (2009) pages 757--784,
with Antonios Zagaris, C.W. Gear,
and Yannis Kevrekidis
[ pdf ]

**
Canards and bifurcation delays
of spatially homogeneous and inhomogeneous types
in reaction-diffusion equations,
**
* Advances in Differential Equations, *
volume 14 (2009) pages 943--962,
with Peter de Maesschalck and Nikola Popovic

**
Front interactions in a three-component system,
**
*
SIAM Journal on Applied Dynamical Systems, *
volume 9 (2010) pages 292--332,
DOI: 10.1137/080744785,
with Peter van Heijster, Arjen Doelman, and Keith Promislow
[ pdf ]

**
A geometric approach to bistable front propagation
in scalar reaction-diffusion equations with cut-off,
**
* Physica D *,
volume 239 (2010) pages 1984--1999,
DOI: 10.1016/j.physd.2010.07.008,
with Nikola Popovic and Freddy Dumortier
[ pdf ]

**
Pinned fronts in heterogeneous media of jump type,
**
* Nonlinearity *,
volume 24 (2011) pages 127-157,
DOI: 10.1088/0951-7715/24/1/007,
with Peter van Heijster, Arjen Doelman, Yasumasa Nishiura,
and Kei-Ichi Ueda
[ pdf ]

**
An elementary model of torus canards,
**
* Chaos *,
volume 21 (2011) article 023131 (17 pages),
DOI: 10.1063/1.3592798,
with G. Nicholas Benes, Anna M. Barry, Mark A. Kramer, and John Burke
[ pdf ]

**
Canards of mixed type in a neural burster,
**
* Physical Review E ,* volume 85 (2012) article 021920,
DOI: 10.1103/PhysRevE.85.021920,
with Mathieu Desroches, John Burke, and Mark Kramer
[ pdf ]

**
A showcase of torus canards in neuronal bursters,
**
* J. Math. Neuroscience*, volume 2 (2012) article number 3, 30 pages,
DOI:10.1186/2190-8567-2-3,
with John Burke, Anna M. Barry, Mathieu Desroches, and Mark A. Kramer,
[ pdf ]

**
Stability and stabilization
of the constrained runs schemes
for equation-free projection to a slow manifold,
**
* Discrete and Continuous Dynamical Systems - A*,
volume 32(8) (2012) 2759--2804,
with Antonios Zagaris, Christophe van de Kerckhoeve, C.W. Gear,
and Yannis Kevrekidis
[ pdf ]

**
Wave speeds for pushed fronts
in scalar reaction-diffusion equations with cut-off,
**
* RIMS Kokyuroku Bessatsu, *
volume B31 (2012) 117--134,
with Freddy Dumortier
[ pdf ]

**
Existence and stability of traveling pulses
in a reaction-diffusion-mechanics system,
**
* Journal of Nonlinear Science,*
volume 23 (2013) 129--177,
DOI: 10.1007/s00332-012-9147-0,
with Matt Holzer and Arjen Doelman
[ pdf ]

**
The dynamics of hybrid metabolic-genetic oscillators
**
* Chaos,*
volume 23 (2013) article no. 013132,
with Ed Reznik and Daniel Segre
[ pdf ]

**
Energy transfer between the shape and volume modes
of a nonspherical gas bubble,
**
* Physics of Fluids, *,
volume 25 (2013) article no. 062101,
with Anthony Harkin and Ali Nadim
[ pdf ]

**
Geometric desingularization
of a cusp singularity in slow-fast systems
with applications to Zeeman's examples,
**
* Journal of Dynamics and Differential Equations *,
volume 25 (2013) 925--958,
DOI 10.1007/s10884-013-9322-5,
with Henk Broer and Martin Krupa
[ pdf ]

**
Adiabatic stability under semi-strong interactions: The weakly damped regime,
**
* Indiana University Mathematics Journal*,
volume 62 (2013) 1809--1859,
DOI 10.1512/iumj_2013.62.5159,
with Thomas Bellsky, Arjen Doelman, and Keith Promislow,
[ pdf ]

**
Mixed-Mode Bursting Oscillations: Dynamics created by a slow passage
through spike-adding canard explosion in a square-wave burster
**
* CHAOS *,
volume 23 (2013) article number 046106,
DOI 1054-1500/2013/23(4)/046106/13/,
with Mathieu Desroches and Martin Krupa
[ pdf ]

**
An analysis of the renormalization group method
for asymptotic expansions
with logarithmic switchback terms,
**
*
Advances in Differential Equations,
*
volume 19 (2014) 245--282,
with Matt Holzer

**
Wave speeds for the FKPP equation with enhancement to the reaction
function,
**
*
Zeitschrift fuer Angewandte und Mathematische Physik,
*
volume 66 (2014), 607--629,
DOI: 10.1007/s00033-014-0422-9
with Freddy Dumortier
[ pdf ]

**
A computational model of BMP movement in sea urchin embryos,
**
*
Journal of Theoretical Biology,
*
volume 363 (2014) 277-289,
DOI: 10.1016/j.jtbi.2014.08.026
with Peter van Heijster, Heather Hardway, and Cynthia Bradham
[ pdf ]

**
A note on a
reaction-diffusion model
describing the bone morphogen protein gradient
in Drosophila embryonic patterning,
**
in
*
"Nonlinear Dynamics in Partial Differential Equations",
Advanced Studies in Pure Mathematics,
*
volume 64
(2015) 209-220,
Mathematical Society of Japan,
with Cynthia Bradham and Peter van Heijster,
ISBN 978-4-86497-022-8

**
Geometry of the computational singular perturbation method,
**
*
Mathematical Modeling of Natural Phenomena
*,
Cambridge University Press,
volume 10(3) (2015) 16-30,
DOI: 10.1051/mmnp/201510303
with Hans G. Kaper and Antonios Zagaris
[ pdf ]

**
Stabilization of ring dark solitons in Bose-Einstein condensates,
**
* Physical Review A*
volume 92 (2015) article no. 033611,
with W. Wang, Panos G. Kevrekidis, Ricardo Carretero-Gonzalez,
Dimitri J. Franzeskakis,
and M. Ma

**
From canards of folded singularities to torus canards
in a forced van der Pol equation,
**
* Journal of Nonlinear Science.*
volume 26 (2016) 405-451,
with John Burke, Mathieu Desroches, Alberto Granados,
Martin Krupa, and Theodore Vo,
doi: 10.1007/s00332-015-9279-0

**
Geometric desingularization of degenerate singularities in the
presence of fast rotation: a new proof of known results for slow passage
through Hopf bifurcations,
**
*
Indagationes Mathematicae,
*
volume 27 (2016) 1184-1203,
with Michael G. Hayes, Peter Szmolyan,
and Martin Wechselberger,
doi: 10.1016/j.indag.2015.11.005

**
Amplitude-modulated bursting: a novel class of bursting rhythms,
**
* Physical Review Letters,*
volume 117 (2016) article number 268101,
with Theodore Vo and Mark Kramer.

**
Analysis of the Approximate Slow Invariant Manifold Method
for reactive flow equations,
**
* Journal of Mathematical Chemistry,*
volume 55 (2017) 1725-1754,
with Xiaoxuan Wu.
DOI: 10.1007/s10910-017-0756-6

**
Dynamical systems analysis of the Maasch-Saltzman
model for glacial cycles,
**
* Physica D,*
volume 359, 2017, pages 1-20,
with Hans Engler, Hans Kaper, and Theodore Vo.
DOI: 10.16/j.physd.2017.08.006

**
Delayed loss of stability
due to slow passage through Hopf bifurcations
in reaction-diffusion equations,
**
* Chaos,*
volume 28 (2018) article number 091103,
with Theodore Vo.
DOI: 10.1063/1.5050508

## Publications in press

## Publications submitted

**
On the existence of and relationship between
canards and torus canards
in forced planar slow/fast equations,
**
with H. Wang and T. Vo, submitted.

####
*For copies, please e-mail
tasso@math.bu.edu
*

####
* The above publications
were completed
with support
from the National Science Foundation.
*

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