 Announcements 
Program 
Workshop on
Bifurcation Analysis and its Applications
Montreal, Canada, July 710, 2010
Part of a Lorenz manifold in the preturbulent regime (by Hinke Osinga)
Announcements
 [New] [September 22, 2010]
 Workshop Pictures.
 Some of the workshop
presentations are available for downloading/viewing. In such cases, the
presentation title is linked to its presentation file in PDF.
 [New] [July 01, 2010]
The program has been updated.
 [April 25, 2010]
The program has been updated.
 [March 05, 2010]
The program has been updated.
 [October 31, 2009] Please watch this section for news on accommodation,
confirmed speakers and deadlines.
Outline
Welcome to the BAA10 Home Page! This fourday workshop will include
introductory and specialized presentations on the development of algorithms for
the analysis of dynamical systems and various applications. In some sense, it is
the continuation of highly successful meetings focused on bifurcation analysis
in Amsterdam, Bielefeld, Bristol, Ghent, Seville, Utrecht and, last year,
Milan.
Organizers
Eusebius Doedel and Lennaert van Veen.
Principal Themes
 Day 1: Algorithms for ODEs and Conservative Systems.
 Day 2: Algorithms for PDEs and Fluid Dynamics.
 Day 3: Biomedical Applications.
 Day 4: Industrial Applications and Mechanical Systems.
Hotel
Le Méridien Versailles.
Location
Engineering and Visual Arts Building (corner rue Ste.Catherine O. and rue
Guy), Concordia University, Montreal, Canada.
Room EV3.309. (After taking the two stairways from the entrance hall to the third floor, turn right,
take the first turn left, and then the first turn right.
Transportation
A taxi stand is in front of the airport; a taxi ride to the Hotel Meridien
Versailles can reach C$40, especially during rush hour. There is also an
airport bus service to
the downtown BerriUQAM metro station, from where one can take the metro to the
GuyConcordia metro station, close to the hotel.
Participation
The workshop is open to the interested public. In order to ensure the workshop
room has enough seats, we kindly request that participants register in advance,
and before June 28, by sending an email to E. J. Doedel (doedel@cse.concordia.ca). A name tag, which
can
be picked up at the entrance of the lecture room, should be worn during the
workshop.
Programme
The titles and speakers are given below. The abstracts are included in the
details. Please select/deselect
the link for the details to show/hide the abstract, respectively.
Please check this page shortly before the workshop, as there may be late
changes due to travel schedules, etc.
Day/Date: Wednesday, July 7. Theme: Algorithms for ODEs and Conservative Systems
10:0010:30 Coffee
Talks
Time 
Title 
Details 
Speaker 
10:3011:00 
Initialization and continuation of homoclinic orbits to equilibria
in MatCont 
Details
Yu.A. Kuznetsov (Utrecht University, The Netherlands)
Joint work with V. De Witte and W. Govaerts (Gent, Belgium), and M. Friedman
(Huntsville, Alabama)
Abstract:
New functionalities of MatCont (a MATLAB continuation package for the
interactive numerical study of parameterized systems of smooth ODEs) related to
the codimensionone orbits homoclinic to hyperbolic and nonhyperbolic
equilibria will be presented. The continuation of such orbits is implemented
using the CISalgorithm, with the Ricatti equation included in the defining
system. It is possible to initiate the homoclinic orbits of both types
interactively, within a GUI, starting from an equilibrium point and using a
homotopy method.
All known codimensiontwo homoclinic bifurcations can be detected and located
during the continuation. Test functions for the inclinationflip bifurcations
are implemented in a new and more efficient way.
Examples illustrating the homotopy method in the cases of 1D and 2Dunstable
invariant manifolds will be presented.

Yuri Kuznetsov 
11:0011:30 
Continuation and bifurcation in conservative systems 
Details
Jorge Galán
(Joint work with F. J Muñoz Almaraz, E. Freire, A. Vanderbauwhede, and E. J.
Doedel)
Abstract:
The usual procedure to study the dynamical behaviour of systems with conserved
quantities and/or symmetries is to reduce the dimension of the problem and
include the integrals of motion as explicit parameters. Some years ago, the
authors proposed an alternative method that instead adds an additional
"unfolding" term to the equations and treats the conserved quantities as
internal parameters, i.e. not present in the equations. In this talk we review
some of the applications of the method paying special attention to
the numerical implementation for the continuation of periodic orbits.

Jorge Galán 
11:3012:00 Coffee, Cake, and Discussion
Time 
Title 
Details 
Speaker 
12:0012:30 
The role of global manifolds in the transition to chaos in the
Lorenz system 
Details
Hinke Osinga
Abstract:
The Lorenz system still fascinates many people because of the
simplicity of the equations that generate such complicated dynamics on
the famous butterfly attractor. This talk addresses the role of the
stable and unstable manifolds in organising the dynamics more
globally. A main object of interest is the stable manifold of the
origin of the Lorenz system, also known as the Lorenz manifold. This
twodimensional manifold and associated manifolds of saddle periodic
orbits can be computed accurately with numerical methods based on the
continuation of orbit segments, defined as solutions of suitable
boundary value problems. This allows us to study bifurcations of
global manifolds as the Rayleigh parameter is changed. We show how
the entire phase space of the Lorenz system is organised and how the
manifolds change dramatically during the transition to chaotic dynamics.

Hinke Osinga 
12:3013:00 
Computing global dynamics of multiparameter systems 
Details
Bill Kalies
Abstract:
A generally applicable method for the efficient computation of a database of
global dynamics of a multiparameter dynamical system is introduced. An outer
approximation of the dynamics for each subset of the parameter range is computed
using rigorous numerical methods and is represented by means of a directed
graph. The dynamics is then decomposed into the recurrent and gradientlike
parts by fast combinatorial algorithms and is classified via Morse
decompositions. These Morse decompositions are compared at adjacent parameter
sets via continuation to detect possible changes in the dynamics. The Conley
index is used to study the structure of isolated invariant sets associated with
the computed Morse decompositions and to detect the existence of certain types
of dynamics. The method is illustrated with an application to the
twodimensional,
densitydependent, Leslie population model.

Bill Kalies 
13:0017:00 Lunch and Discussions
Time 
Title 
Details 
Speaker 
17:0017:30 
Breakdown of Analyticity: From rigorous results to numerics 
Details
Renato Calleja, McGill University
Abstract:
We formulate and justify rigorously a numerically efficient criterion for the
computation of the analyticity breakdown of quasiperiodic solutions in
Symplectic maps and 1D Statistical Mechanics models. Depending on the physical
interpretation of the model, the analyticity breakdown may correspond to the
onset of mobility of dislocations, or of spin waves (in the 1D models) and to
the onset of global transport in symplectic twist maps. The criterion we propose
here is based on the blowup of Sobolev norms of the hull functions. The
justification of the criterion suggests fast numerical algorithms that we have
implemented in several examples.

Renato Calleja 
Demonstration
Time 
Title 
Details 
Demonstrator 
17:3018:00 
Quality of numerics: TIDES. Application to the Lorenz and Rössler models 
Details
Roberto Barrio, Departamento de Matemática Aplicada, Universidad de Zaragoza, E50009 Zaragoza, Spain
(Joint work with Fernando Blesa, Marcos Rodríguez, and Sergio Serrano)
Abstract:
In the numerical study of Dynamical Systems we have to deal with Differential
Equations, and we have to solve them with different goals. Sometimes we have to
integrate very long time and we need speed, or we need to study the evolution of
partial derivatives along the solution or we deal with chaotic problems and we
need high precision to follow the orbit, 100 digits, 1000 digits, or more. In
all these cases we can use the new integrator tool TIDES
[1,3], acronym of Taylor Series Integrator of Differential EquationS. The
Taylor series method applied to the integration of Ordinary Differential
Equations can be adapted easily to these special demands which hardly can be
obtained using other standard methods such as RungeKutta.
In this new software we have implemented all these features and we have
created the program providing easytouse interfaces. A simple line in Mathematica
generates automatically the necessary files in C or FORTRAN
to solve the problem. Besides, changing a simple option and linking with the
library lTIDES
also provided in this software we can integrate without difficulties partial
derivatives and/or arbitrary high precision via the multiple precision library MPFR [6].

Roberto Barrio 
18:0018:30 Discussions and Demonstrations
Day/Date: Thursday, July 8. Theme: Algorithms for PDEs and Fluid Dynamics
09:0009:30 Coffee
Talks
Time 
Title 
Details 
Speaker 
09:3010:00 
Lagrangian Coherent Structures, Hyperbolicity and Lyapunov
exponents: A Unified View 
Details
George Haller
Abstract:
We review the fundamental physical motivation behind the definition of
Lagrangian Coherent Structures (LCS) and show how it leads to the concept of
finitetime hyperbolicity in nonautonomous dynamical systems. Using this
concept of hyperbolicity, we obtain a selfconsistent criterion for the
existence of attracting and repelling material surfaces in unsteady fluid flows
and in other nonautonomous dynamical systems. The existence of LCS is often
postulated in terms of features of the FiniteTime Lyapunov Exponent (FTLE)
field associated with the system. As simple examples show, however, the FTLE
field does not necessarily highlight LCS, or may highlight structures that are
not LCS. Under appropriate nondegeneracy conditions, we
show that ridges of the FTLE field indeed coincide with LCS in volumepreserving
flows. For general flows, we obtain a more general scalar field whose ridges
correspond to LCS. We illustrate the results on applications to atmospheric and
oceanic flow data.

George Haller 
10:0010:30 
Equivariant PDEs and the freezing method 
Details
WolfJürgen Beyn
Abstract:
We consider semilinear parabolic systems in one and two space dimensions
which exhibit localized moving patterns, such as traveling, rotating and spiral
waves. These patterns may be viewed as relative equilibria of an equivariant
timedependent PDE. We discuss nonlinear stability and instability of such
relative equilibria in terms of (generally continuous) spectra of
linearizations.
The results are closely related to a numerical approach, called the freezing
method or the method of slices. It uses equivariance of the PDE and transforms
the given system into a partial differential algebraic equation (PDAE) which is
subsequently discretized by truncation to a bounded domain with finite boundary
conditions. The approach allows to freeze the target pattern in a comoving frame
while simultaneously providing information on the motion of the group variables.
For several examples we show how Lyapunov stability of the resulting PDAE and
its numerical analog relate to stability with
asymptotic phase for the original pattern.

WolfJürgen Beyn 
10:3011:00 Coffee, Cake, and Discussion
Time 
Title 
Details 
Speaker 
11:0011:30 
Secondary flow transitions in a differentially heated rotating
channel of fluid 
Details
Greg Lewis
(Joint work with Matt Hennessy)
Abstract:
A preliminary investigation of the secondary fluid flow transitions that
occur in an O(2)symmetric differentially heated rotating periodic channel is
presented. In the channel, the primary transition occurs when a timeindependent
flow that is uniform along the channel bifurcates to a stationary wave flow.
Using numerical continuation, we search for the parameter values at which
transitions occur from this wave solution.
We discuss a strategy for computing steady solutions of the three dimensional
NavierStokes equations in the Boussinesq approximation and for determining flow
transitions. In particular, the NavierStokes equations are solved using a
velocityvorticity method in combination with a secondorder centered
finitedifference scheme on an unstaggered and nonuniform grid. The resulting
nonlinear system of algebraic equations is solved using Newton's method. GMRES
with block Jacobi preconditioning is used to iteratively solve the large sparse
linear systems that result from the discretization.

Greg Lewis 
11:3012:00 
Localized edge states and homoclinic snaking in plane Couette flow 
Details
Tobias Schneider
(Joint work with John F. Gibson and John Burke)
Abstract:
Inducing turbulence in linearly stable shear flows such as pipe and plane
Couette flow requires perturbations of finite amplitude. The shape and size of
critical perturbations can be determined using the edge tracking algorithm (PRL
96, 174101 (2006); PRL 99, 034502 (2007)). For small, periodically continued
domains both critical
perturbations and the turbulent dynamics itself have recently been linked to
exact equilibrium and traveling wave solutions of the NavierStokes equations.
In large domains, where localized perturbations are observed to induce spatially
localized patches of turbulence which slowly invade the surrounding laminar
flow, these periodic solutions can however not capture the full spatiotemporal
dynamics but suggest the existence of their localized counterparts. We here
examine a new class of localized solutions to plane Couette flow that under
continuation in Reynolds number exhibit a sequence of saddlenode bifurcations
strikingly similar to the "homoclinic snaking" phenomenon observed the
SwiftHohenberg equation. These localized solutions originate from bifurcations
off the spatially periodic equilibria discovered by Nagata
and others and retain their physical structure, demonstrating the relevance of
exact solutions to turbulent flows in spatially extended domains.

Tobias Schneider 
12:0014:30 Lunch and Discussions
Time 
Title 
Details 
Speaker 
14:3015:00 
A lowdimensional model of separation bubbles 
Details
Rouslan Krechetnikov
Abstract:
In this work, motivated by the problem of modelbased predictive control of
separated flows, we identify the key variables and the requirements on a
modelbased observer and construct a prototype lowdimensional model to be
embedded in control applications.
Namely, using a phenomenological physicsbased approach and dynamical systems
and singularity theories, we uncover the lowdimensional nature of the complex
dynamics of actuated separated flows and capture the crucial bifurcation and
hysteresis inherent in separation phenomena. This new look at the problem
naturally leads to several important implications, such as, firstly, uncovering
the physical mechanisms for hysteresis, secondly, predicting a finite amplitude
instability of the bubble, and, thirdly, to new issues to be studied
theoretically and tested experimentally.

Rouslan Krechetnikov 
15:0015:30 
Computation of invariant tori by NewtonKrylov methods in
largescale dissipative systems 
Details
Juan Sánchez (Joint work with M. Net and C. Simó)
Abstract:
Invariant tori arise in general continuous or discrete dynamical systems,
usually, when a branch of periodic orbits loses stability at a NeimarkSacker
bifurcation, although in Hamiltonian systems they appear typically around a
(totally or partially) elliptic fixed point, periodic orbit or lower dimensional
torus.
A method to compute invariant tori in highdimensional systems, obtained as
discretizations of PDEs, by continuation and NewtonKrylov methods will be
described. Invariant tori are found as fixed points of a generalized Poincare
map so that the dimension of the system of equations to be solved is that of the
original system of ODEs. Therefore there is no prohibitive increase in the size
of the problem. Due to the dissipative nature of the systems studied, the
convergence of the linear solvers is extremely fast. The computation of periodic
orbits inside the Arnold's tongues is also considered.
The thermal convection of a binary mixture of fluids, in a rectangular
cavity, has been used to test the method. The branch of tori obtained, starts at
a NeimarkSacker bifurcation, and ends at a pitchfork bifurcation. Later on, the
stable branches of invariant tori undergo a cascade of perioddoubling
bifurcations leading to chaos. In this case, as typically happens, only a finite
number of period doublings is found before the chaotic range is reached.
Although the method is suitable for stable tori, we have been able to compute an
small branch of unstable invariant tori.

Juan Sánchez 
15:3016:00 
Coherent structures emerging from turbulence in the nonlocal complex
GinzburgLandau equation 
Details
Robert Hölzel
Abstract:
The nonlocal complex GinzburgLandau equation (NCGLE), an extension of the
complex GinzburgLandau equation (CGLE), is a normalform approach modelling the
nonlocal migration coupling that arises in electrochemical systems [V.
GarciaMorales and K. Krischer, Phys. Rev. Lett. 100, 054101 (2008)]. A
peculiarity of migration coupling is that the range of the coupling can be tuned
experimentally. The limit of negligible coupling range corresponds to diffusion
coupling, and in this limit, the NCGLE becomes the CGLE.
We show with numerical simulations of the NCGLE that with increasing range of
the nonlocal coupling coherent patterns develop in the turbulent parameter
region of the CGLE, and WE present a bifurcation analysis of the NCGLE with the
coupling range as main bifurcation parameter. We discuss in detail a series of
bifurcations involving heteroclinic connections of stationary states and of
limit cycles.

Robert Hölzel 
16:0016:30 Drinks, Snacks, and Discussion
Demonstration
Time 
Title 
Details 
Demonstrator 
16:3017:00 
Using OOMPHLIB to study bifurcation phenomena in fluid flows 
Details
Andrew Hazel
Abstract:
I shall demonstrate the use of spatial adaptivity, continuation and
bifurcation detection within OOMPHLIB, an opensource, C++ software library
(developed jointly with Matthias Heil) designed for the robust solution of
multiphysics problems, see http://www.oomphlib.org. Examples will be presented
from an ongoing project (joint with Rich Hewitt and Phil Haines) to investigate
the relationship between the nonlinear behaviour of similarity solutions to the
NavierStokes equations and the corresponding
full solutions in finite domains.
Fluid flows often exhibit features on many different lengthscales. A uniform
numerical discretisation based on the smallest lengthscale in the problem can
be extremely inefficient if the smallscale features are localised in space. One
solution, known as spatial adaptivity, is to solve the problem with a relatively
coarse discretisation and then to use an error estimator to select the regions
that contain finescale features. The discretisation in the selected regions is
refined and the problem is solved again. The process continues iteratively until
the error estimate is below a prescribed tolerance, or the computer runs out of
memory! If the error estimator is "good", the final discretisation will be
"adapted" to the particular problem and usually contains far fewer degrees of
freedom than the uniform discretisation required to achieve equivalent accuracy.
When assessing the stability of a fluid flow and/or seeking
bifurcations the associated eigenfunctions of the linearised system must also be
accurately resolved. In many cases, particularly near Hopf bifurcations of
reasonably large frequency, the flow and the eigenfunction can have quite
different spatial structures and using a discretisation based on the flow alone
can lead to large errors in the location of the bifurcation.

Andrew Hazel 
17:0018:00 Discussions and Demonstrations
Day/Date: Friday, July 9. Theme: Biomedical Applications
09:0009:30 Coffee
Talks
Time 
Title 
Details 
Speaker 
09:3010:00 
New insights into dynamic bifurcation problems with application to
neuronal and chemical systems 
Details
Steven Baer
Abstract:
When a parameter slowly ramps through a Hopf bifurcation, stability loss is
delayed considerably when compared to classical static theory. Inherent to
biological, chemical, and physical systems, but often overlooked or
misunderstood in the literature are nonlinear ramp problems where a parameter
slowly accelerates or deaccelerates through the bifurcation point. In this talk
I will briefly review, from a neuroscience perspective, the importance of the
dynamic bifurcation problem. I will then present recent results that show,
numerically and analytically, how slow nonlinear ramps can significantly
increase or decrease the onset threshold, changing profoundly our understanding of stability
loss delay in dynamic bifurcation problems. I will apply the results to membrane
accommodation in nerves, the formation of pacemakers in the BelousovZhabotinsky
reaction, and neuronal elliptic bursting. At the end to the talk I will discuss
ongoing research and several open problem areas.

Steven Baer 
10:0010:30 
Correlates between bifurcations and physiological attributes in the
cortex: a dynamicist approach 
Details
Federico Frascoli
(Joint work with L. van Veen, D. T. J. Liley, and I. Bojak)
Abstract:
Mean field models (MFMs) of cortical tissues focus on salient, average
features of populations of neurons, with the scope of understanding the
foundations of electrical brain activity. One of the common aspects of MFMs is
the presence of a high dimensional parameter space, which tries to capture the
essential neurobiological attributes of the cortex.
The parameter space of a MFMs of brain electrocortical activity is here
investigated, in relation to the effects in brain dynamics induced by general
anesthetics (GAs). It is possible to discover meaningful correlations through
the study of bifurcations of the anesthetized cortex. The generality of our
approach is also well suited to probe the parameter spaces of other existing
MFMs, in search of correlations between physiological attributes and dynamical
responses.
In particular, we show the presence of correlates among measurable attributes
of the brain, EEG spectral powers and dynamical patterns in the model. These
links are not accessible by standard linear or nonlinear parameter analyses, but
emerge when the parameter space is partitioned according to the dynamical
responses to GAs. These responses belong to two archetypal categories or
"families", which are investigated and characterized in depth. It also emerges
that families can be influenced dramatically by the actions of exhogeneous
stimuli, like those driven by the thalamic input.
The complexity of oscillatory activity within each family is not equivalent,
with diverse repertoires of sustained orbits. Distributions of some cortical
attributes are also antithetic, with some parameters exerting a strong control
on the dynamics of the reaction to GAs.
Finally, the role of inhibition in affecting the nature of the overall cortical
activity appears to be very strong.

Federico Frascoli 
10:3011:00 Coffee, Cake, and Discussion
Time 
Title 
Details 
Speaker 
11:0011:30 
Polyrhythms in dynamical models of multifunctional central pattern
generator networks 
Details
Andrey Shilnikov
Abstract:
A Central Pattern Generator is a neural network controlling various vital
repetitive locomotive functions including respiration and walking of humans,
swimming and crawling of leeches etc. This talk is focused on modeling
polyrhythmic dynamics in a multifunctional CPG, that is able to generate
multiple rhythms associated with distinct types of locomotive activity. We
describe synergetic mechanisms of the emergence of several synchronous behaviors
in in mutually inhibitory motifs being a network's building block.

Andrey Shilnikov 
11:3012:00 
Phase resetting biological oscillations  a topological theorem vs.
the real world 
Details
Leon Glass
Joint work with Michael Guevara, Alvin Shrier, Art Winfree, Thomas Gedeon,
Mark Josephson, Trine KroghMadsen, Eusebius Doedel, Bart Borek and Bart Oldeman
Abstract:
One of the classic ways to characterize biological or other oscillators is to
measure the resetting of the oscillation as a function of the phase of the
stimulus. To do this, we define the phase transition curve which gives the new
phase of the oscillation as a function of the phase of the stimulus. Provided
the stimulus leaves the oscillator in the same basin of attraction for all
stimuli phases, using topological arguments, the phase transition curve should
be a continuous map of the unit circle onto itself. Careful experimental tests
of the predictions are rare. However, in the 1980s, Michael Guevara did an
experiment that appeared to contradict the topological result. I will consider
mathematical and experimental issues involved with the resetting of oscillations
associated with limit cycles in ordinary differential equations representing
biological oscillations, partial differential equations representing the
propagation of an excitation wave in a one dimensional ring of excitable medium,
and partial differential equations representing a one dimension excitable medium
in which a pacemaker is embedded. These problems are relevant to problems
associated with the initiation and termination of serious cardiac arrhythmias.

Leon Glass 
12:0014:30 Lunch and Discussions
Time 
Title 
Details 
Speaker 
14:3015:00 
Dynamical models for the cell cycle 
Details
Willy Govaerts
(Joint work with Charlotte Sonck)
Abstract:
Basic knowledge of the cell cycle of various types of cells consists of the
substances involved in the process (cyclins, APC's etcetera) and the regulatory
network.
Dynamical systems models based on this regulatory network are fairly recent,
and they are not the only possible approach. Competing models use logical
dynamic modeling, a simpler approach that in some cases is quite successful by
showing that most trajectories funnel into a path which steps through the cell
cycle in a robust way.
On the other hand, models based on differential equations allow to use
mathematical tools such as bifurcation theory and numerical continuation under
parameter variation. In this way, multistability, periodic behaviour etc. can be
derived and studied in a fairly standard way.
A survey of the already impressive literature was recently published by A.
CsikaszNagy ("Computational systems biology of the cell cycle", Briefings in
Bioinformatics, Vol. 10 no 4, 424434). The leading group appears to be that of
John. J. Tyson and Bela Novak. Among other things, they built the most detailed
model so far of cellcycle regulation by
describing the control network of budding yeast /Saccharomyces cerevisiae./
In most cases, these models exhibit an even richer bifurcation structure than
is usually taken into account, e.g. TakensBogdanov points, periodic
oscillations internally in the G1 state of the system etc. It is not always
obvious how much of this is relevant for the applications.
We discuss a computational and bifurcation study of the budding yeast model
that includes a few phenomena that were so far not discussed in this context.
This includes the existence of different limit cycles, born at different Hopf
points that (approximately) merge in a single bigger limit cycle and a rather
unexpected relation between the growth
rate of the cell and the mass increase after DNAreplication.
We further discuss the implications of the funneling effect for the cell
cycle as a boundary value problem, and the computation of this cycle as the
fixed point of a map.

Willy Govaerts 
15:0015:30 
Modelling with MassAction Kinetics and Beyond 
Details
Markus Kirkilionis
Abstract:
Massaction kinetics is a powerful tool to describe events created by
collision of molecules or individuals in a wellmixed environment giving them
locally the same probability to meet each other. Moreover this probability is
only dependent on the concentration of the mutual partners. Mass action systems
can be found in chemistry, cell biology, but also game theory and economics.
Mathematically this gives rise to dynamical systems of a special type, more
specific of polynomial type. I will give an overview how this property can be
used to determine different types of bifurcations, for example the occurrence of
bistability, or oscillations via a Hopf bifurcation. All tools will be borrowing
methods from algebraic geometry. Finally I will give an outlook what usually
goes wrong in the modelling part while using massaction kinetics if biochemical
or cellular molecular events are considered.

Markus Kirkilionis 
15:3016:00 
Torus canards in a reduced neuron model 
Details
Mark Kramer
Abstract:
Neurons can exhibit a variety of dynamical states, including rapid spiking
and bursting. We describe these two states  and their transitions  in a
reduced mathematical model of a cerebellar Purkinje cell. We find at the
transition a canard phenomena that follows temporarily a branch of repelling
limit cycles. To explore these dynamics, we have recently developed a simpler
mathematical model of the canard phenomena, which we will also discuss.

Mark Kramer 
16:0016:30 Drinks, Snacks, and Discussion
Demonstration
Time 
Title 
Details 
Demonstrator 
16:3017:00 
Numerical continuation of orbit segments 
Details
Bart Oldeman
Abstract:
The method of successive continuation of orbit segments (also known as the
homotopy method) is a very powerful method for determining how a trajectory
varies as initial conditions change. Instead of shooting from a set of initial
conditions we pose a boundary value problem (BVP). First we compute one orbit
segment that satisfies one initial condition, which can actually be done by
continuation in its 'period' T, from 0 until the orbit satisfies a certain end
condition, for instance, for its length, time, or its intersection with a plane.
Next the initial condition of the orbit can be varied, keeping the end condition
from the first step.
Keeping the initial and end conditions wellposed the orbit is then continued
as a whole, which means that all points on the orbit are taken into account, and
not just the initial conditions. For example, sometimes the initial conditions
might not vary numerically but the segment changes a lot at its end or
elsewhere, and this method has no problems with that.
This method can be implemented in standard continuation software such as
AUTO, and has been used to compute, for example, the Lorenz manifold. I show
here its application to compute a simple unstable manifold and a phase resetting
curve for a Van der Polstyle system.

Bart Oldeman 
17:0018:00 Discussions and Demonstrations
Day/Date: Saturday, July 10. Theme: Industrial Applications and Mechanical
Systems
09:0009:30 Coffee
Talks
Time 
Title 
Details 
Speaker 
09:3010:00 
Task embedding  a paradigm for modular construction of composite
continuation problems and an example realization in a multipoint,
boundaryvalueproblem, collocation toolbox for the continuation of sets
of constrained orbit segments 
Details
Harry Dankowicz
Abstract:
A central theme in the recent development of the continuation package COCO by
Frank Schilder and the speaker is the construction of extended continuation
problems that afford runtime flexibility in deploying a particular covering
algorithm and selectively constraining the continuation. The corresponding
mathematical formulation naturally supports the idea of task embedding, a shared
responsibility for constructing the extended continuation problem across several
toolboxes. According to this notion, functionality afforded by distinct
toolboxes should be formulated in such a way that it can be combined to solve
composite continuation problems without code modification or reimplementation.
As argued in this talk, such an extended formulation enables innovative
computations that are not supported in a similar way as 'builtin' functionality
by any existing core implementations.
Specifically, this talk illustrates the notion of task embedding through the
implementation of a COCOcompatible collocation toolbox for multipoint,
boundaryvalue problems, which supports the generalpurpose parameter
continuation of sets of constrained orbits segments, such as i) segmented
trajectories in hybrid dynamical systems, for example, mechanical systems with
impacts, friction, and switching control, ii) homoclinic orbits represented by
an equilibrium point and a finitetime trajectory that starts and ends near this
equilibrium point, and iii) collections of trajectories that represent
quasiperiodic invariant tori. The collocation algorithm allows for
segmentdependent meshing and nontrivial boundary conditions involving internal
mesh points and includes a full discretization
of the corresponding variational equations, making it straightforward to track
solution branches associated with critical combinations of eigenvalues.

Harry Dankowicz 
10:0010:30 
Bifurcations with impact and friction; why it is easier to drag
chalk than push it 
Details
Alan Champneys
(Joint work with Arne Nordmark and Harry Dankowicz)
Abstract:
Newtonian impact and coulomb friction, when treated seperately are known to lead
to consistent formulations in terms of piecewisesmooth dynamical systems. Nevertheless,
socalled discontinuityinduced bifurcations such as grazing and sliding bifurcations can occur.
This talk considers what happens when friction is taken into account during an impact
event. The analysis is restricted to 2D, where the canonical example is that of a
slender body allowed to contact a rigid frictional surface (a piece of chalk
on a blackboard). First it is shown how to derive consistent impact laws that
generalise the coefficient of resitution law. These lead to the possibilities of
discontinuityinduced bifurcations as the sequences of sticking or slipping changes
during an impact event. Next it is shown that for contacting motion the socalled
Painleve paradox of nonuniqueness can be resolved by smoothing and passing to the
limit. However, there remains the possibility of reverse chatter, where infinite numbers
of impacts accumulate in reverse time, that cannot be ruled out at transition points between
stick and slip. The existence of such motion shows the fundamental nonuniqueness in
forward time simulations of rigid formulations of impact with Coulomb friction.

Alan Champneys 
10:3011:00 
Dynamics of aircraft as ground vehicles 
Details
Bernd Krauskopf, University of Bristol
(Joint work with James Rankin, Mark Lowenberg, and Mathieu Desroches)
Abstract:
Aircraft are meant to fly, but they also need to operate efficiently
and safely on the ground. As ground vehicles, commercial aircraft are quite
special; in particular, they feature a tricycle configuration
and very strong deformations of the tyres.
The talk will demonstrate how the standard ground manoeuvre of
aircraft turning can be studied by means of a bifurcation analysis of a
nonlinear aircraft model. In particular, it is shown that the
phenomenon of a sudden loss of tyre holding force when entering a spin is
mathematically a canard explosion.

Bernd Krauskopf 
11:0011:30 Coffee, Cake, and Discussion
Time 
Title 
Details 
Speaker 
11:3012:00 
Formation of asteroids: a simple mechanical model for complicated
shapes 
Details
Gabor Domokos
(Joint work with A. A. Sipos, Gy Szabo, and P. Varkonyi)
Abstract:
While the number of asteroids with known shapes has drastically increased
over the past few years, little is known on the the timeevolution of shapes and
the underlying physical processes. One apparent common feature of these shapes
is the existence of large, relatively flat areas separated by edges. Here we
propose a simple abrasion model based on multiple collisions with small objects,
which accounts for the emergence of such polyhedral shapes. We show that the
model is realistic, since the bulk of the collisions falls into this category.
We point out other effects, not captured by our model as well as an analogy to
pebble shapes.

Gabor Domokos 
12:0012:30 
Stable manifolds as thresholds for multipulse excitability 
Details
Pablo Aguirre
(Joint work with Bernd Krauskopf and Hinke Osinga)
Abstract:
Invariant manifolds of saddletype equilibria and periodic orbits are a key
ingredient to understand the global dynamics in many applied vector fields. They
act as boundaries that organize the overall dynamics in phase space [1]. While
it is fairly easy to study the related onedimensional objects, the same is not
true for (un)stable manifolds of higher dimension. These manifolds can now be
computed with high accuracy with numerical methods based on continuation of
orbit segments, defined as solutions of suitable boundary value problems [1, 4].
As an example we consider a model for a laser with optical injection [6]. Under
certain conditions, the laser dynamics presents excitable behavior just before
the creation of a homoclinic orbit to a saddlefocus p (also known as Shilnikov
homoclinic [3, 5]). The twodimensional stable manifold Ws(p) of this
equilibrium becomes an excitability threshold for the laser. A small
perturbation above W^{s}(p) makes an
excursion following the unstable manifold W^{u}(p)
before converging to an attractor q, originating a pulselike response in the
state variables. Multipulse behavior is also possible, meaning several pulses or
responses are generated from a single perturbation above this boundary.
Implementing our calculations with AUTO [2] we show the topological and
geometrical changes the twodimensional stable manifold W^{s}(p)
undergoes through a Shilnikov homoclinic bifurcation. In this way, we are able
to understand how the phase space from the laser dynamics is organized.

Pablo Aguirre 
Demonstration
Time 
Title 
Details 
Demonstrator 
12:3013:00 

Details
Harry Dankowicz
Abstract:
This demonstration illustrates the fundamental elements of the constructor
formalism of the continuation package COCO and its support of task embedding 
the shared responsibility for formulating and processing an extended
continuation problem across several toolboxes. Examples demonstrate a wide range
of functionality and adaptability, including flexible event handling,
continuation of sets of constrained trajectory segments and the associated
variational problem, and multidimensional manifold covering.

Harry Dankowicz 
Acknowledgments
This workshop is supported by the
Applied Mathematics Laboratory of the
Centre de Recherches Mathématiques (CRM),
Montreal, Canada; by the Centre for
Applied Mathematics in Bioscience and Medicine (CAMBAM), McGill University,
Montreal, Canada; and by the Aid to Research Related Events Programme of the
Office of the VicePresident, Research and Graduate Studies of Concordia
University.
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