Research Overview
Motivated by the study of physical structures such as crystals, grids of neurons and population patches,
an increasing interest has arisen in mathematical modelling techniques that reflect the underlying spatial
discreteness. My main research goal is to develop tools that can be used to characterize the differences and
similarities between such discrete models and their more traditional continuous counterparts. Although most
of my work is analytical, I have also worked on numerical algorithms to help visualize the theoretical concepts.
Some of my results also turned out to be applicable in the seemingly unrelated area of economic modelling.
As a consequence, my interests have also extended to this field.
Waves and Patterns on Two Dimensional Lattices
Embedding the standard square grid into the plane breaks the natural isotropy
of space that is often taken for granted, as two preferred lattice directions are chosen.
In particular, patterns that travel in a straight line will experience
different effects depending on the orientation of their trajectory
versus the grid. In this project we study the existence and stability
of travelling waves and explore how the direction dependence
comes into play. We use both comparison principles
and spectral methods in our stability analysis.
This allows us to consider
small perturbations in a nonmonotonic setting and large perturbations
in a monotonic setting.
Papers

A. Hoffman, H.J. Hupkes and E. S. Van Vleck,
Entire Solutions for Bistable Lattice Differential Equations with Obstacles
Memoirs of the AMS, to appear [PDF].
 Large basin stability analysis based on comparison principle.
 Holes and impurities in the lattice can be accomodated.
 Inspired by Berestycki, Hamel and Matano (2009).

A. Hoffman, H.J. Hupkes and E. S. Van Vleck,
MultiDimensional Stability of Waves Travelling through Rectangular Lattices in Rational Directions
Transactions of the AMS, to appear [PDF].
 Local stability analysis based on spectral methods.
 No comparison principle required.
 Different decay rates for different directions.
 Inspired by Kapitula (1997).

H.J. Hupkes and E. S. Van Vleck,
Negative Diffusion and Travelling Waves in High Dimensional Lattice Systems
SIAM Journal on Mathematical Analysis 45, 10681135 (2013) [PDF].
 Existence of travelling checkerboards for lattices with periodic spatial structure.
 Extension of Chen's (1997) classic travelling wave construction for scalar
systems to multicomponent systems.
 Extension of MalletParet's (1999) Fredholm framework to multicomponent systems.
Talks

Travelling around Obstacles in Planar Anisotropic Spatial Systems, Amsterdam, June 2013
[PDF].

MultiDimensional Stability of Travelling Waves through Rectangular Lattices, Minneapolis MN, December 2012
[PDF].

Negative Diffusion in High Dimensional Lattice Systems  Travelling Waves, Oberwolfach, December 2012
[PDF].
Waves and Patterns on One Dimensional Lattices
Many results concerning the existence and stability
of travelling waves in one dimensional lattices
are based on comparison principles.
This project explores alternative techniques to
analyze travelling patterns,
by providing an infinite dimensional analytical
underpinning to classic geometrical ideas.
Papers

C.H.S. Hamster and H.J. Hupkes,
Stability of Travelling Waves for
ReactionDiffusion Equations with Multiplicative Noise
submitted [PDF] (updated July 2018).
 Nonlinear stability approach for stochastic travelling waves.
 Prediction of changes to waveprofile and speed.

W. Schouten and H.J. Hupkes,
Nonlinear stability of pulse solutions for the discrete FitzHughNagumo equation
with infiniterange interactions
submitted [PDF].
 Existence and stability of pulse solutions in nearcontinuum regime.
 Based on spectral convergence technique developed by Bates and coworkers.

H.J. Hupkes and B. Sandstede,
Stability of Pulse Solutions for the Discrete FitzHughNagumo System
Transactions of the AMS, to appear [PDF].
 Stability analysis based on an analytical version of the exchange lemma
and exponential dichotomies for functional differential equations of mixed type.
 Circumvents the use of the Evan's function.

H.J. Hupkes, B. Sandstede,
Travelling Pulses for the Discrete FitzHughNagumo System
SIAM Journal on Applied Dynamical Systems 9, 827882 (2010) [PDF].

Pulses for FitzHughNagumo LDE obtained by gluing together front and back solutions to Nagumo LDE.

Develops an analytical variant of geometric singular perturbation theory,
based on Lin's method and exponential dichotomies.

Inspired by Krupa, Sandstede and Szmolyan (1997).

M. Beck, H.J. Hupkes, B. Sandstede and K. Zumbrun,
Nonlinear Stability of Semidiscrete Shocks for TwoSided Schemes,
SIAM Journal on Mathematical Analysis 42, 857903 (2010) [PDF].

Stability analysis for systems with a conservation law,
where the essential spectrum touches the imaginary axis.

Based on meromorphic continuation of Green's function through essential spectrum.

H.J. Hupkes and B. Sandstede,
Modulated Wave Trains for Lattice Differential Systems,
Journal of Dynamics and Differential Equations 21, 417485 (2009) [PDF].

Construction of modulated travelling waves that connect wave trains with slighly different frequencies.

Based on analytic construction of global center manifold around ring of equilibria,
circumventing graph transform techniques.

Construction of global center manifold inspired by Sakomoto (1990) and Yi (1993).

Construction of modulated waves inspired by Doelman, Sandstede, Scheel and Schneider (2009).
Talks

Existence and Stability of Fast Pulses for the Discrete FitzHughNagumo System,
Snowbird UT, May 2011
[PDF].

Travelling Pulses for the Discrete FitzHughNagumo System,
Providence RI, November 2009
[PDF].

Modulated Travelling Waves in Discrete Reaction Diffusion Systems,
Snowbird UT, May 2009
[PDF].
Numerical Aspects
In this project we explore the role that discretization schemes and numerical
algorithms play when studying spatial evolution systems.

H.J. Hupkes and E. Van Vleck,
Travelling Waves for Adaptive Grid Discretizations of
ReactionDiffusion Systems,
submitted [PDF].
 Existence of travelling wave solutions for discretizations on arclengthequidistributed meshes.
 Based on spectral convergence technique developed by Bates and coworkers.

H.J. Hupkes and E. Van Vleck
Travelling waves for Complete Discretizations of Reaction Diffusion Systems
JDDE, to appear. [PDF].

Explores effect of full spatialtemporal discretizations of reactiondiffusion PDEs
on the existence and uniqueness of travelling wave solutions.

Uses a singular perturbation technique to study bifurcations between difference operators
and differentialdifference operators.

H.J. Hupkes, D. Pelinovsky and B. Sandstede,
Propagation failure in the discrete Nagumo equation
Proceedings of the AMS 139, 35373551 (2011) [PDF].

Explores nonstandard discretizations of reactiondiffusion PDEs
that do NOT exhibit propagation failure.

H.J. Hupkes and S.M. Verduyn Lunel,
Analysis of Newton's Method to Compute Travelling Waves in Discrete Media,
Journal of Dynamics and Differential Equations 17, 523572 (2005) [PDF,PS].

Theoretical underpinning of common numerical technique used to solve (singular) MFDEs.

Discusses algorithm originally described by Elmer and Van Vleck (2002).
Talks

Travelling Waves vs Discretization Schemes
Eindhoven, Sep 2017
[PDF].

Travelling Waves for Fully Discretized Bistable ReactionDiffusion Problems,
Madrid, July 2014
[PDF].

Propagation Failure in the Discrete Nagumo Equation,
Athens GA, April 2011
[PDF].
Initial Value Problems involving MFDEs with Economic Applications
A common assumption in economic theory is that agents have
perfect foresight.
While not entirely realistic, it is of course true that
expectations play
an important role when making decisions, which can lead to wellknown phenomena
such as
selffulfilling prophecies. Functional differential equations
of mixed type (MFDEs) arise naturally in certain macroeconomic agentbased models
as they can incorporate such
expectations. Models of this type
suffer from an incompatibility between the
mathematical state space
and the (smaller)
modelling state space. This project
investigates the consequences of this mismatch, focussing on
the effects of incomplete information on decisionmaking.
Papers

H.J. Hupkes and E. AugeraudVéron,
WellPosedness of Initial Value Problems for VectorValued Functional Differential Equations of Mixed Type
Preprint [PDF].

WienerHopf factorizations and exponential dichotomies for MFDEs posed on Hilbert spaces.

Practical application: Nash equilibria for competitive games with time delays.

H. d'Albis, E. AugeraudVéron and H.J. Hupkes,
Discontinuous Initial Value Problems for Functional DifferentialAlgebraic Equations of Mixed Type
Journal of Differential Equations, to appear [PDF].

In what sense can jumps at single points in time stabilize otherwise unstable
solutions to MFDEs?

Practical application: can central banks control runaway inflation
by hiking interest rates?

Also covers differentialalgebraic systems.

H.J. Hupkes and E. AugeraudVéron,
WellPosedness of Initial Value Problems for Functional Differential and Algebraic Equations of Mixed Type
Discrete and Continuous Dynamical Systems A 30, 737765 (2011) [PDF].

Inspired by the WienerHopf factorization developed by Verduyn Lunel and MalletParet (to appear).

Explores practical ways in which wellposedness questions for MFDEs can be answered
even if an explicit WienerHopf factorization is not available.

Also covers differentialalgebraic systems.
Talks

Wellposedness of Initial Value Problems for Functional DifferentialAlgebraic Equations of Mixed Type,
Loughborough, August 2011
[PDF].
MFDEs: bifurcation theory near equilibria and periodic solutions
Functional differential equations of mixed type (MFDEs) are differential
equations with both retarded and advanced arguments. As such, they can
be seen as a natural generalization of delay differential equations.
In this project we show how a number of classical tools can be adapted
for use in the MFDE world.
Papers

H.J. Hupkes and S.M. Verduyn Lunel,
Lin's Method and Homoclinic Bifurcations for Functional Differential Equations of Mixed Type,
Indiana University Mathematics Journal 58, 24332488 (2009) [PDF].

Adaptation of Lin's method for use in the infinitedimensional world of MFDEs.

Exponential dichotomies allow bifurcation equations to be posed on finitedimensional
subspaces.

Inspired by the exponential splittings developed by Verduyn Lunel and MalletParet (to appear).

H.J. Hupkes and S.M. Verduyn Lunel,
Center Manifolds for Periodic Functional Differential Equations of Mixed Type,
Journal of Differential Equations 245, 15261565 (2008)
(original MI Report 200708) [PDF,PS].

Center manifolds are constructed around periodic solutions to MFDEs.

H.J. Hupkes, E. AugeraudVéron and S.M. Verduyn Lunel,
Center Projections for Smooth Difference Equations of Mixed Type,
Journal of Differential Equations 244, 803835 (2008)
(original MI Report 200701) [PDF,PS].

Center manifolds are constructed for differentialalgebraic versions of MFDEs.

Key ingredient is the use of jets to ensure that the differentialalgebraic
structure does not interfere with the cutoff operators.

H.J. Hupkes and S.M. Verduyn Lunel,
Center Manifold Theory for Functional Differential Equations of Mixed Type,
Journal of Dynamics and Differential Equations 19, 497560 (2007)
(original MI Report 200603) [PDF,PS].

Local center manifolds are constructed around equilibrium solutions to MFDEs.

Generalizes the construction of center manifolds for delay differential equations
(see Chapter 9  Diekmann, Van Gils, Verduyn Lunel and Walther).

Circumvents use of sunstar calculus, which is not (readily) available for MFDEs.
Talks

Lin's Method and Homoclinic Bifurcations for Functional Differential Equations of Mixed Type,
Arlington TX, May 2008
[PDF].

Invariant Manifolds and Applications for Functional DifferentialAlgebraic Equations of Mixed Type,
Vienna, August 2007
[PDF].

Center Manifold Theory for Functional Differential Equations of Mixed Type,
Poitiers, June 2006
[PDF].