Undergraduate Student Research Projects

I take on summer students, through the NSERC Undergraduate Student Research Awards (USRA) programme (Canadian residents only), and also through the new McGill SURA awards (international students also elligible). See the

• McGill Mathematics and Statistics,
• McGill Faculty of Science and
• NSERC

Usually I'd expect you to be a Math Honours or Majors student, reasonably advanced in your degree, with some analysis courses and a Numerical Analysis course in your resume, and a cumulative GPA well above 3.00.

The deadline for applications is usually in late January. If interested contact me early January, or better still before Christmas. But I am interested in applications right up to the deadline. Most of my current research revolves around analysis and numerical analysis of delay and advance-delay equations, and recent projects have explored aspects of that. If anything below interests you, contact me.

Available Projects

Analysis and/or Numerics for Delay and Advance-Delay Differential Equations

My current research interests centre around dynamical systems and numerical analysis, particularly in the area of delay differential equations and advance-delay differential equations. A very simple example of a delay equation is

u'(t)=-γu(t)-κu(t-a)

Where γ>0 and a>0 and κ≥0 are constants. If κ=0 this is a very trivial ordinary differential equation. But for κ>0 it becomes very interesting. If you haven't seen an equation like this before, think about it for a while. If you assume that u(t)=ce^λt is a solution, you ought to be able to find an equation that λ satisfies (which doesnt involve t). A harder exercise is to show that this equation has infinitely many linearly independent complex solutions. Now since the original equation was linear, any linear combination of solutions is a solution and so the solution space is infinite dimensional. The stability of the trivial solution u(t)=0 is also interesting; it loses stability as κ increases.

Adding nonlinear terms makes the equation much more interesting, now bifurcations can occur. Adding extra delays or non-constant delays add extra complications, especially if the delays are state-dependent (that is the delay depends on the state of the system).

I don't expect prior knowledge of delay equations (they are not taught in undergraduate courses), and so a significant component of the project would be learning about these fascinating and complicated differential equations. Additional facets of the project could be analytical or computational and would depend on your interests and expertise also. Interesting problems include, but are not limited to, periodic orbits for state-dependent delay differential equations and the associated bifurcation structures, which can either be solved algebraically in a singular limit, or numerically in the nonsingular case. Also of interest is the development and analysis of numerical methods for delay differential equations.

If some of this sounds potentially interesting and challenging, contact me.

Previous Projects

Summer 2011: Michael Snarski

Periodic Orbits of a Singularly Perturbed Two State Dependent Delay Differential Equation

Many processes are modelled by differential equations subject to delays. In most models and mathematical theory this time delay is fixed, but in application areas there is much evidence of state dependent delays. However, the state dependency is usually suppressed when deriving mathematical models of these processes, because there is little mathematical theory of such equations. In this project we will study a model multiple delay nonlinear differential equation, with linearly state-dependent delays which undergoes Hopf bifurcations. Of particular interest are the periodic orbits created in the Hopf bifurcations. Last year we extended simple geometric techniques for the single delay problem to the multiple delay problem and hence identified some families of periodic orbits for the two delay problem in a singularly perturbed limit. The current project will build on the previous work to find additional families of periodic orbits, in particular studying the orbits created in a period doubling bifurcation identified in the previous project, and other possible bifurcations in the system.

Summer 2011: Namdar Homayounfar

Numerical Study Of A Family Of A State Dependent Delay Differential Equations Close To A Singular Limit

Numerical study using ddebiftool to find Hopf bifurcations and the resulting branches of periodic solutions in a family of state dependent delay differential equations close to a singularly perturbed limit. Secondary fold and period doubling bifurcations and the resulting solution branches are also studied. Bifurcations and transitions between unimodal, bimodal, trimodal and quadrimodal periodic solutions are also identified.

Summer 2010: Daniel Bernucci

Singularly Perturbed Multiple State Dependent Delay Differential Equations

Many processes are modelled by differential equations subject to delays, where the delays can arise in a number of contexts such as maturation time in population models. In most models and mathematical theory this delay time is fixed, but in application areas there is much evidence of state dependent delays. For example, when the body has deficit of white blood cells, production of new ones is ramped up and they are actually matured faster. However, the state dependency is usually suppressed when deriving mathematical models of these processes, because there is little mathematical theory of such equations. In this project we will study a model multiple delay nonlinear differential equation, with linearly state-dependent delays which undergoes Hopf bifurcations.

In this project the student will learn the basic theory of delay differential equations, and some of the numerical methods used to study them (applying existing matlab packages), and apply this knowledge to a study of singularly perturbed state-dependent equations of the form mentioned above. Of particular interest are periodic orbits which exist in and close to a singularly perturbed limit, which have been observed to exist numerically. In the case of a single delay, simple geometry determines the form of the orbit. The student will use a generalisation of these simple techniques to study the possible periodic orbits in the singular limit for two state-dependent delays, and how they depend on the parameters in the system, and will also undertake a numerical investigation of these orbits.

Summer 2009: Orianna DeMasi Analysis and Numerical Analysis of State Dependent Delay Differential Equations Many processes are modelled by differential equations subject to delays, where the delays can arise in a number of contexts such as maturation time in population models. In most models and mathematical theory this delay time is fixed, but in application areas there is much evidence of state dependent delays. For example, when the body has deficit of white blood cells, production of new ones is ramped up and they are actually matured faster. However, the state dependency is usually suppressed when deriving mathematical models of these processes, because there is little mathematical theory of such equations. In this project we will study a model multiple delay nonlinear delay differential equation, with linearly state-dependent delays which can undergo Hopf bifurcations, and also has inaccessible regions of phase space (where the equations become advance-delayed and are no longer well posed as initial value problems). In this project the student will learn the basic theory of delay differential equations, and some of the numerical methods used to study them (applying existing matlab packages), and apply this knowledge to a study of state-dependent equations of the form mentioned above. Of particular interest are the Hopf bifurcations, the evolution of the periodic orbits created in the bifurcations as parameters are varied, and the possibility of multiple co-existing periodic orbits, or even multiple Hopf bifurcations. The overall aim is to increase the understanding of model state-dependent problems, so that in future more applicable equations can be tackled. Finn and Orianna's projects led a paper which will appear in Discrete and Continuous Dynamical Systems - Series A.

Summer 2008: Finn Upham

Model Problems in State Dependent Delay Equations

Many processes are modelled by differential equations subject to delays, where the delays can arise in a number of contexts such as maturation time in population models, or communication time between spatially located components of a system. In most models and mathematical theory this delay time is fixed, but in application areas there is much evidence of state dependent delays. For example, when the body has deficit of white blood cells, production of new ones is ramped up and they are actually matured faster. However, the state dependency is usually suppressed when deriving mathematical models of these processes, because there is little mathematical theory of such equations. There is a model problem due to Mallet-Paret and co-workers where the delay is linearly state dependent, which leads to a region of phase space which is inaccessible, because the delay term becomes advanced there. Multiple delay versions of this problem, are in general illposed as retarded equations because solutions may enter the inaccessible region. In this project the student will learn the basic theory of delay differential equations and study model state dependent delay equations, such as the one mentioned, and seek to identify parameter regions for which the initial value problem is well posed as a retarded delay equation. She will also conduct some numerical studies (applying existing matlab packages) on the behaviour of solutions in the case of retarded equations, and finally investigate the possibility of bounded solutions in parameter regimes where the state-dependent arguments become advanced.

Summer 2004: Myriam Berube

Travelling Waves and Differential Equations with Advances and Retards

Functional Differential Equations (FDEs) with advances and retards such as the discrete Fitzhugh-Nagumo (dFN) equation

-cu'(t)=u(t+1)-2u(t)+u(t-1)-f(u(t))-v(t)

-cv'(t)=b(u(t)-rv(t))

arise naturally as boundary value problems when seeking travelling waves or pulses for lattice differential equations, and in other applications. Whilst equations where the velocity depends on both the past and future positions may seem crazy when you first see them, they are incredibly fascinating really. The dFN equations are used to model nerve impulses in mylineated nerve fibres, and understanding their behaviour is an important first step before modelling demylineation, a factor in MS. Solving the equations analytically is usually impossible except for piecewise linear nonlinearities.

In this project the student will learn about travelling waves and FDEs and numerically compute travelling pulse solutions to the dFN equations with the naturally arising cubic nonlinearity. These solutions will be found by homotopying from known solutions in the idealised piecewise linear case, using the supervisors previously developed collocation code for solution and parameter continuation of FDE boundary value problems. This is an exciting project; Elmer & Van Vleck have only recently found the pulse solutions in the piecewise linear dFN, and the cubic case has not previously been studied in detail.