Dynamical Systems, Bifurcation Theory, Delay-Differential Equations
Prof. LeBlanc is an applied mathematician who works mainly in the analysis of dynamical systems, with emphasis on their bifurcations. Most of the classes of equations in which he is interested (e.g reaction-diffusion partial differential equations, delay-differential equations) are immediately relevant to neural and electrophysiological modelling. For reaction-diffusion partial differential equations, he is interested in the dynamics and bifurcations of spiral waves. In cardiac tissue, these waves are believed to be a precursor to several fatal types of arrythmias. LeBlanc and his collaborators Marty Golubitsky (U. Houston), Ian Melbourne (U. Surrey), Claudia Wulff (U. Surrey) and his students have used a dynamical systems approach to explain (and predict) many of the dynamical features of spiral waves. For delay-differential equations (DDEs), LeBlanc has been working on the construction of a bifurcation theory which exploits symmetries of the associated system. This theoretical work will be important to modellers who use DDEs, since one of the goals of this research program is to identify and classify the range of possible dynamics accessible from within the parameter space of a given DDE model.
Victor LeBlanc was born and raised in Dieppe, NB, Canada. He received an honours B.Sc. Physics from the Université de Moncton in 1989, and then moved on to the University of Waterloo on an NSERC 1967 Science and Engineering Scholarship, where he received an M.Math (1991) and Ph.D. (1995), both in Applied Mathematics. His M.Math. thesis (supervised by John Wainwright) dealt with a dynamical systems analysis of orthogonal spatially homogeneous cosmological models with perfect fluid and magnetic field sources. His doctoral thesis, which was supervised by William Langford, was on the analysis of the 1:2 resonant Hopf bifurcation in differential equations. He was awarded an NSERC Postdoctoral Fellowship, which he undertook at the University of Houston under the supervision of Marty Golubitsky. In 1996, he joined the Department of Mathematics and Statistics of the University of Ottawa as an Assistant Professor, and was promoted to Associate Professor in 2000. He has refereed papers for several of the most prestigious journals in dynamical systems: e.g. Physica D, Nonlinearity, Journal of Nonlinear Science, Journal of Differential Equations, SIAM Journal on Applied Dynamical Systems.
Choi, Y. and LeBlanc, V.G. (2005) Toroidal normal forms for bifurcations in retarded functional differential equations II: saddle-node/multiple Hopf interaction, 30 pages. Submitted.
Boily, P., LeBlanc, V.G. and Matsui, E.T. (2006) Spiral anchoring in media with multiple inhomogeneities: a dynamical system approach, 31 pages. Submitted.
Choi, Y. and LeBlanc, V.G. (2006) Toroidal normal forms for bifurcations in retarded functional differential equations I: multiple Hopf and transcritical/multiple Hopf interaction, Journal of Differential Equations (to appear), 46 pages.
Buono, P.L. and LeBlanc, V.G. (2005) Equivariant versal unfoldings for linear retarded functional differential equations, Discrete and Continuous Dynamical Systems-A 12, pp. 283-302
Bourgault, Y., Ethier, M. and LeBlanc, V.G. (2003) Simulation of Electrophysiological Waves with an Unstructured Finite Element Method, ESAIM: Mathematical Modelling and Numerical Analysis 37, pp. 649-662
Buono, P.L. and LeBlanc, V.G. (2003) Versal unfoldings for linear retarded functional differential equations, Journal of Differential Equations 193, pp. 307-342
LeBlanc, V.G. and Roth, B.J. (2003) Meandering of spiral waves in anisotropic tissue, Dynamics of Continuous, Discrete and Impulsive Systems, Series B10, pp. 29-42
LeBlanc, V.G. (2002) Rotational symmetry-breaking for spiral waves, Nonlinearity 15, pp. 1179-1203
Redmond, B., LeBlanc, V.G. and Longtin, A. (2002) Bifurcation analysis of a class of first-order nonlinear delay-differential equations with reflectional symmetry, Physica D 166, pp. 131-146
LeBlanc, V.G. (2001) The role of symmetry in spiral wave dynamics, Physics in Canada 57, pp. 121-128
LeBlanc, V.G. and Wulff, C. (2000) Translational symmetry-breaking for spiral waves, Journal of Nonlinear Science 10, pp. 569-601