Canard Cycles

Peter De Maesschalck, born in 1975, has been at Hasselt University, Belgium, for much of his career. His research focuses on slow-fast systems in low dimensional systems both from a qualitative point of view and from the point of view of asymptotic expansions. Part of his research is inspired by theoretical questions such as Hilbert's 16th problem on limit cycles of polynomial systems, another part is motivated by applications of slow-fast systems in, e.g., neurological models. Freddy Dumortier, born in 1947, emeritus professor at Hasselt University, is former president of the Belgian Mathematical Society and is currently permanent secretary of the Royal Flemish Academy of Belgium for Science and the Arts. He is the author of many papers and his main results deal with singularities and their unfolding, bifurcation theory, Lienard equations, Hilbert's 16th problem, slow-fast systems and the wave speed in reaction-diffusion equations. Robert Roussarie, born in 1944, is emeritus professor of the University of Bourgogne-Franche Comte. After a career at the CNRS he was professor at the Institut de Mathematique de Bourgogne. He worked on the theory of foliations, of singularities in differential geometry, bifurcations of vector fields and finally slow-fast systems. He also contributed to applied research on ferro-resonance in electrical networks, systems of ecological populations, systems in control theory and free interface problems in combustion theory.

Part I Basic Notions.- 1 Basic Definitions and Notions.- 2 Local Invariants and Normal Forms.- 3 The Slow Vector Field.- 4 Slow-Fast Cycles.- 5 The Slow Divergence Integral.- 6 Breaking Mechanisms.- 7 Overview of Known Results.- Part II Technical Tools.- 8 Blow-Up of Contact Points.- 9 Center Manifolds.- 10 Normal Forms.- 11 Smooth Functions on Admissible Monomials and More.- 12 Local Transition Maps.- Part III Results and Open Problems.- 13 Ordinary Canard Cycles.- 14 Transitory Canard Cycles with Slow-Fast Passage Through a Jump Point.- 15 Transitory Canard Cycles with Fast-Fast Passage Through a Jump Point.- 16 Outlook and Open Problems.- Index.- References.