Abstract

In these lectures we want to study systems depending on multiple time scales and present geometric techniques to study such systems (by “geometric” we mean coming from “regular” dynamical systems theory). In fact we will focus on two different time scales, hence on so-called “slow-fast”systems. We will essentially limit to two-dimensional systems, however depending on an arbitrary number of parameters. Our attention will primarely go to periodic orbits, called relaxation oscillations, and the bifurcations that they undergo. Special attention will be given to Liénard equations.

Some points of attention:

• center manifolds near normally hyperbolic singular curves
• role of slow divergence integral
• canard solutions
• (family) blow up of contact points
• generic breaking mechanisms and entry-exit relations
• multi-layer canard cycles and translated power functions
• slow dynamics with singularities
• birth of canard cycles
• Hilbert’s 16th problem, Smale’s 13th problem and Roussarie’s program
• compactification of spaces of polynomial Liénard equations

Course notes will be distributed. The notes are based on different papers about singular perturbations by Robert Roussarie, Peter De Maesschalck and myself , that have already been published (see e.g. MathSciNet) or that will be made available during the course week. Prerequisites can be found in the chapters 1,2,3,5 and 7 of the book:
F.Dumortier, J.Llibre, J.C.Artés: “Qualitative theory of planar differential systems”, Universitext, Springer-Verlag, Berlin, 2006, xvi+298pp (ISBN:3-540-32893-9).

Outline

1. Examples of nonsmooth systems. Introduction to phase plane dynamics, simple ideas of flows and maps. Existence of equilibira, stability and bifurcations
2. Piecewise linear maps in the plane, analysis and classification of equilibria, their existence, stability and bifurcations. Period adding bifurcations. Relationship to flows.
3. Three dimensional piecewise linear maps; classification of simple bifurcations
4. General n-dimensional piecewise linear maps.
5. Other piecewise smooth maps in the plane (including ‘map with a gap’).
6. Smoothing of piecewise linear maps and effect on period adding bifurcations.
7. Application: global dynamics of low immersion high-speed milling.
8. Nonsmooth flows; discontinuity induced bifurcations, Filippov system.
9. One-parameter bifurcations of planar Filippov systems.
10. Grazing and chattering; discontinuity mappings near grazing.
11. Application: DC/DC converter.
12. Towards a nonsmooth catastrophe theory (if time).

References

• (For background in nonlinear dynamics) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry and engineering. Strogatz, S. H. Perseus Books.
• One-parameter bifurcations of planar Filippov systems Kuznetsov, Y.A. et al Int. J. Bifurcation and Chaos 13, 2157- 2188 (2003).
• Piecewise smooth dynamical systems: theory and applications. Di Bernardo et alSpringer, Applied Mathematical Sciences vol. 163.

Outline

1. Examples of hyperbolic sets: horseshoe, solenoid, Anosov, derivatives of Anosov and Plykin atractor.
2. Transversal homoclinic intersections.
3. Persistence and stability of hyperbolic sets; shadowing lemma.
4. Stability of globally hyperbolic diffeomorphisms (Anosov).
5. Filtrations and spectral decomposition of diffeomorphisms. Axiom A. Omega-stability theorem.
6. Comments on the stability and omega-stability conjectures.
7. Elements of bifurcation theory.
8. Homoclinic tangencies. Newhouse phenomenon.
9. Some notions on robust transitivity and partially hyperbolic systems.

References

• A. Katok, B. Hasselblatt, Introduction to the moderm theory of dynamical systems; Cambridge University Press, 1995.
• Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations; Cambridge University Press, 1995.
• E. R. Pujals, M. Sambarino, Topics on homoclinic bifurcation, dominated splitting,robust transitivity and related results; Handbook of dynamical systems vol 1B, Elsevier (2005) 327-378.
• M. Shub, Global Stability of Dynamical Systems; Springer-Verlag, 1987.