Titlebook: Canard Cycles; From Birth to Transi Peter De Maesschalck,Freddy Dumortier,Robert Rouss Book 2021 The Editor(s) (if applicable) and The Auth

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The Slow Divergence IntegralA crucial notion in the study of the dynamics near a slow-fast cycle is the slow divergence integral. It is introduced in an intrinsic way and it is proved to be invariant under smooth time changes. Attention goes to calculating it in practice.

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Breaking MechanismsIn generating closed orbits from slow-fast cycles it is necessary to make use of so called breaking mechanisms. The chapter deals with the most generic ones, namely the jump breaking mechanism and the Hopf breaking mechanism. A description, accompanied by examples, is given.

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Local Transition MapsUsing the normal forms introduced in Chap. . the transitions near the singularities, showing up after blowing up the contact points, are described in terms of the functions that are introduced in Chap. .. Together with Chap. . on the blow-up technique this chapter forms the heart of the book.

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Outlook and Open ProblemsThe chapter contains a number of open problems and questions in view of a possible generalization of the theory, restricting to two dimensions. A quite ambitious programme for further study is presented, focusing on a number of steps that have already been taken in that direction.

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Basic Definitions and Notionsthe variation of situations that is dealt with, as well as examples that are beyond the scope of the book. Attention also goes to the more common and more restricted definitions. The chapter ends by stressing the big difference between singular points that are normally hyperbolic and the other ones, that are called contact points.