CORIA - Université de Rouen
F-76801 Saint-Etienne du Rouvray cedex
This lecture will be devoted to the main route to chaos and few techniques to characterize chaotic behaviors. Among the numerous route to chaos, period-doubling cascade and intermittency will be briefly described. Then it will be explained how to choose a Poincaré section, how to compute a first-return map and what may be learnt from its structure. In particular, in some school cases, it will be explained in which way the critical points of the map (corresponding to its extrema) can be used to make a partition of the attractor and how each domain is different from the topological point of view .
Periodic orbits will be described in terms of symbolic dynamics as well as using linking numbers. It will be explained how it is possible to deduce a compatible template (or knot-holder) from these numbers. Once the template is obtained, all the topological properties are explicit and the attractor is characterized. Few attractors with different topologies  will be presented and it will be explained how these attractors could be classified using bounding tori 
Some open cases will be briefly introduced. For instance, the tri-dimensional system published by Edward Lorenz in 1984 , few Rössler systems and a Chua circuit.
 C. Letellier, E. Roulin & O. E. Rössler, Inequivalent topologies of chaos in simple equations, Chaos, Solitons & Fractals, 28, 337-360, 2006.
 T. D. Tsankov& R. Gilmore, Strange attractors are classified by bounding tori, Physical Review Letters, 91 (13), 134104, 2003.