By Zeraoulia Elhadj

This e-book is a accomplished choice of identified effects concerning the Lozi map, a piecewise-affine model of the Henon map. Henon map is likely one of the so much studied examples in dynamical platforms and it draws loads of recognition from researchers, but it is hard to research analytically. easier constitution of the Lozi map makes it superior for such research. The booklet is not just an excellent advent to the Lozi map and its generalizations, it additionally summarizes of significant strategies in dynamical structures conception equivalent to hyperbolicity, SRB measures, attractor kinds, and extra.

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**Example text**

2) (e) The Lebesgue measure is defined by m(A) = m*(A) for any Lebesgue measurable set A. 2) if the set A is a topologically transitive attractor of a dynamical system. 1, and also for hyperbolic diffeomorphisms and flows studied in Sinai (1972), Ruelle (1976), Ruelle & Bowen 1975(a-b). The common definition of the SRB measure is given for discrete time systems by: Definition 2 Let A be an attractor for a map f, and µ be an f-invariant probability measure of A. e. 3) We have m* (S\A) = 0 if set A is a topologically transitive attractor of a dynamical system.

In this case, the homoclinic tangency of stable and unstable manifolds of saddle points in the Poincaré section is the principal cause of this complexity as shown in [Gavrilov & Shil’nikov (1972-1973), Afraimovich (1984–1989–1990)]. On the other hand, for the quasi-attractors the basins of attraction of co-existing limit sets is very narrow and it can have fractal boundaries. Hence, rigorous mathematical description of quasi-attractors are still an open problem, because almost non-hyperbolic attractors are obscured by noise.

From the above presentation and examples, we can conclude that the hyperbolic attractors have the following properties: 1. They are the limit sets for which Smale’s Axiom A is satisfied. 2. They are structurally stable. 3. , the same dimension for their stable and unstable manifolds. In conclusion, we note that the previous properties are results of a rigorous axiomatic foundation that exploits the notion of hyperbolicity [Ott (1993), Katok & Hasselblatt (1995)]. In fact, hyperbolic chaos is often called true chaos because it is characterized by a homogeneous and topologically stable structure as shown in [Anosov (1967), Smale (1967), Ruelle & Takens (1971), Guckenheimer & Holms (1981)].