ABSTRACT
In this chapter, we discuss the most interesting results concerned with maps obtained from the idea of generalizing the Lozi maps (3.1) or (3.2) to different forms with different purposes. Some simple piecewise linear models for the zones of instability are presented in Section 4.1. In Section 4.2, we present a special case of the Lozi maps where it is possible to calculate rigorously their fractal dimensions using the usual covering. The 3-D piecewise linear noodle map is presented in Section 4.3. This map can display additional phenomena expected from four-dimensional fl ows and not possible in two-dimensional maps, i.e., hyperchaos phenomenon1. A class of systems called generalized hyperbolic attractors is discussed in Section 4.4. This class includes the 2-D hyperbolic attractors of Belykh and Lozi mappings (3.1) or (3.2) and the Lorenz attractor given by (1.49) and have further ergodic properties such as Smale spectral decomposition and some important statistical properties. In Section 4.5, the global periodicity property of some generalized Lozi mappings is discussed. The generalized discrete Halanay inequality and the global stability are presented in Section 4.6. Some global behaviors of some (max) difference equations are discussed in Section 4.7. We note that in Sections 4.5, 4.6 and 4.7, each difference equation can be reduced to a special case of the Lozi mapping. In Section 4.8, a class of piecewise-linear area-preserving plane maps is presented and discussed. The main issue here is to look to the properties of rotation number and the
associated circle map. Several analytical results are obtained and discussed. In Section 4.9, a smooth version of the Lozi mappings (3.1) is presented in order to make a comparison with piecewise dynamics. In Section 4.10, a map with border-collision period doubling scenario is introduced. This route to chaos is relatively new in the literature and usually not taken for the Lozi map (3.1). In Section 4.11, a rigorous proof of chaos in a 2-D piecewise linear map is based on the equivalence of the matrices defi ning its formula. Since, only one route to chaos was observed for the Lozi map (3.1), we present in Section 4.12 a 2-D map with two different routes to chaos, namely, period doubling and border collision bifurcations. Section 4.13 is devoted to the generation of multifold chaotic attractors, this including C1-multifold chaotic attractors in Section 4.13.1 and C∞-multifold chaotic attractors in Section 4.13.2. In Section 4.14, a simple 2-D piecewise linear map is presented. This map differs from the Lozi map in that it has a much wider variety of attractors. Another 2-D piecewise linear map called the discrete hyperchaotic double scroll is presented in Section 4.15. This map is obtained from the Lozi map (3.2) using the characteristic function of the Chua circuit (1.50)-(1.51). Since all the generalized Lozi mappings are piecewise linear, we present in Section 4.16, a short description of the theory of 2-D piecewise smooth maps along with the non-invertible piecewise smooth case. Some relevant results about the normal form and the occurrence of robust chaos are also discussed in some detail. Finally, we note that notations in this chapter are not unifi ed in the whole text because the subject of each section is different from the others. So the unifi cation of notations is done only for each section of this chapter.
