Download Lozi Mappings: Theory and Applications by Zeraoulia Elhadj PDF
By Zeraoulia Elhadj
This publication is a entire selection of identified effects in regards to the Lozi map, a piecewise-affine model of the Henon map. Henon map is among the so much studied examples in dynamical structures and it draws loads of cognizance from researchers, but it is tough to investigate analytically. easier constitution of the Lozi map makes it enhanced for such research. The booklet is not just an outstanding creation to the Lozi map and its generalizations, it additionally summarizes of vital suggestions in dynamical platforms conception equivalent to hyperbolicity, SRB measures, attractor forms, and extra.
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1978), Sinai (1979), Devaney, Plykin (1984–1989), Shilnikov (1993), Ott (1993), Katok & Hasselblatt (1999), Hunt (2000), Kuznetsov, (2001), Anishchenko, et al. (2003), Hunt & MacKay (2003), Belykh, et al. (2005)]. 49) Hence, these attractors are closest in their structure and properties to robust hyperbolic attractors. 49) is called some times quasi-hyperbolic [Afraimovich, et al. (1977), Mischaikow & Mrozek (1995–1998)]. On the other hand, Lorenz-type attractors were considered as examples of truly strange attractors [Shil’nikov (1980), Williams (1977), Cook & Roberts (1970)] and there are a finite number of these attractors in the literature.
The proof of Theorem 4 is based on the analysis of the one-dimensional map x 1 − ax2 using a very long treatment. This proof is based on a modified version of the proof given in [Benedicks & Carleson (1985)] of Jakobson’s Theorem. In order to obtain the chaotic behavior with simplified proof, the authors replace a property called basic assumption (BA) by a simple rule and similarly in the so called binding condition (BC). The main result using these assumptions is that the derivatives grow exponentially for a model case, and then they prove that the Hénon map behave in an analogous way.
Theorem 2 ⎛ 1 1 ⎞ implies that this map is chaotic if |a| > max ⎜ , ⎟ and |b|> max ⎝ | m1 | | m0 | ⎠ ⎞ ⎛ | am | | am0 | 1 ⎜ ⎟. 6 Ergodic theory The ergodic theory was motivated by problems of statistical physics. The most important results in ergodic theory are the ergodic theorems of Birkhoff and von Neumann. Its central aspect is the investigation of the behavior of a dynamical system when it is allowed to run for a long period of time taking into account that the time average is the same for almost all initial points and that ergodic systems has stronger properties, such as mixing and equidistribution.