doi:10.1016/S0960-0779(00)00192-2 
Copyright © 2001 Elsevier Science Ltd. All rights reserved.
Dynamics of a Hénon–Lozi-type map
M. A. Aziz-Alaoui
, 
, a, Carl Robertb and Celso Grebogic
a Département de mathématiques, L. M., Fac. Sc. Tech., BP 540, 76058 Le Havre Cedex, France
b Department of Physics, University of California, Santa Barbara, CA 93106, USA
c Institute for Plasma Research, Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MA 20742, USA
Accepted 4 August 2000
Available online 5 July 2001.
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Abstract
We present and analyze a smooth version of the piecewise linear Lozi map. The principal motivation for this work is to develop a map, which is better amenable for an analytical treatment as compared to the Hénon map and is one that still possesses the characteristics of a Hénon-type dynamics. This paper is a first step. It does the comparison of the Lozi map (which is a piecewise linear version of the Hénon map) with the map that we introduce. This comparison is done for fixed parameters and also through global bifurcation by changing a parameter. If 
measures the degree of smoothness, we prove that, as → 0, the stability and the existence of the fixed points are the same for both maps. We also numerically compare the chaotic dynamics, both in the form of an attractor and of a chaotic saddle.
Fig. 2. Comparison of the Lozi and mappings: (a) fixed points of the Lozi mapping; (b) fixed points of the mapping.
Fig. 3. (a) Stability of the fixed points of the Lozi map. (b) Stability of the fixed points of .
Fig. 4. Simple global behavior of .
Fig. 5. Appearance of the attractor: the transformation is applied to a square (digitized by an array of 100 by 100 regularly spaced points): (a) 1; (b) 2; (c) 3; (d) 4; (e) 5; (f) 6; (g) 7; (h) 8; (i) 12 times with the parameters =0.05, a=1.7 and b=0.5.
Fig. 6. Enlargement of a strange attractor for =0.01, with a=1.7 and b=0.5. (a–c) show a zoom in the linear region, and (d–f) show a zoom in the nonlinear region. This figure, along with Fig. 7 and Fig. 11 have been done with the help of the software DYNAMICS [14].
Fig. 7. Bifurcation diagrams for: (a) Lozi map; (b) with =0.1, a=1.7 and b=0.5; (c) zoom in (a), there is no period-doubling route to chaos; (d) zoom in (b), there is period-doubling route to chaos.
Fig. 8. The attractors for transformation, with a=1.7, b=0.5. is given in each figure.
Fig. 9. Lyapunov exponent of mapping with a=1.7, b=0.5 and for: (a) =0.1; (b) =0.01. Values of the Lyapunov exponent h greater than 0 show evidence of chaos.
Fig. 10. Boundary crisis curve. Crossing this curve in one direction (from left to right) leads to the sudden disappearance of a chaotic attractor. Crossing the curve in the opposite direction leads to the sudden creation of a chaotic attractor.
Fig. 11. Comparison of (a) and (b) before (a=1.7, b=0.5) and after (a=2.2, b=0.5) the boundary crisis. Before crisis: chaotic attractor with its basin of attraction, see (a1) and (b1). After crisis: only a chaotic saddle remains, see (a2) and (b2). Chaotic saddle with part of the stable manifold of the fixed point to show how the gaps are formed in the saddle, see (a3) and (b3). Note: only part of the stable manifold is being computed in order to help visualizing the mechanism.
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