A two-dimensional mapping with a strange attractor
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Lorenz (1963) has investigated a system of three first-order differential equations, whose solutions tend toward a “strange attractor”. We show that the same properties can be observed in a simple mapping of the plane defined by:xi+1=y i +1−ax i 2 ,yi+1=bx i . Numerical experiments are carried out fora=1.4,b=0.3. Depending on the initial point (x0,y0), the sequence of points obtained by iteration of the mapping either diverges to infinity or tends to a strange attractor, which appears to be the product of a one-dimensional manifold by a Cantor set.
KeywordsDifferential Equation Neural Network Manifold Statistical Physic Complex System
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- Birkhoff, G. D.: Trans. Amer. Math. Soc.18, 199 (1917)Google Scholar
- Engel, W.: Math. Annalen130, 11 (1955)Google Scholar
- Engel, W.: Math. Annalen136, 319 (1958)Google Scholar
- Hénon, M.: Quart. Appl. Math.27, 291 (1969)Google Scholar
- Lanford, O.: Work cited by Ruelle, 1975Google Scholar
- Lorenz, E. N.: J. atmos. Sci.20, 130 (1963)Google Scholar
- Pomeau, Y.: to appear (1976)Google Scholar
- Ruelle, D., Takens, F.: Comm. math. Phys.20, 167;23, 343 (1971)Google Scholar
- Ruelle, D.: Report at the Conference on “Quantum Dynamics Models and Mathematics” in Bielefeld, September 1975Google Scholar