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Birkhoﬀ had found a ‘remarkable curve’ in a 2D map; it appeared to be non–diﬀerentiable and it would be nice to see if a smooth ﬂow could generate such a curve. The work of Cartwright and Littlewood lead to the work of N. Levinson, which in turn provided the basis for the horseshoe construction of Steve Smale. In Russia, Aleksandr M. Lyapunov paralleled the methods of Poincar´e and initiated the strong Russian dynamical systems school. A. Andronov 11 carried on with the study of nonlinear oscillators and in 1937 introduced together with Lev S.

In this notation, the intersection of V2 and H2 consists of 16 squares, one of which is ΛAB•BB = f 2 (AB) ∩ BB. All the points in ΛAB•BB are in B and will continue to be in B for at least one more iteration. Their previous trajectory before landing in BB was A followed by B. Any one of the intersections ΛP •F of a horizontal strip with a vertical strip, where P and F are sequences of As and Bs, is an aﬃne transformation of a small region in V1 . If P has k symbols in it, and if f −k (ΛP •F ) and ΛP •F intersect, then the region ΛP •F will have a ﬁxed–point.

The quantity x is proportional the circulatory ﬂuid particle velocity; the quantities y and z are related to the temperature proﬁle; σ, b and r are dimensionless parameters. Lorenz studied the case with σ = 10 and b = 8/3 at varying r (which is proportional to the Rayleigh number). It is easy to see by linear analysis that the ﬁxed–point (0, 0, 0) is stable for r < 1. 74. 11. In this ‘strange’, chaotic regime one has the so called sensitive dependence on initial conditions. Consider two trajectories, x(t) and x (t), initially very close and denote with Δ(t) = ||x (t)−x(t)|| their separation.