## Algebra und Geometrie 2. Moduln und Algebren by von Oniscik A.L., Sulanke R.

By von Oniscik A.L., Sulanke R.

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Extra resources for Algebra und Geometrie 2. Moduln und Algebren

Example text

The points in these horizontal strips come from vertical strips in the original square. Let S0 be the original square, map it forward n times, and consider only the points that fall back into the square S0 , which is a set of horizontal stripes Hn = f n (S0 ) ∩ S0 . The points in the horizontal stripes came from the vertical stripes Vn = f −n (Hn ), which are the horizontal strips Hn mapped backwards n times. 19). Fig. 19. Iterated horseshoe map: pre–images of the square region. Now, if a point is to remain indeﬁnitely in the square, then it must belong to an invariant set Λ that maps to itself.

18 1 Basics of Nonlinear and Chaotic Dynamics Fig. 10. Examples of regular attractors: ﬁxed–point (left) and limit cycle (right). Note that limit cycles exist only in nonlinear dynamics. After the seminal works of Poincar´e, Lorenz, Smale, May, and H´enon (to cite only the most eminent ones) it is now well established that the so called chaotic behavior is ubiquitous. 7) z˙ = xy − bz This system is related to the Rayleigh–B´enard convection under very crude approximations. 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.

As the driving increases even more, the so–called fractal–ﬁngers created by the homoclinic tangling, make a sudden incursion into the safe basin. At that point, the integrity of the in–well motions is lost [TS01]. 24), let X be the point of intersection, with X ahead of X on one manifold and ahead of X 42 1 Basics of Nonlinear and Chaotic Dynamics Fig. 22. Motion of a damped particle in a potential well, driven by a periodic force F cos(wt),. Up: potential (x − V )−plot, with V = x2 /2 − x3 /3; down: the corresponding phase (x − x)−portrait, ˙ showing the safe basin of attraction – if the driving is switched oﬀ (F = 0).