By von Oniscik A.L., Sulanke R.

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**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).