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Extra resources for Concepts and Results in Chaotic Dynamics: A Short Course (Theoretical and Mathematical Physics)

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In other words, if we consider a ball B of radius r > 0 around the origin, the only points whose trajectory stays forever in B 48 4 Hyperbolicity W s (x0 ) W u (x0 ) x0 Fig. 1. The hyperbolic fixed point x0 for a linear map f , with its stable and unstable manifolds W s (x0 ) and W u (x0 ) (which are linear) and a few points of four orbits of f are those on the x(2) axis. In addition, these are also the only points whose orbit converges to the origin. Therefore, for our example, the x(2) axis is called the stable manifold of the fixed point.

2), and I0 , I1 as before, but now for the rotation map, we find that Nn = 2(n + 1), which is much smaller than in the preceding example. We call this the “Swiss Chocolate Theorem”: Swiss chocolate has 24 pieces, and to get them all you must break the tablet 23 times, since each break produces exactly one new piece, no matter in which order you break. For the case of the map f , since the preimages are unique, and the boundaries of the Qn ’s are those points which land on either 0(= 1) or 12 , we see that there are exactly 2(n + 1) of them (as each preimage “breaks” exactly one of the intervals which already exist, and we start with 2 pieces).

One of the most studied examples is the so-called Lotka–Volterra system, which describes the time evolution of the concentration of a predator–prey system. The phase space is R 2+ and the flow of the simplest (normalized) model is given by dx = x(1 − y) , dt dy = y(x − 1) . dt This model shows the famous oscillation for the concentration of predators (y) and preys (x). We refer to (Volterra 1990; Murray 2002; 2003) for more references and examples. 37. Partial Differential Equations The time evolution of many physical systems can be described by a nonlinear partial differential equation (PDE).

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