By B. Aulbach

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**Example text**

In a given coordinate system). If we change the parameterization,the dynamical equations should be modified accordingly so that the same physical states are connected by the evolution laws. , zn+l = f(x,)]. 17) simply expresses that on the one hand, yn+l = h(x,+l) = h ( f ( z n ) )and , on the other hand, yn+l = g(yn) = g(h(xn)). 17) is recovered by comparing the two paths from xn to yn+l. , see [38]). Conjugacy, or smooth conjugacy, corresponds to the case where h is a diffeomorphism. If h is a homeomorphism, one has topological conjugacy.

For higher periods, (1) the periodic window is too narrow compared to the scale of the plot, and (2) the number of samples is sufficiently large that the window cannot be distinguished from the chaotic regimes. Ideally, we would like to determine for each periodic solution the range of parameter values over which it is stable. 9 2 a Fig. 4 Enlarged view of the chaotic zone of the bifurcation diagram of Fig. 3. Inside periodic windows of period up to 8, vertical lines indicate the parameter values where the corresponding orbits are most stable, with the period indicated above the line.

It can be shown that Lyapunov exponents are independent of the initial condition 50, except perhaps for a set of measure zero [39]. Since the distance between infinitesimally close states grows exponentially as 6xn enX6xo,sensitivity to initial conditions is associated with a strictly positive Lyapunov exponent. 25) is X = In 2. 3 Sensitivity to Initial Conditionsand Mixing Sensitivityto initial conditions can also be expressedin a way that is more topological, without using distances. 29)]. 3 1).