By Marco Thiel, Jürgen Kurths, M. Carmen Romano, György Károlyi, Alessandro Moura

This ebook is a set of contributions on a number of facets of lively frontier study within the box of dynamical platforms and chaos.

Each bankruptcy examines a particular learn subject and, as well as reviewing fresh effects, additionally discusses destiny views.

The result's a useful photograph of the nation of the sector through a few of its most vital researchers.

The first contribution during this booklet, "How did you get into Chaos?", is absolutely a set of private bills by way of a couple of special scientists on how they entered the sector of chaos and dynamical platforms, that includes reviews and reminiscences through James Yorke, Harry Swinney, Floris Takens, Peter Grassberger, Edward Ott, Lou Pecora, Itamar Procaccia, Michael Berry, Giulio Casati, Valentin Afraimovich, Robert MacKay, and final yet no longer least, Celso Grebogi, to whom this quantity is devoted.

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**Example text**

As we saw above, when n > 2 and |λ| is small enough, λ lies in the McMullen domain and so the corresponding Julia sets are always Cantor sets of concentric simple closed curves. But when n = 2, the Julia sets vary wildly; often, but not always, they are Sierpinski curves. For example, in Fig. 5, note that there is a copy of the Mandelbrot set whose “tail” actually extends to the origin. Whenever λ is chosen in this set, J(Fλ ) contains small pieces that are homeomorphic to the corresponding Julia set from the quadratic family z2 + c together with infinitely many preimages of ∂Bλ (as well as other buried points).

4c. The vertex V m represents the cluster state Pm and vertices are connected by edges when there are heteroclinic connections between the corresponding cluster states. Observe that there exist switching dynamics between cluster states but the network structure is trivial: the heteroclinic network splits into two disconnected components. For N ≥ 5 the network becomes connected and much more complex as will be shown in the next section. 4 Heteroclinic Networks for Odd Numbers of Oscillators Here we consider some of the scaling properties of the robust heteroclinic networks discussed in the previous section, for more larger numbers of oscillators.

Here the situation involves the structure of the Julia sets of Fλ when λ is close to 0. As we saw above, when n > 2 and |λ| is small enough, λ lies in the McMullen domain and so the corresponding Julia sets are always Cantor sets of concentric simple closed curves. But when n = 2, the Julia sets vary wildly; often, but not always, they are Sierpinski curves. For example, in Fig. 5, note that there is a copy of the Mandelbrot set whose “tail” actually extends to the origin. Whenever λ is chosen in this set, J(Fλ ) contains small pieces that are homeomorphic to the corresponding Julia set from the quadratic family z2 + c together with infinitely many preimages of ∂Bλ (as well as other buried points).