Download Nonlinear Dynamics and Chaos: Advances and Perspectives by Marco Thiel, Jürgen Kurths, M. Carmen Romano, György PDF

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

This e-book 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 contemporary effects, additionally discusses destiny views.

The result's a useful picture of the nation of the sector by way of a few of its most vital researchers.

The first contribution during this publication, "How did you get into Chaos?", is de facto a suite of private money owed via a couple of distinct scientists on how they entered the sector of chaos and dynamical platforms, that includes reviews and memories by way of 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 now not least, Celso Grebogi, to whom this quantity is devoted.

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Additional info for Nonlinear Dynamics and Chaos: Advances and Perspectives

Example text

This is particularly the case if there is weak coupling between the oscillators, or more specifically if the timescale associated with the coupling is much longer than the timescale associated with relaxation onto the limit cycle [6, 10, 17]. Moreover, phase models are typically useful even far from the weak coupling limit, especially concerning their predictions for generic bifurcations and attractors. Clearly, the structure of coupling between oscillators is critical to determine what sort of dynamics is possible on a network of coupled oscillators, and there is a vast literature looking at the topology of coupling and the influence this has on the network dynamics; see for instance [1] and references therein.

8 The Case n=2 As mentioned above, the situation in the case n = 2 is very different. We no longer have a McMullen domain. Rather, the following result is shown in [12]: Theorem 6 Suppose n = 2. Then, in every neighborhood of the origin in the parameter plane, there are infinitely many disjoint open sets Oj , j = 1, 2, 3, . , containing parameters having the following properties: 1. If λ ∈ Oj , then the Julia set of Fλ is a Sierpinski curve, so that if λ ∈ Oj and μ ∈ Ok , the Julia sets of Fλ and Fμ are homeomorphic; 2.

5 The parameter plane for the family z2 + λ/z2 and a magnification centered at the origin Singular Perturbations of Complex Analytic Dynamical Systems Fλ2 (cλ ) = 4λ + 27 1 4 and so λ → Fλ2 (cλ ) is an analytic function of λ that is a homeomorphism. If −1/16 < λ < 0, then one checks easily that the critical circle is mapped strictly inside itself. It follows that J(Fλ ) is a connected set and Bλ and Tλ are disjoint. In particular, the second image of the critical point lands on the real axis and lies in the complement of Bλ in R.

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