Jorge Buescu

Bubbling of attractors and synchronisation of chaotic oscillators

We present a system of two coupled identical chaotic electronic circuits that exhibit a blowout bifurcation resulting in loss of stability of the synchronised state. We introduce the concept of bubbling of an attractor, a new type of intermittency that is triggered by low levels of noise, and demonstrate numerical and experimental examples of this behaviour. In particular we observe bubbling near the synchronised state of two coupled chaotic oscillators. We give a theoretical description of the behaviour associated with locally riddled basins, emphasising the role of invariant measures. In general these are non-unique for a given chaotic attractor, which gives rise to a spectrum of Lyapunov exponents. The behaviour of the attractor depends on the whole spectrum. In particular, bubbling is associated with the loss of stability of an attractor in a dynamically invariant subspace, and is typical in such systems.

Liapunov stability and adding machines

In Chapter 1 we discussed several notions of stability for compact invariant sets of dynamical systems. Here we shall prove that, under very general hypotheses, the set of connected components of a stable set of a discrete dynamical system possesses a tightly constrained structure. More precisely, suppose that X is a locally compact, locally connected metric space, f: X → X is a continuous mapping (not necessarily invertible) and A is a compact transitive set. Let

Robust simulations of turing machines with analytic maps and flows

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L/sup 2/(R) nonstationary processes and the sampling theorem

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Computational bounds on polynomial differential equations

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Computability, noncomputability, and hyperbolic systems

K be the set of connected components of A and let : K → K be the map induced by f We proved in § 1.3 that either K is finite or a Cantor set; in either case f acts transitively on K. Our main result (Theorem 2.3.1 below) is that, if A is Liapunov stable and has infinitely many connected components, then

Liapunov stability and adding machines revisited

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Inequalities for differentiable reproducing kernels and an application to positive integral operators

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Computability and Dynamical Systems

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A minimum distance: arithmetic and harmonic means in a geometric dispute

acts on K as a ‘generalized adding machine’, which we describe in a moment. We remark that imposing the stronger condition of asymptotic stability destroys the Cantor structure altogether and K must be finite — which is the content of Theorem 1.4.6. Thus adding machines can be Liapunov stable but never asymptotically stable. This Theorem may be strengthened to a version that does not require transitivity but the weaker property of being a stable ω-limit set.