Suppose that a dynamical system possesses an invariant submanifold, and the restriction of the system to this submanifold has a chaotic attractor A. Under which conditions is A an attractor for the original system, and in what sense?We characterize the transverse dynamics near A in terms of the normal Liapunov spectrum of A. In particular, we emphasize the role of invariant measures on A. Our results identify the points at which A: (1) ceases to be asymptotically stable, possibly developing a locally riddled basin; (2) ceases to be an attractor; (3) becomes a transversely repelling chaotic saddle. We show, in the context of what we call 'normal parameters' how these transitions can be viewed as being robust. Finally, we discuss some numerical examples displaying these transitions.

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In this paper, we show that there are Initial Value Problems defined with polynomial ordinary differential equations that can simulate universal Turing machines in the presence of bounded noise. The polynomial ODE defining the IVP is explicitly obtained and the simulation is performed in real time.

Abstract. In this paper, we show that closed-form analytic maps and flows can simulate Turing machines in an error-robust manner. The maps and ODEs defining the flows are explicitly obtained and the simulation is performed in real time.

We study positive integral operators K in L 2 (R) with continuous kernel k(x, y). We show that if k(x, x) ∈ L 1 (R) the operator is compact and Hilbert-Schmidt. If in addition k(x, x) → 0 as |x| → ∞, k is represented by an absolutely and uniformly convergent bilinear series of uniformly continuous eigenfunctions and K is trace class. Replacing the first assumption by the stronger k 1/2 (x, x) ∈ L 1 (R) then k ∈ L 1 (R 2 ) and the bilinear series converges also in L 1 . Sharp norm bounds are obtained and Mercer's theorem is derived as a special case.