Lower bounds for the Hausdorff dimension of attractors (original) (raw)

Abstract

The existence of certain m-dimensional structures in a dynamical system implies that the Hamdorff dimension of its attractor is at least m + 1. A Bendixson criterion for the nonexistence of periodic orbits for systems in Hilbert spaces is found.

Figures (3)

and the Poincaré-Bohl theorem (Lloyd, 1978, Theorem 2.1.5), imply deg(Rg, U, p)=deg(Rgo, U, p)= £1, if pe Ng(vo), so that Ns(v9) < p(U) as asserted. Thus N,(v,) ¢ 2y( 0) < U,AB,, if p(U) < U;B,, and, since BB, is contained in a ball of radius 2 ||ZB,||,

and the Poincaré-Bohl theorem (Lloyd, 1978, Theorem 2.1.5), imply deg(Rg, U, p)=deg(Rgo, U, p)= £1, if pe Ng(vo), so that Ns(v9) < p(U) as asserted. Thus N,(v,) ¢ 2y( 0) < U,AB,, if p(U) < U;B,, and, since BB, is contained in a ball of radius 2 ||ZB,||,

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