Eden's hypothesis for a Lorenz system
Let \(K\) denote the attractor of the Lorenz system \[ \dot x= -\sigma x+ \sigma y,\quad \dot y= rx- y- xz,\quad \dot z= - bz+ xy.\tag{1} \] \textit{A. Eden} [RAIRO, Modelisation Math. Anal. Numer. 23, No. 3, 405-413 (1989; Zbl 0684.58022)] suggested that for the Hausdorff dimension of \(K\) the estimate \(\dim K\leq 2- (\lambda_1(0)+ \lambda_0(0))/\lambda_3(0)\) holds where \(\lambda_1(0)\geq \lambda_2(0)\geq \lambda_3(0)\) are the eigenvalues of the symmetrized Jacobian matrix \({1\over 2}(\partial f(u)/\partial u+ (\partial f(u)/\partial u)^*)\) of the right-hand side \(f(u)\) of system (1) when \(u= 0\). The authors prove the estimate \[ \dim K\leq 3- {2(\sigma+ b+ 1)\over \sqrt{(\sigma- 1)^2+ 4r\sigma+ \sigma+1}} \] which implies that Eden's hypothesis is valid for a broad range of parameters \(r\), \(b\), \(\sigma\). The proof is based on results from the first author's paper [Vestn. Leningr. Univ., Math. 24, No. 4, 9-11 (1991); translation from Vestn. Leningr. Univ., Ser. I 1991, No. 4, 11-14 (1991; Zbl 0752.34030)].
scientific article; zbMATH DE number 815608