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On the Statistical Stability of Families of Attracting Sets and the Contracting Lorenz Attractor

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Abstract

We present criteria for statistical stability of attracting sets for vector fields using dynamical conditions on the corresponding generated flows. These conditions are easily verified for all singular-hyperbolic attracting sets of \(C^2\) vector fields using known results, providing robust examples of statistically stable singular attracting sets (encompassing in particular the Lorenz and geometrical Lorenz attractors). These conditions are shown to hold also on the persistent but non-robust family of contracting Lorenz flows (also known as Rovella attractors), providing examples of statistical stability among members of non-open families of dynamical systems. In both instances, our conditions avoid the use of detailed information about perturbations of the one-dimensional induced dynamics on specially chosen Poincaré sections.

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Notes

  1. We write \(A+B\) the union of the disjoint subsets A and B.

  2. We write \(f(x)=O(g(x))\) at \(x=x_0\) if there exists \(M,\delta \) such that \(|f(x)|\le M |g(x)|\) when \(0<|x-x_0|<\delta \).

  3. This technical condition was strongly used to derive the stated results; see [42, Remarks, p. 240].

  4. This depends only on the neighborhood \({\mathcal V}\) through d.

  5. This follows from the invariance of the stable manifolds of all points in U together with the closeness of \(y_1\) and \(x_1\), together with the value of d in the neighborbood \({\mathcal V}\).

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Acknowledgements

I thank the Mathematics Department at UFBA; CAPES-Brazil and CNPq-Brazil for the basic support of research activities; and also the anonymous referee for many suggestions that helped to improve the text.

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  1. Instituto de Matemática e Estatística, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110, Salvador, Brazil

    Vitor Araujo

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Correspondence to Vitor Araujo.

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Communicated by Eric A. Carlen.

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The author was partially supported by CNPq-Brazil (Grant 300985/2019-3).

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Araujo, V. On the Statistical Stability of Families of Attracting Sets and the Contracting Lorenz Attractor. J Stat Phys 182, 53 (2021). https://doi.org/10.1007/s10955-021-02729-x

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