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Numerical Study of Discrete Lorenz-Like Attractors

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Abstract

We consider a homotopic to the identity family of maps, obtained as a discretization of the Lorenz system, such that the dynamics of the last is recovered as a limit dynamics when the discretization parameter tends to zero. We investigate the structure of the discrete Lorenz-like attractors that the map shows for different values of parameters. In particular, we check the pseudohyperbolicity of the observed discrete attractors and show how to use interpolating vector fields to compute kneading diagrams for near-identity maps. For larger discretization parameter values, the map exhibits what appears to be genuinely-discrete Lorenz-like attractors, that is, discrete chaotic pseudohyperbolic attractors with a negative second Lyapunov exponent. The numerical methods used are general enough to be adapted for arbitrary near-identity discrete systems with similar phase space structure.

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Notes

  1. In 3D flow systems bifurcations of homoclinic tangencies inevitably lead to the appearance of stable periodic orbits [17, 18], thus, an attractor cannot be pseudohyperbolic in principle.

  2. This condition means that any possible contraction in \(E_{1}(x)\) is uniformly weaker than any contraction in \(E_{2}(x)\), and any expansion in \(E_{1}(x)\) is uniformly stronger than any possible expansion in \(E_{2}(x)\). Note that it guarantees the persistence of the invariant families of linear subspaces \(E_{1}\) and \(E_{2}\) (and the attractor \({\cal A}\)) under \(\mathcal{C}^{1}\) perturbations of the system.

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Funding

A. V. and A. M. have been supported by the Spanish grant PID2021-125535NB-I00 (MICINN/AEI/FEDER, UE) and the Catalan grant 2021-SGR-01072. A. V. also acknowledges the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M). The work of A. K. and K. Z. (numerical verification of pseudohyperbolicity conditions) has been supported by the RSF grant 23-71-30008.

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Authors and Affiliations

  1. National Research University Higher School of Economics, ul. Bolshaya Pecherskaya 25/12, 603155, Nizhny Novgorod, Russia

    Alexey Kazakov & Kirill Zaichikov

  2. Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, 08007, Barcelona, Spain

    Ainoa Murillo & Arturo Vieiro

  3. Centre de Recerca Matematica (CRM), 08193, Bellaterra, Spain

    Arturo Vieiro

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  1. Alexey Kazakov
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Kazakov, A., Murillo, A., Vieiro, A. et al. Numerical Study of Discrete Lorenz-Like Attractors. Regul. Chaot. Dyn. 29, 78–99 (2024). https://doi.org/10.1134/S1560354724010064

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