Estimating of the inertial manifold dimension for a chaotic attractor of complex Ginzburg-Landau equation using a neural network

Pavel V. Kuptsov; Anna V. Kuptsova

doi:10.1117/12.2523235

3 June 2019 Estimating of the inertial manifold dimension for a chaotic attractor of complex Ginzburg-Landau equation using a neural network

Pavel V. Kuptsov, Anna V. Kuptsova

Author Affiliations +

Proceedings Volume 11067, Saratov Fall Meeting 2018: Computations and Data Analysis: from Nanoscale Tools to Brain Functions; 110670N (2019) https://doi.org/10.1117/12.2523235
Event: International Symposium on Optics and Biophotonics VI: Saratov Fall Meeting 2018, 2018, Saratov, Russian Federation

Abstract

Dimension of an inertial manifold for a chaotic attractor of spatially distributed system is estimated using autoencoder neural network. The inertial manifold is a low dimensional manifold where the chaotic attractor is embedded. The autoencoder maps system state vectors onto themselves letting them pass through an inner state with a reduced dimension. The training processes of the autoencoder is shown to depend dramatically on the reduced dimension: a learning curve saturates when the dimension is too small and decays if it is sufficient for a lossless information transfer. The smallest sufficient value is considered as a dimension of the inertial manifold, and the autoencoder implements a mapping onto the inertial manifold and back. The correctness of the computed dimension is confirmed by its remarkable coincidence with the one obtained as a number of covariant Lyapunov vectors with vanishing pairwise angles. These vectors are called physical modes. Unlike never having zero angles residual ones they are known to span a tangent subspace for the inertial manifold.

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Pavel V. Kuptsov and Anna V. Kuptsova "Estimating of the inertial manifold dimension for a chaotic attractor of complex Ginzburg-Landau equation using a neural network", Proc. SPIE 11067, Saratov Fall Meeting 2018: Computations and Data Analysis: from Nanoscale Tools to Brain Functions, 110670N (3 June 2019); https://doi.org/10.1117/12.2523235

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