https://doi.org/10.1140/epjs/s11734-024-01340-x
Regular Article
Deep learning-based stability of quasi-integrable and non-resonant Hamiltonian systems driven by fractional Gaussian noise
Department of Mechanics, Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, Zhejiang University, 310027, Hangzhou, China
Received:
15
May
2024
Accepted:
16
September
2024
Published online:
2
October
2024
The study of the stability of dynamic systems excited by fractional Gaussian noise (fGn) is very difficult since the response of the system is not a Markov process, which results in the classical diffusion process method not being applicable and very low efficiency of numerical simulation. In the present paper, a procedure based on deep learning for calculating the largest Lyapunov exponent to determine the asymptotic Lyapunov stability with probability one of quasi-integrable and non-resonant Hamiltonian systems under parametric excitations of fGn is first proposed. First, fGn is regarded approximately as a wide-band process. Next, the original system is approximated as averaged SDEs with fewer dimensions using stochastic averaging method, where the drift and diffusion coefficients of the averaged SDEs were obtained using BPNN (back propagation neural network). Then, the expression for the largest Lyapunov exponent of the averaged SDEs is obtained by generalizing Khasminskii's procedure and the stochastic stability of the original systems is determined approximately. Two example of MODF nonlinear system is carried out to illustrate the proposed procedure. The results are compared with those from simulation of original system. The comparison shows the effectiveness of the proposed procedure.
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© The Author(s), under exclusive licence to EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2024. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.