An operator methodology for the global dynamic analysis of stochastic nonlinear systems

Benedetti, Kaio C. B.; Gonçalves, Paulo B.; Lenci, Stefano; Rega, Giuseppe

doi:10.1016/j.taml.2022.100419

Cited by 11 publications

(6 citation statements)

References 27 publications

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“…The adaptative discretization reduces the analysis' computational cost without losing precision in these important regions. This economy is expected to be greater if the important regions are localized in small phase-space areas, as showed in [45,46].…”

Section: Deterministic Global Analysismentioning

confidence: 95%

“…As an example, figure 10 shows a phase-space final discretization by applying the adaptative algorithm described in [45,46]. Some regions, particularly attractors and basins boundaries, after several interactions, have a much higher boxresolution.…”

Section: Deterministic Global Analysismentioning

confidence: 99%

“…The framework for the analysis of nondeterministic dynamics though an operator approach has been addressed recently in [45,46], see also appendix B. There, pairs of dual global operators, each pair constituted by a transfer and a composition operator, are employed for each type of dynamical system.…”

Section: Nondeterministic Global Analysismentioning

confidence: 99%

“…In the framework of an operator approach for the analysis of global dynamics, a concise description of the construction of the discretized operators is summarized here, along with the information given by them. Detailed expositions can be found in [45,46]. Initially, a phase-space window of interest X is partitioned into a finite collection of n disjoint boxes B = {b 1 , .…”

Section: Appendix B Discretized Transfer Operator Approach For Nondet...mentioning

confidence: 99%

“…Basins of attraction were highly affected for large uncertainty levels, and the computational cost increased significantly with them. To reduce this cost, adaptative phase-space discretization techniques, designed for nondeterministic global dynamic analysis, are an interesting field of investigation [45]. Some recent results quantified the effects of random parameters and noise on the basins, attractors, and dynamic integrity of nonlinear one degree of freedom (dof) oscillators [46], obtained by applying an operator perspective for the nondeterministic analysis.…”

Section: Introductionmentioning

confidence: 99%

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Parameter uncertainty and noise effects on the global dynamics of an electrically actuated microarch

Benedetti

Gonçalves

Lenci

et al. 2023

J. Micromech. Microeng.

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This work aims to study the effect of uncertainties and noise on the nonlinear global dynamics of a micro-electro-mechanical arch obtained from an imperfect microbeam under an axial load and electric excitation. An adaptative phase-space discretization strategy based on an operator approach is proposed. The Ulam method, a classical discretization of flows in phase-space, is extended here to nondeterministic cases. A unified description is formulated based on the Perron-Frobenius, Koopman, and Foias linear operators. Also, a procedure to obtain global structures in the mean sense of systems with parametric uncertainties is presented. The stochastic basins of attraction and attractors’ distributions replace the usual basin and attractor concepts. For parameter uncertainty cases, the phase-space is augmented with the corresponding probability space. The microarch is assumed to be shallow and modelled using a nonlinear Bernoulli-Euler beam theory and is discretized by the Galerkin method using as interpolating function the linear vibration modes. Then, from the discretized multi degree of freedom model (mdof) model, an accurate single degree of freedom (sdof) reduced order model, based on theory of nonlinear normal modes, is derived. Several competing attractors are observed, leading to different (acceptable or unacceptable) behaviours. Extensive numerical simulations are performed to investigate the effect of noise and uncertainties on the coexisting basins of attraction, attractors` distributions, and basins boundaries. The appearance and disappearance of attractors and stochastic bifurcation are observed, and the time-dependency of stochastic responses is demonstrated, with long-transients influencing global behaviour. To consider uncertainties and noise in design, a dynamic integrity measure is proposed via curves of constant probability, which give quantitative information about the changes in structural safety. For each attractor, the basin robustness as a function of a stochastic parameter is investigated. The weighted basin area can quantify the integrity of nondeterministic cases, being also the most natural generalization of the global integrity measure. While referring to particular MEMS, the relevance of the dynamical integrity analysis for stochastic systems to quantify tolerances and safety margins is underlined here.

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Section: Deterministic Global Analysismentioning

confidence: 95%

Section: Deterministic Global Analysismentioning

confidence: 99%

Section: Nondeterministic Global Analysismentioning

confidence: 99%

Section: Appendix B Discretized Transfer Operator Approach For Nondet...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Parameter uncertainty and noise effects on the global dynamics of an electrically actuated microarch

Benedetti

Gonçalves

Lenci

et al. 2023

J. Micromech. Microeng.

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Global Nonlinear Dynamics: Challenges in the Analysis and Safety of Deterministic or Stochastic Systems

Rega

2024

Exploiting the Use of Strong Nonlinearity in Dynamics and Acoustics

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Koopman-Based Spectral Clustering of Directed and Time-Evolving Graphs

Klus

Conrad

2022

J Nonlinear Sci

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While spectral clustering algorithms for undirected graphs are well established and have been successfully applied to unsupervised machine learning problems ranging from image segmentation and genome sequencing to signal processing and social network analysis, clustering directed graphs remains notoriously difficult. Two of the main challenges are that the eigenvalues and eigenvectors of graph Laplacians associated with directed graphs are in general complex-valued and that there is no universally accepted definition of clusters in directed graphs. We first exploit relationships between the graph Laplacian and transfer operators and in particular between clusters in undirected graphs and metastable sets in stochastic dynamical systems and then use a generalization of the notion of metastability to derive clustering algorithms for directed and time-evolving graphs. The resulting clusters can be interpreted as coherent sets, which play an important role in the analysis of transport and mixing processes in fluid flows. Graphic Abstract

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An operator methodology for the global dynamic analysis of stochastic nonlinear systems

Cited by 11 publications

References 27 publications

Parameter uncertainty and noise effects on the global dynamics of an electrically actuated microarch

Parameter uncertainty and noise effects on the global dynamics of an electrically actuated microarch

Global Nonlinear Dynamics: Challenges in the Analysis and Safety of Deterministic or Stochastic Systems

Koopman-Based Spectral Clustering of Directed and Time-Evolving Graphs

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