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Six-dimensional basins of attraction computation on small clusters with semi-parallelized SCM method

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Abstract

In modern engineering applications it is needed to determine the robustness of stable steady states, which depends on the shape and size of the associated basins of attraction. Once the basins are known, the quantitative measures can be computed by means of dynamical integrity tools and arguments. Those numerical techniques applied to strongly nonlinear systems, with six of more phase-space variables, demand considerable amounts of computing power, available only on High Performance Computing platforms. With the aim to minimize utilization of computer resources, we developed a software to adapt basin computations to small, affordable, clusters. It is based on Simple Cell Mapping method, modified to reduce the memory load, adjust integration time and overcome discretization discontinuities. Resource intensive part of computations is parallelized with Message Passing Interface and less demanding operations are kept serial, due to their inherit sequential nature. As intended, the program computes full six-dimensional basins of attraction at the adequate accuracy to distinguish compact parts of basins, but not to fully disclose fractalities or chaos. Disadvantages of the SCM method are addressed and the effectiveness of proposed solutions are demonstrated and discussed by the paradigmatic example composed of three coupled Duffing oscillators.

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Abbreviations

2D:

Two-dimensional

3D:

Three-dimensional

6D:

Six-dimensional

BoA:

Basins of attraction

CM:

Cell mapping

CPA:

Connecting post-processing algorithm

CPU:

Central processing unit

CSCM:

Clustered Simple Cell Mapping

DOF:

Degree of Freedom

FCVDP:

Forced coupled Van der Pol oscillators

FDO:

Forced Duffing oscillators

GB:

Giga-byte

GoS:

Grid of Starts

GPU:

Graphical processing unit

HPC:

High Performance Computing

MPI:

Message Passing Interface standard

NSSCM:

Not So Simple Cell Mapping

OpenMP:

Open multi-processing application programming interface

PSCM:

Parallel Simple Cell Mapping

RAM:

Random access memory

SCM:

Simple cell mapping

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Acknowledgements

Authors would like to thank to Franco Moglie, Polytechnic University of Marche, Ancona, Italy, Radu Serban and Dan Negrut, University of Wisconsin-Madison, USA, for help with HPC.

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Authors and Affiliations

  1. Polytechnic University of Marche, Ancona, Italy

    Nemanja Andonovski & Stefano Lenci

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  1. Nemanja Andonovski
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  2. Stefano Lenci
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Correspondence to Nemanja Andonovski.

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Andonovski, N., Lenci, S. Six-dimensional basins of attraction computation on small clusters with semi-parallelized SCM method. Int. J. Dynam. Control 8, 436–447 (2020). https://doi.org/10.1007/s40435-019-00557-2

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