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


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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Basins of attraction


Cell mapping


Connecting post-processing algorithm


Central processing unit


Clustered Simple Cell Mapping


Degree of Freedom


Forced coupled Van der Pol oscillators


Forced Duffing oscillators




Grid of Starts


Graphical processing unit


High Performance Computing


Message Passing Interface standard


Not So Simple Cell Mapping


Open multi-processing application programming interface


Parallel Simple Cell Mapping


Random access memory


Simple cell mapping


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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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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).

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  • Basin of attraction
  • Global analysis
  • Simple Cell Mapping