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Robust Approximation Algorithms for the Detection of Attraction Basins in Dynamical Systems

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Abstract

A particular solution of a dynamical system is completely determined by its initial condition. When the omega limit set reduces to a point, the solution settles at steady state. The possible steady states of the system are completely determined by its parameters. However, with the same parameter set, it is possible that several steady states can originate from different initial conditions (multi-stability). In that case the outcome depends on the chosen initial condition. Therefore, it is important to assess the domain of attraction for each possible attractor. The algorithms presented here are general and robust enough so as to solve the problem of reconstructing the basin of attraction of each stable equilibrium point. In order to have a graphical representation of the separatrix manifolds, we focus on systems of two and three ordinary differential equations exhibiting bi- or tri-stability. For this purpose we have implemented several Matlab functions for the approximation of the points lying on the curves or on the surfaces determining the basins of attraction and for the reconstruction of such curves and surfaces. We approximate the latter with the implicit partition of unity method using radial basis functions as local approximants. Numerical results, obtained with a Matlab package made available to the scientific community, support our findings.

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Notes

  1. In case of bistability the manifold through the origin and a saddle point partitions the phase space into two regions. In case of a system with three equilibria instead, more saddles are involved in the dynamics. But the three separating manifolds all intersect only at one saddle with all nonnegative populations.

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Acknowledgments

The authors sincerely thank the two anonymous referees for helping to significantly improve our paper. This work was supported by the University of Turin via grant “Metodi numerici nelle scienze applicate”.

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Authors and Affiliations

  1. Department of Mathematics “G. Peano”, University of Turin, via Carlo Alberto 10, 10123, Turin, Italy

    Roberto Cavoretto, Alessandra De Rossi, Emma Perracchione & Ezio Venturino

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  1. Roberto Cavoretto
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  2. Alessandra De Rossi
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  3. Emma Perracchione
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  4. Ezio Venturino
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Correspondence to Roberto Cavoretto.

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Cavoretto, R., De Rossi, A., Perracchione, E. et al. Robust Approximation Algorithms for the Detection of Attraction Basins in Dynamical Systems. J Sci Comput 68, 395–415 (2016). https://doi.org/10.1007/s10915-015-0143-z

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  • DOI: https://doi.org/10.1007/s10915-015-0143-z

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