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No Access Submitted: 31 May 2018 Accepted: 22 August 2018 Published Online: 10 September 2018

Time dependent stability margin in multistable systems

Chaos 28, 093104 (2018); https://doi.org/10.1063/1.5042310
P. Brzeski1,a), J. Kurths2,3,b), and P. Perlikowski1,c)
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ABSTRACT
We propose a novel technique to analyze multistable, non-linear dynamical systems. It enables one to characterize the evolution of a time-dependent stability margin along stable periodic orbits. By that, we are able to indicate the moments along the trajectory when the stability surplus is minimal, and even relatively small perturbation can lead to a tipping point. We explain the proposed approach using two paradigmatic dynamical systems, i.e., Rössler and Duffing oscillators. Then, the method is validated experimentally using the rig with a double pendulum excited parametrically. Both numerical and experimental results reveal significant fluctuations of sensitivity to perturbations along the considered periodic orbits. The proposed concept can be used in multiple applications including engineering, fluid dynamics, climate research, and photonics.

Acknowledgments

This work is funded by the National Science Center Poland based on the Decision No. DEC-2015/16/T/ST8/00516. We would like to thank Tomasz Kapitaniak, Juliusz Grabski, and Jerzy Wojewoda for support.

References
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  1. 1. S. Auer, F. Hellmann, M. Krause, and J. Kurths, “Stability of synchrony against local intermittent fluctuations in tree-like power grids,” Chaos Interdiscip. J. Nonlinear Sci. 27, 127003 (2017). https://doi.org/10.1063/1.5001818, Google ScholarScitation, ISI
  2. 2. S. Auer, K. Kleis, P. Schultz, J. Kurths, and F. Hellmann, “The impact of model detail on power grid resilience measures,” Eur. Phys. J. Spec. Top. 225, 609–625 (2016). https://doi.org/10.1140/epjst/e2015-50265-9, Google ScholarCrossref
  3. 3. P. Belardinelli and S. Lenci, “A first parallel programming approach in basins of attraction computation,” Int. J. Non Linear Mech. 80, 76–81 (2016). https://doi.org/10.1016/j.ijnonlinmec.2015.10.016, Google ScholarCrossref
  4. 4. P. Brzeski, M. Lazarek, T. Kapitaniak, J. Kurths, and P. Perlikowski, “Basin stability approach for quantifying responses of multistable systems with parameters mismatch,” Meccanica 51, 2713–2726 (2016). https://doi.org/10.1007/s11012-016-0534-8, Google ScholarCrossref
  5. 5. P. Brzeski, J. Wojewoda, T. Kapitaniak, J. Kurths, and P. Perlikowski, “Sample-based approach can outperform the classical dynamical analysis-experimental confirmation of the basin stability method,” Sci. Rep. 7, 6121 (2017). https://doi.org/10.1038/s41598-017-05015-7, Google ScholarCrossref
  6. 6. T. Coletta, R. Delabays, I. Adagideli, and P. Jacquod, “Topologically protected loop flows in high voltage AC power grids,” New J. Phys. 18, 103042 (2016). https://doi.org/10.1088/1367-2630/18/10/103042, Google ScholarCrossref
  7. 7. A. Daza, A. Wagemakers, B. Georgeot, D. Guéry-Odelin, and M. A. F. Sanjuán, “Basin entropy: A new tool to analyze uncertainty in dynamical systems,” Sci. Rep. 6, 31416 (2016). https://doi.org/10.1038/srep31416, Google ScholarCrossref
  8. 8. A. Daza, B. Georgeot, D. Guéry-Odelin, A. Wagemakers, and M. A. F. Sanjuán, “Chaotic dynamics and fractal structures in experiments with cold atoms,” Phys. Rev. A 95, 013629 (2017). https://doi.org/10.1103/PhysRevA.95.013629, Google ScholarCrossref
  9. 9. R. de la Llave, “A tutorial on KAM theory,” in Proceedings of Symposia in Pure Mathematics 69, edited by A. Katok et al. (American Mathematical Society, 2001), pp. 175–292. Google Scholar
  10. 10. E. Eschenazi, H. G. Solari, and R. Gilmore, “Basins of attraction in driven dynamical systems,” Phys. Rev. A 39, 2609 (1989). https://doi.org/10.1103/PhysRevA.39.2609, Google ScholarCrossref
  11. 11. H. Fang and K. W. Wang, “Piezoelectric vibration-driven locomotion systems—exploiting resonance and bistable dynamics,” J. Sound Vib. 391, 153–169 (2017). https://doi.org/10.1016/j.jsv.2016.12.009, Google ScholarCrossref
  12. 12. J. Huang and A. C. J. Luo, “Analytical solutions of period-1 motions in a buckled, nonlinear Jeffcott rotor system,” Int. J. Dyn. Control 4, 376–383 (2016). https://doi.org/10.1007/s40435-015-0149-2, Google ScholarCrossref
  13. 13. G. Iooss and D. D. Joseph, Elementary Stability and Bifurcation Theory (Springer Science & Business Media, Berlin, 2012). Google Scholar
  14. 14. Y.-B. Kim, “Quasi-periodic response and stability analysis for non-linear systems: A general approach,” J. Sound Vib. 192, 821–833 (1996). https://doi.org/10.1006/jsvi.1996.0220, Google ScholarCrossref
  15. 15. H. Kim, S. H. Lee, and P. Holme, “Community consistency determines the stability transition window of power-grid nodes,” New J. Phys. 17, 113005 (2015). https://doi.org/10.1088/1367-2630/17/11/113005, Google ScholarCrossref
  16. 16. H. Kim, S. H. Lee, and P. Holme, “Building blocks of the basin stability of power grids,” Phys. Rev. E 93, 062318 (2016). https://doi.org/10.1103/PhysRevE.93.062318, Google ScholarCrossref
  17. 17. V. Kohar, P. Ji, A. Choudhary, S. Sinha, and J. Kurths, “Synchronization in time-varying networks,” Phys. Rev. E 90, 022812 (2014). https://doi.org/10.1103/PhysRevE.90.022812, Google ScholarCrossref
  18. 18. Y. Kuznetsov, Elements of Applied Bifurcation Theory (Springer-Verlag, New York, 1995). Google ScholarCrossref
  19. 19. A. N. Lansbury, J. M. T. Thompson, and H. B. Stewart, “Basin erosion in the twin-well duffing oscillator: Two distinct bifurcation scenarios,” Int. J. Bifurcat. Chaos 2, 505–532 (1992). https://doi.org/10.1142/S0218127492000677, Google ScholarCrossref
  20. 20. S. Lenci and G. Rega, “Competing dynamic solutions in a parametrically excited pendulum: Attractor robustness and basin integrity,” J. Comput. Nonlinear Dyn. 3, 041010 (2008). https://doi.org/10.1115/1.2960468, Google ScholarCrossref
  21. 21. S. Lenci and G. Rega, “Experimental versus theoretical robustness of rotating solutions in a parametrically excited pendulum: A dynamical integrity perspective,” Physica D 240, 814–824 (2011). https://doi.org/10.1016/j.physd.2010.12.014, Google ScholarCrossref
  22. 22. S. Leng, W. Lin, and J. Kurths, “Basin stability in delayed dynamics,” Sci. Rep. 6, 21449 (2016). https://doi.org/10.1038/srep21449, Google ScholarCrossref
  23. 23. O. V. Maslennikov, V. I. Nekorkin, and J. Kurths, “Basin stability for burst synchronization in small-world networks of chaotic slow-fast oscillators,” Phys. Rev. E 92, 042803 (2015). https://doi.org/10.1103/PhysRevE.92.042803, Google ScholarCrossref
  24. 24. P. J. Menck, J. Heitzig, N. Marwan, and J. Kurths, “How basin stability complements the linear-stability paradigm,” Nat. Phys. 9, 89–92 (2013). https://doi.org/10.1038/nphys2516, Google ScholarCrossref
  25. 25. C. Mitra, J. Kurths, and R. V. Donner, “An integrative quantifier of multistability in complex systems based on ecological resilience,” Sci. Rep. 5, 16196 (2015). https://doi.org/10.1038/srep16196, Google ScholarCrossref
  26. 26. G. Rega and S. Lenci, “Identifying, evaluating, and controlling dynamical integrity measures in non-linear mechanical oscillators,” Nonlinear Anal. Theory Methods Appl. 63, 902–914 (2005). https://doi.org/10.1016/j.na.2005.01.084, Google ScholarCrossref
  27. 27. B. Schäfer, C. Grabow, S. Auer, J. Kurths, D. Witthaut, and M. Timme, “Taming instabilities in power grid networks by decentralized control,” Eur. Phys. J. Spec. Top. 225, 569–582 (2016). https://doi.org/10.1140/epjst/e2015-50136-y, Google ScholarCrossref
  28. 28. P. Schultz, J. Heitzig, and J. Kurths, “Detours around basin stability in power networks,” New J. Phys. 16, 125001 (2014). https://doi.org/10.1088/1367-2630/16/12/125001, Google ScholarCrossref
  29. 29. J. Strzalko, J. Grabski, J. Wojewoda, M. Wiercigroch, and T. Kapitaniak, “Synchronous rotation of the set of double pendula: Experimental observations,” Chaos Interdiscip. J. Nonlinear Sci. 22, 047503 (2012). https://doi.org/10.1063/1.4740460, Google ScholarScitation, ISI
  30. 30. J. M. T. Thompson, “Chaotic phenomena triggering the escape from a potential well,” Proc. R. Soc. Lond. A 421, 195–225 (1989). https://doi.org/10.1098/rspa.1989.0009, Google ScholarCrossref
  31. 31. J. M. T. Thompson and M. S. Soliman, “Fractal control boundaries of driven oscillators and their relevance to safe engineering design,” Proc. R. Soc. Lond. A 428, 1–13 (1990). https://doi.org/10.1098/rspa.1990.0022, Google ScholarCrossref
  32. 32. J. A. Yorke and H. E. Nusse, Dynamics: Numerical Explorations (Springer-Verlag, New York, 1998). Google Scholar
  33. 33. Y. Zou, T. Pereira, M. Small, Z. Liu, and J. Kurths, “Basin of attraction determines hysteresis in explosive synchronization,” Phys. Rev. Lett. 112, 114102 (2014). https://doi.org/10.1103/PhysRevLett.112.114102, Google ScholarCrossref
  1. © 2018 Author(s). Published by AIP Publishing.

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