Predicting Tipping Points in a Family of PWL Systems: Detecting Multistability via Linear Operators Properties

Year 2024, Volume: 6 Issue: 2, 73 - 82, 30.06.2024

J. L. Echenausía-monroy Rıcardo Cuesta-garcía Hector Gilardi-velázquez Sishu Shankar Muni Joaquin Alvarez-gallegos

https://doi.org/10.51537/chaos.1376123

Abstract

The study of dynamical systems is based on the solution of differential equations that may exhibit various behaviors, such as fixed points, limit cycles, periodic, quasi-periodic attractors, chaotic behavior, and coexistence of attractors, to name a few. In this paper, we present a simple and novel method for predicting the occurrence of tipping points in a family of Piece-Wise Linear systems (PWL) that exhibit a transition from monostability to multistability with the variation of a single parameter, without the need to compute time series, i.e., without solving the differential equations of the system. The linearized system of the model is analyzed, the stable and unstable manifolds are taken to be real vectors in space, and the changes suffered by these vectors as a result of the modification of the parameter are examined using such simple metrics as the magnitude of a vector or the angle between two vectors in space. The results obtained with the linear analysis of the system agree well with those obtained with the numerical resolution of the dynamical system itself. The work presented here is an extension of previous results on this topic and contributes to the understanding of the mechanisms by which a system changes its stability by fragmenting its basin of attraction. This, in turn, enriches the field by providing an alternative to numerical resolution to identify quantitative changes in the dynamics of complex systems without having to solve the differential equation system.

Keywords

Nonlinear dynamics, Chaotic system, Multistability, PWL system, Bifurcation, Tipping point

Ethical Statement

The authors state that there are no ethical conflicts in publishing this paper since it does not involve experiments with living beings.

Supporting Institution

J.L.E.M. thanks CONAHCYT for financial support (CVU-706850), and to CICESE.

Project Number

CVU-706850

Thanks

J.L.E.M. thanks CONAHCYT for financial support (CVU-706850), and to J.A.G. for the opportunity to do a postdoctoral fellowship at CICESE.

References

• Anzo-Hernández, A., H. Gilardi-Velázquez, and E. Campos- Cantón, 2018 On multistability behavior of unstable dissipative systems. Chaos: An Interdisciplinary Journal of Nonlinear Science 28: 033613.
• Awal, N. M. and I. R. Epstein, 2021 Period-doubling route to mixedmode chaos. Physical Review E 104: 024211.
• Biggs, R., S. R. Carpenter, andW. A. Brock, 2009 Turning back from the brink: detecting an impending regime shift in time to avert it. Proceedings of the National academy of Sciences 106: 826–831.
• Campos-Cantón, E., 2015 Switched systems based on unstable dissipative systems. IFAC-PapersOnLine 48: 116–121.
• Campos-Cantón, E., J. G. Barajas-Ramirez, G. Solis-Perales, and R. Femat, 2010 Multiscroll attractors by switching systems. Chaos: An Interdisciplinary Journal of Nonlinear Science 20.
• Chen, L., F. Nazarimehr, S. Jafari, E. Tlelo-Cuautle, and I. Hussain, 2020 Investigation of early warning indexes in a threedimensional chaotic system with zero eigenvalues. Entropy 22: 341.
• Echenausía-Monroy, J., J. Cuesta-García, and J. Ramirez-Pena, 2022 The wonder world of complex systems. Chaos Theory and Applications 4: 267–273.
• Echenausía-Monroy, J., H. Gilardi-Velázquez, N.Wang, R. Jaimes- Reátegui, J. García-López, et al., 2022a Multistability route in a pwl multi-scroll system through fractional-order derivatives. Chaos, Solitons & Fractals 161: 112355.
• Echenausía-Monroy, J., S. Jafari, G. Huerta-Cuellar, and H. Gilardi- Velázquez, 2022b Predicting the emergence of multistability in a monoparametric pwl system. International Journal of Bifurcation and Chaos 32: 2250206.
• Echenausía-Monroy, J. L., J. García-López, R. Jaimes-Reátegui, D. López-Mancilla, and G. Huerta-Cuellar, 2018 Family of bistable attractors contained in an unstable dissipative switching system associated to a snlf. Complexity 2018.
• Echenausía-Monroy, J. L., G. Huerta-Cuellar, R. Jaimes-Reátegui, J. H. García-López, V. Aboites, et al., 2020 Multistability emergence through fractional-order-derivatives in a pwl multi-scroll system. Electronics 9: 880.
• Fang, S., S. Zhou, D. Yurchenko, T. Yang, andW.-H. Liao, 2022 Multistability phenomenon in signal processing, energy harvesting, composite structures, and metamaterials: A review. Mechanical Systems and Signal Processing 166: 108419.
• Gilardi-Velázquez, H. E., R. d. J. Escalante-González, and E. Campos-Cantón, 2018 Bistable behavior via switching dissipative systems with unstable dynamics and its electronic design. IFAC-PapersOnLine 51: 502–507.
• Gilardi-Velázquez, H. E., L. Ontañón-García, D. G. Hurtado- Rodriguez, and E. Campos-Cantón, 2017 Multistability in piecewise linear systems versus eigenspectra variation and round function. International Journal of Bifurcation and Chaos 27: 1730031.
• Guan, S., C.-H. Lai, and G.Wei, 2005 Bistable chaos without symmetry in generalized synchronization. Physical Review E 71: 036209.
• Jiang, J., A. Hastings, and Y.-C. Lai, 2019 Harnessing tipping points in complex ecological networks. Journal of the Royal Society Interface 16: 20190345.
• Jiang, J., Z.-G. Huang, T. P. Seager,W. Lin, C. Grebogi, et al., 2018 Predicting tipping points in mutualistic networks through dimension reduction. Proceedings of the National Academy of Sciences 115: E639–E647.
• Jung, H. and J. W. Ager, 2023 A tipping point for solar production of hydrogen? Joule 7: 459–461.
• Kele¸s, Z., G. Sonugür, and M. Alçin, 2023 The modeling of the rucklidge chaotic system with artificial neural networks. Chaos Theory and Applications 5: 59–64.
• Kleiner, I., 2007 A history of abstract algebra. Springer Science & Business Media.
• Lane, S. N., 2011 The tipping point: How little things can make a big difference. Geography 96: 34–38.
• Lenton, T. M., H. Held, E. Kriegler, J. W. Hall, W. Lucht, et al., 2008 Tipping elements in the earth’s climate system. Proceedings of the national Academy of Sciences 105: 1786–1793.
• Lohmann, J., D. Castellana, P. D. Ditlevsen, and H. A. Dijkstra, 2021 Abrupt climate change as a rate-dependent cascading tipping point. Earth System Dynamics 12: 819–835.
• Lorenz, E. N., 1963 Deterministic nonperiodic flow. Journal of atmospheric sciences 20: 130–141.
• Moghadam, N. N., R. Ramamoorthy, F. Nazarimehr, K. Rajagopal, and S. Jafari, 2022 Tipping points of a complex network biomass model: Local and global parameter variations. Physica A: Statistical Mechanics and its Applications 592: 126845.
• Moore, J. C., 2018 Predicting tipping points in complex environmental systems. Proceedings of the National Academy of Sciences 115: 635–636.
• Nazarimehr, F., S. Jafari, S. M. R. Hashemi Golpayegani, M. Perc, and J. C. Sprott, 2018 Predicting tipping points of dynamical systems during a period-doubling route to chaos. Chaos: An Interdisciplinary Journal of Nonlinear Science 28.
• Ott, E., 2002 Chaos in dynamical systems. Cambridge university press.
• O’Regan, S. M., E. B. O’Dea, P. Rohani, and J. M. Drake, 2020 Transient indicators of tipping points in infectious diseases. Journal of the Royal Society Interface 17: 20200094.
• Peng, X., M. Small, Y. Zhao, and J. M. Moore, 2019 Detecting and predicting tipping points. International Journal of Bifurcation and Chaos 29: 1930022.
• Rial, J. A., R. A. Pielke, M. Beniston, M. Claussen, J. Canadell, et al., 2004 Nonlinearities, feedbacks and critical thresholds within the earth’s climate system. Climatic change 65: 11–38.
• Safavi, S. and P. Dayan, 2022 Multistability, perceptual value, and internal foraging. Neuron.
• Scheffer, M., S. Carpenter, J. A. Foley, C. Folke, and B. Walker, 2001 Catastrophic shifts in ecosystems. Nature 413: 591–596.
• Tsakonas, S., M. Hanias, and L. Magafas, 2022 Application of the moving lyapunov exponent to the s&p 500 index to predict major declines. Journal of Risk.
• Wunderling, N., A. von der Heydt, Y. Aksenov, S. Barker, R. Bastiaansen, et al., 2023 Climate tipping point interactions and cascades: A review. EGUsphere 2023: 1–45.