Abstract
We show that planar continuous alternating systems, which can be used to model systems with seasonality, can exhibit a type of Parrondo’s dynamic paradox, in which the stability of an equilibrium, common to all seasons is reversed for the global seasonal system. As a byproduct of our approach we also prove that there are locally invertible orientation preserving planar maps that cannot be the time-1 flow map of any smooth planar vector field.
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References
Arrowsmith, D.K., Place, C.M.: An introduction to Dynamical Systems. Cambridge University Press, Cambridge (1990)
Bass, H., Meisters, G.: Polynomial flows in the plane. Adv. Math. 55, 173–208 (1985)
Buonomo, B., Chitnis, N., d’Onofrio, A.: Seasonality in epidemic models: a literature review. Ricerche di Matematica 67, 7–25 (2018)
Cánovas, J.S., Linero, A., Peralta-Salas, D.: Dynamic Parrondo’s paradox. Phys. D 218, 177–184 (2006)
Cid, B., Hilker, F.M., Liz, E.: Harvest timing and its population dynamic consequences in a discrete single-species model. Math. Biosci. 248, 78–87 (2014)
Cima, A., Gasull, A., Mañosa, V.: Non-autonomous \(2\)-periodic Gumovski–Mira difference equations. Int. J. Bifurc. Chaos Appl. Sci. Eng. 22, Paper No 1250264 (14 pages) (2012)
Cima, A., Gasull, A., Mañosa, V.: Integrability and non-integrability of periodic non-autonomous Lyness recurrences. Dyn. Syst. 28, 518–538 (2013)
Cima, A., Gasull, A., Mañosa, V.: Parrondo’s dynamic paradox for the stability of non-hyperbolic fixed points. Discrete Contin. Dyn. Syst. Ser. A 38, 889–904 (2018)
Fretwell, S.D.: Populations in a Seasonal Environment. Princeton University Press, Princeton (1972)
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1990)
Harmer, G.P., Abbott, D.: Losing strategies can win by Parrondo’s paradox. Nature (London) 402(6764), 864 (1999)
Hsu, S.B., Zhao, X.Q.: A Lotka–Volterra competition model with seasonal succession. J. Math. Biol. 64, 109–130 (2012)
Ji, H., Strugarek, M.: Sharp seasonal threshold property for cooperative population dynamics with concave nonlinearities. Bull. Des Sci. Math. 147, 58–82 (2018)
Kot, M., Schaffer, W.M.: The effects of seasonality on discrete models of population growth. Theor. Popul. Biol. 26, 340–360 (1984)
Liz, E.: Effects of strength and timing of harvest on seasonal population models: stability switches and catastrophic shifts. Theor. Ecol. 10, 235–244 (2017)
Parrondo, J.M.R.: How to cheat a bad mathematician. In EEC HC&M Network on Complexity and Chaos (#ERBCHRX-CT940546), ISI, Torino, Italy. Unpublished (1996)
Palis, J.: Vector fields generate few diffeomorphisms. Bull. Am. Math. Soc. 80, 503–505 (1974)
Pireddu, M., Zanolin, F.: Chaotic dynamics in the Volterra predator–prey model via linked twist maps. Opusc. Math. 28, 567–592 (2008)
Xiao, D.: Dynamics and bifurcations on a class of population model with seasonal constant-yield harvesting. Discrete Contin. Dyn. Syst. Ser. B 21, 699–719 (2016)
Xu, C., Boyce, M.S., Daley, D.J.: Harvesting in seasonal environments. J. Math. Biol. 50, 663–682 (2005)
Zhang, X.: Embedding smooth diffeomorphisms in flows. J. Differ. Equ. 248, 1603–1616 (2010)
Zhang, X.: The embedding flows of \({\cal{C}}^\infty \) hyperbolic diffeomorphisms. J. Differ. Equ. 250, 2283–2298 (2011)
Funding
The authors are supported by Ministry of Economy, Industry and Competitiveness–State Research Agency of the Spanish Government through Grants MTM2016-77278-P (MINECO/AEI/FEDER, UE, first and second authors) and DPI2016-77407-P (MINECO/AEI/FEDER, UE, third author). The first and second authors are also supported by the Grant 2017-SGR-1617 from AGAUR, Generalitat de Catalunya. The third author acknowledges the group’s research recognition 2017-SGR-388 from AGAUR, Generalitat de Catalunya.
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Cima, A., Gasull, A. & Mañosa, V. A dynamic Parrondo’s paradox for continuous seasonal systems. Nonlinear Dyn 102, 1033–1043 (2020). https://doi.org/10.1007/s11071-020-05656-w
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DOI: https://doi.org/10.1007/s11071-020-05656-w
Keywords
- Continuous dynamical systems with seasonality
- Non-hyperbolic critical points
- Local asymptotic stability
- Parrondo’s dynamic paradox