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A dynamic Parrondo’s paradox for continuous seasonal systems


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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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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Correspondence to Armengol Gasull.

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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).

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  • Continuous dynamical systems with seasonality
  • Non-hyperbolic critical points
  • Local asymptotic stability
  • Parrondo’s dynamic paradox

Mathematics Subject Classification

  • Primary 37C75
  • 34D20
  • Secondary 37C25