Stability pockets of a periodically forced oscillator in a model for seasonality. (English) Zbl 1382.37019
Summary: A periodically forced oscillator in a model for seasonality shows chains of stability pockets in the parameter plane. The frequency of the oscillator and the length of the photoperiod in the Zeitgeber are the two parameters. The present study is intended as a theoretical complement to the numerical study of Schmal et al. [loc. cit.] of stability pockets (or Arnol’d onions in their terminology). We construct the Poincaré map of the forced oscillator and show that the Arnol’d tongues are taken into chains of stability pockets by a map with a number of folds. This number is related to the rational point \((\frac{p}{q}, 0)\) on the frequency axis from which a chain of \(p\) pockets emanates. Stability pockets are already observed in an article by B. van der Pol and M. J. O. Strutt [Philos. Mag., VII. Ser. 5, 18–38 (1928; JFM 54.0469.02)] and later explained by H. Broer and M. Levi [Arch. Ration. Mech. Anal. 131, No. 3, 225–240 (1995; Zbl 0840.34047)].
MSC:
| 37C10 | Dynamics induced by flows and semiflows |
| 70K05 | Phase plane analysis, limit cycles for nonlinear problems in mechanics |
| 70K40 | Forced motions for nonlinear problems in mechanics |
| 37C50 | Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics |
| 34B30 | Special ordinary differential equations (Mathieu, Hill, Bessel, etc.) |
References:
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| [2] | Broer, H. W., Normal forms in perturbation theory, (Meyers, Robert A., Encyclopedia of Complexity and Systems Science (2009), Springer: Springer New York), 6310-6329 |
| [3] | Broer, H. W.; Levi, M., Geometrical aspects of stability theory for Hill’s equations, Arch. Ration. Mech. Anal., 131, 225-240 (1995) · Zbl 0840.34047 |
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| [10] | Schmal, Christoph; Myung, Jihwan; Herzel, Hanspeter; Bordyugov, Grigory, A theoretical study on seasonality, Front. Neurol., 6, 94 (2015) |
| [11] | Simó, C., Averaging under fast quasiperiodic forcing, (Seimenis, I., NATO-ARW Integrable and Chaotic Behaviour in Hamiltonian Systems, Torun, Poland, 1993 (1994), Plenum Pub. Co.: Plenum Pub. Co. New York), 13-34 |
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| [13] | van der Pol, B.; Strutt, M. J.O., On the stability of the solutions of Mathieu’s equation, Phil. Mag., 5, 18-38 (1928) · JFM 54.0469.02 |
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