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Effective reduction of a three-dimensional circadian oscillator model

  • * Corresponding author: Shuang Chen

    * Corresponding author: Shuang Chen 
This work was partly supported by the NSFC grants 11531006, 11771449, 11771161, and the Hubei provincial postdoctoral science and technology activity project
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  • We investigate the dynamics of a three-dimensional system modeling a molecular mechanism for the circadian rhythm in Drosophila. We first prove the existence of a compact attractor in the region with biological meaning. Under the assumption that the dimerization reactions are fast, in this attractor we reduce the three-dimensional system to a simpler two-dimensional system on the persistent normally hyperbolic slow manifold.

    Mathematics Subject Classification: Primary: 34C45, 34A26; Secondary: 34C20.

    Citation:

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  • Figure 1.  The mechanism for the circadian oscillator model (1). Adapted from [33]

    Figure 2.  The attraction of the slow manifold. The surface is the critical manifolds $ \mathcal{M}_{0} $, which is the zeroth-order approximation of the slow manifold, and the discrete orbits, respectively, start from $ (10,10,2) $, $ (15,15,2) $, $ (20,20,2) $, $ (10,20,2) $ and $ (20,10,2) $. Here $ k_{a} = 20000 $, $ k_{d} = 100 $ and the remaining parameters in (2) are chosen as in [33,Table 1,p.2414], that is, $ v_{m} = 1 $, $ k_{3} = k_{m} = 0.1 $, $ v_{p} = 0.5 $, $ k_{1} = 10 $, $ k_{2} = 0.03 $, $ P_{c} = 0.1 $ and $ J_{p} = 0.05 $. System (16) with $ \widetilde{k}_{2} = 1 $ has small parameter $ \varepsilon = 0.0003 $

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