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Weak pullback mean random attractors for non-autonomous $ p $-Laplacian equations
Effective reduction of a three-dimensional circadian oscillator model
1. | School of Mathematics and Statistics & Center for Mathematical Sciences, Huazhong University of Sciences and Technology, Wuhan, Hubei 430074, China |
2. | Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA |
3. | School of Mathematics and Statistics, Huazhong University of Sciences and Technology, Wuhan, Hubei 430074, China |
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.
References:
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S. Boie, V. Kirk, J. Sneyd and M. Wechselberger,
Effects of quasi-steady-state reduction on biophysical models with oscillations, J. Theoret. Biol., 393 (2016), 16-31.
doi: 10.1016/j.jtbi.2015.12.011. |
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S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer, New York, 1982. |
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B. Deng,
The Šil'nikov problem, exponential expansion, strong $\lambda$-lemma, $C^{1}$ linearization, and homoclinic bifurcation, J. Differential Equations, 79 (1989), 189-231.
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B. Deng and G. Hines,
Food chain chaos due to transcritical point, Chaos, 13 (2003), 578-585.
doi: 10.1063/1.1576531. |
[5] |
Z. Du, J. Li and X. Li,
The existence of solitary wave solutions of delayed Camassa-Holm equation via a geometric approach, J. Funct. Anal., 275 (2018), 988-1007.
doi: 10.1016/j.jfa.2018.05.005. |
[6] |
F. Dumortier and R. Roussarie, Canard Cycles and Center Manifolds, Mem. Amer. Math. Soc., Vol. 577, Providence, 1996.
doi: 10.1090/memo/0577. |
[7] |
F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, Berlin, 2006. |
[8] |
J. Dunlap,
Molecular bases for circadian clocks, Cell, 96 (1999), 271-290.
doi: 10.1016/S0092-8674(00)80566-8. |
[9] |
N. Fenichel,
Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971), 193-226.
doi: 10.1512/iumj.1972.21.21017. |
[10] |
N. Fenichel,
Asymptotic stability with rate conditions, Indiana Univ. Math. J., 23 (1974), 1109-1137.
doi: 10.1512/iumj.1974.23.23090. |
[11] |
N. Fenichel,
Asymptotic stability with rate conditions. II, Indiana Univ. Math. J., 26 (1977), 81-93.
doi: 10.1512/iumj.1977.26.26006. |
[12] |
N. Fenichel,
Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.
doi: 10.1016/0022-0396(79)90152-9. |
[13] |
D. B. Forger, Biological Clocks, Rhythms, and Oscillations, MIT Press, Cambridge, MA, 2017.
![]() |
[14] |
A. Goeke, S. Walcher and E. Zerz,
Determining "small parameters" for quasi-steady state, J. Differential Equations, 259 (2015), 1149-1180.
doi: 10.1016/j.jde.2015.02.038. |
[15] |
D. Gonze,
Modeling circadian clocks: From equations to oscillations, Cent. Eur. J. Bio., 6 (2011), 699-711.
doi: 10.2478/s11535-011-0061-5. |
[16] |
D. A. Goussis and H. N. Najm,
Model reduction and physical understanding of slowly oscillating processes: The circadian cycle, Multiscale Model. Simul., 5 (2006), 1297-1332.
doi: 10.1137/060649768. |
[17] |
G. Hek,
Geometric singular perturbation theory in biological practice, J. Math. Biol., 60 (2010), 347-386.
doi: 10.1007/s00285-009-0266-7. |
[18] |
C. K. R. T. Jones, Geometric Singular Perturbation Theory, in Dynamical Systems (eds. R. Johnson), Lecture Notes in Math., Springer, Berlin, 1609 (1995), 44–118.
doi: 10.1007/BFb0095239. |
[19] |
C. K. R. T. Jones, T. J. Kaper and N. Kopell,
Tracking invariant manifolds up to exponentially small errors, SIAM J. Math. Anal., 27 (1996), 558-577.
doi: 10.1137/S003614109325966X. |
[20] |
H. G. Kaper and T. J. Kaper,
Asymptotic analysis of two reduction methods for systems of chemical reactions, Phys. D, 165 (2002), 66-93.
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[21] |
J. Keener and J. Sneyd, Mathematical Physiology, Int. Appl. Math., 8, Springer-Verlag, New York, 1998.
doi: 10.1007/978-0-387-79388-7. |
[22] |
I. Kosiuk and P. Szmolyan,
Geometric analysis of the Goldbeter minimal model for the embryonic cell cycle, J. Math. Biol., 72 (2016), 1337-1368.
doi: 10.1007/s00285-015-0905-0. |
[23] |
M. Krupa and P. Szmolyan,
Extending geometric singular perturbation theory to nonhyperbolic points-fold and canard points in two dimensions, SIAM J. Math. Anal., 33 (2001), 286-314.
doi: 10.1137/S0036141099360919. |
[24] |
M. Krupa and P. Szmolyan,
Relaxation oscillation and canard explosion, J. Differential Equations, 174 (2001), 312-368.
doi: 10.1006/jdeq.2000.3929. |
[25] |
C. Kuehn, Multiple Time Scale Dynamics, Appl. Math. Sci., 191, Springer, Swizerland, 2015.
doi: 10.1007/978-3-319-12316-5. |
[26] |
C. Li and H. Zhu,
Canard cycles for predator-prey systems with Holling types of functional response, J. Differential Equations, 254 (2013), 879-910.
doi: 10.1016/j.jde.2012.10.003. |
[27] |
W. Liu,
Exchange lemmas for singular perturbation problems with certain turning points, J. Differential Equations, 67 (2000), 134-180.
doi: 10.1006/jdeq.2000.3778. |
[28] |
U. Maas and S. Pope,
Simplifying chemical kinetics: Intrinsic low dimensional manifolds in composition space, Combust. Flame, 88 (1992), 239-264.
doi: 10.1016/0010-2180(92)90034-M. |
[29] |
D. McMillen, N. Kopell, J. Hasty and J. Collins, Synchronizing genetic relaxation oscillators by intercell signaling, Proc. Natl. Acad. Sci. USA, 99 (2002), 679-684. Google Scholar |
[30] |
J. E. Rubin and D. Terman, Geometric singular perturbation analysis of neuronal dynamics, in Handbook of Dynamical Systems, North-Holland, Amsterdam, 2 (2002), 93–146.
doi: 10.1016/S1874-575X(02)80024-8. |
[31] |
S. Schecter,
Exchange lemmas 1: Deng's lemma, J. Differenital Equations, 245 (2008), 392-410.
doi: 10.1016/j.jde.2007.08.011. |
[32] |
S. Schecter,
Exchange lemmas 2: General exchange lemma, J. Differenital Equations, 245 (2008), 411-441.
doi: 10.1016/j.jde.2007.10.021. |
[33] |
J. Tyson, C. Hong, C. Thron and B. Novak,
A simple model of circadian rhythms based on dimerization and proteolysis of PER and TIM, Biophys. J., 77 (1999), 2411-2417.
doi: 10.1016/S0006-3495(99)77078-5. |
[34] |
S. Wiggins, Normally Hyperbolic Invariant Manifolds in Dynamical Systems, Appl. Math. Sci., 105, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4612-4312-0. |
[35] |
Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equations, Transl. Math. Monographs, 101, Amer. Math. Soc., Providence, 1992.
doi: 10.1007/978-1-4757-4969-4_4. |
show all references
References:
[1] |
S. Boie, V. Kirk, J. Sneyd and M. Wechselberger,
Effects of quasi-steady-state reduction on biophysical models with oscillations, J. Theoret. Biol., 393 (2016), 16-31.
doi: 10.1016/j.jtbi.2015.12.011. |
[2] |
S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer, New York, 1982. |
[3] |
B. Deng,
The Šil'nikov problem, exponential expansion, strong $\lambda$-lemma, $C^{1}$ linearization, and homoclinic bifurcation, J. Differential Equations, 79 (1989), 189-231.
doi: 10.1016/0022-0396(89)90100-9. |
[4] |
B. Deng and G. Hines,
Food chain chaos due to transcritical point, Chaos, 13 (2003), 578-585.
doi: 10.1063/1.1576531. |
[5] |
Z. Du, J. Li and X. Li,
The existence of solitary wave solutions of delayed Camassa-Holm equation via a geometric approach, J. Funct. Anal., 275 (2018), 988-1007.
doi: 10.1016/j.jfa.2018.05.005. |
[6] |
F. Dumortier and R. Roussarie, Canard Cycles and Center Manifolds, Mem. Amer. Math. Soc., Vol. 577, Providence, 1996.
doi: 10.1090/memo/0577. |
[7] |
F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, Berlin, 2006. |
[8] |
J. Dunlap,
Molecular bases for circadian clocks, Cell, 96 (1999), 271-290.
doi: 10.1016/S0092-8674(00)80566-8. |
[9] |
N. Fenichel,
Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971), 193-226.
doi: 10.1512/iumj.1972.21.21017. |
[10] |
N. Fenichel,
Asymptotic stability with rate conditions, Indiana Univ. Math. J., 23 (1974), 1109-1137.
doi: 10.1512/iumj.1974.23.23090. |
[11] |
N. Fenichel,
Asymptotic stability with rate conditions. II, Indiana Univ. Math. J., 26 (1977), 81-93.
doi: 10.1512/iumj.1977.26.26006. |
[12] |
N. Fenichel,
Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.
doi: 10.1016/0022-0396(79)90152-9. |
[13] |
D. B. Forger, Biological Clocks, Rhythms, and Oscillations, MIT Press, Cambridge, MA, 2017.
![]() |
[14] |
A. Goeke, S. Walcher and E. Zerz,
Determining "small parameters" for quasi-steady state, J. Differential Equations, 259 (2015), 1149-1180.
doi: 10.1016/j.jde.2015.02.038. |
[15] |
D. Gonze,
Modeling circadian clocks: From equations to oscillations, Cent. Eur. J. Bio., 6 (2011), 699-711.
doi: 10.2478/s11535-011-0061-5. |
[16] |
D. A. Goussis and H. N. Najm,
Model reduction and physical understanding of slowly oscillating processes: The circadian cycle, Multiscale Model. Simul., 5 (2006), 1297-1332.
doi: 10.1137/060649768. |
[17] |
G. Hek,
Geometric singular perturbation theory in biological practice, J. Math. Biol., 60 (2010), 347-386.
doi: 10.1007/s00285-009-0266-7. |
[18] |
C. K. R. T. Jones, Geometric Singular Perturbation Theory, in Dynamical Systems (eds. R. Johnson), Lecture Notes in Math., Springer, Berlin, 1609 (1995), 44–118.
doi: 10.1007/BFb0095239. |
[19] |
C. K. R. T. Jones, T. J. Kaper and N. Kopell,
Tracking invariant manifolds up to exponentially small errors, SIAM J. Math. Anal., 27 (1996), 558-577.
doi: 10.1137/S003614109325966X. |
[20] |
H. G. Kaper and T. J. Kaper,
Asymptotic analysis of two reduction methods for systems of chemical reactions, Phys. D, 165 (2002), 66-93.
doi: 10.1016/S0167-2789(02)00386-X. |
[21] |
J. Keener and J. Sneyd, Mathematical Physiology, Int. Appl. Math., 8, Springer-Verlag, New York, 1998.
doi: 10.1007/978-0-387-79388-7. |
[22] |
I. Kosiuk and P. Szmolyan,
Geometric analysis of the Goldbeter minimal model for the embryonic cell cycle, J. Math. Biol., 72 (2016), 1337-1368.
doi: 10.1007/s00285-015-0905-0. |
[23] |
M. Krupa and P. Szmolyan,
Extending geometric singular perturbation theory to nonhyperbolic points-fold and canard points in two dimensions, SIAM J. Math. Anal., 33 (2001), 286-314.
doi: 10.1137/S0036141099360919. |
[24] |
M. Krupa and P. Szmolyan,
Relaxation oscillation and canard explosion, J. Differential Equations, 174 (2001), 312-368.
doi: 10.1006/jdeq.2000.3929. |
[25] |
C. Kuehn, Multiple Time Scale Dynamics, Appl. Math. Sci., 191, Springer, Swizerland, 2015.
doi: 10.1007/978-3-319-12316-5. |
[26] |
C. Li and H. Zhu,
Canard cycles for predator-prey systems with Holling types of functional response, J. Differential Equations, 254 (2013), 879-910.
doi: 10.1016/j.jde.2012.10.003. |
[27] |
W. Liu,
Exchange lemmas for singular perturbation problems with certain turning points, J. Differential Equations, 67 (2000), 134-180.
doi: 10.1006/jdeq.2000.3778. |
[28] |
U. Maas and S. Pope,
Simplifying chemical kinetics: Intrinsic low dimensional manifolds in composition space, Combust. Flame, 88 (1992), 239-264.
doi: 10.1016/0010-2180(92)90034-M. |
[29] |
D. McMillen, N. Kopell, J. Hasty and J. Collins, Synchronizing genetic relaxation oscillators by intercell signaling, Proc. Natl. Acad. Sci. USA, 99 (2002), 679-684. Google Scholar |
[30] |
J. E. Rubin and D. Terman, Geometric singular perturbation analysis of neuronal dynamics, in Handbook of Dynamical Systems, North-Holland, Amsterdam, 2 (2002), 93–146.
doi: 10.1016/S1874-575X(02)80024-8. |
[31] |
S. Schecter,
Exchange lemmas 1: Deng's lemma, J. Differenital Equations, 245 (2008), 392-410.
doi: 10.1016/j.jde.2007.08.011. |
[32] |
S. Schecter,
Exchange lemmas 2: General exchange lemma, J. Differenital Equations, 245 (2008), 411-441.
doi: 10.1016/j.jde.2007.10.021. |
[33] |
J. Tyson, C. Hong, C. Thron and B. Novak,
A simple model of circadian rhythms based on dimerization and proteolysis of PER and TIM, Biophys. J., 77 (1999), 2411-2417.
doi: 10.1016/S0006-3495(99)77078-5. |
[34] |
S. Wiggins, Normally Hyperbolic Invariant Manifolds in Dynamical Systems, Appl. Math. Sci., 105, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4612-4312-0. |
[35] |
Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equations, Transl. Math. Monographs, 101, Amer. Math. Soc., Providence, 1992.
doi: 10.1007/978-1-4757-4969-4_4. |

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