Slow Invariant Manifold Analysis in a Mitotic Model of Frog Eggs via Flow Curvature Method

Full Text (PDF, 475KB), PP.41-48

Views: 0 Downloads: 0


A. K. M. Nazimuddin 1,* Md. Showkat Ali 2

1. Department of Mathematical and Physical Sciences, East West University, Dhaka-1212, Bangladesh.

2. Department of Applied Mathematics, University of Dhaka, Dhaka-1000, Bangladesh.

* Corresponding author.


Received: 10 Mar. 2022 / Revised: 14 Apr. 2022 / Accepted: 21 May 2022 / Published: 8 Oct. 2022

Index Terms

Mitotic Model, Darboux Theorem, Slow-Fast System, Differential Geometry, Flow Curvature Manifold


A slow-fast dynamical systems can be investigated qualitatively and quantitatively in the study of nonlinear chaotic dynamical systems. Slow-fast autonomous dynamical systems exhibit a dichotomy of motion, which is alternately slow and quick, according to experiments. Some investigations show that slow-fast dynamical systems have slow manifolds, which is supported by theory. The goal of the proposed study is to show how differential geometry may be used to examine the slow manifold of the dynamical system known as the mitotic model of frog eggs. The algebraic equation of the flow curvature manifold is obtained using the flow curvature technique applied to the dynamical mitosis model. Using the Darboux invariance theorem, we then argue that this slow manifold equation is invariant with regard to the flow.

Cite This Paper

A. K. M. Nazimuddin, Md. Showkat Ali," Slow Invariant Manifold Analysis in a Mitotic Model of Frog Eggs via Flow Curvature Method", International Journal of Mathematical Sciences and Computing(IJMSC), Vol.8, No.4, pp. 41-48, 2022. DOI: 10.5815/ijmsc.2022.04.04


[1]Nikolay Karabutov (2017). Adaptive Observers with Uncertainty in Loop Tuning for Linear Time-Varying Dynamical  Systems,  International  Journal  of  Intelligent  Systems and Applications, 9(4) :1-13.

[2]Ruisong  Ye,  Huiqing  Huang,  Xiangbo  Tan (2014). A  Novel  Image  Encryption  Scheme  Based  on Multi-orbit    Hybrid  of  Discrete  Dynamical  System,  I.J. Modern Education and Computer Science, 6(10) : 29-39.

[3]Ping Sun (2011). Solid Launcher Dynamical Analysis and Autopilot Design, I.J. Image, Graphics and Signal Processing, 3(1) : 53-60.

[4]Tikhonov, A.N. (1948). On the dependence of solutions of differential equations on a small parameter, Mat. Sbornik N. S., 31:575–586.

[5]Andronov, A.A., Chaikin, S.E (1937). Plane Theory of Oscillators, I, Moscow. 

[6]Levinson, N., (1949). A second-order differential equation with singular solutions, Ann. Math, 50:127–153.

[7]Fenichel, N.  (1971). Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J, 21:193–225.

[8]Fenichel, N. (1974). Asymptotic stability with rate conditions, Indiana Univ. Math. J, 23:1109–1137. 

[9]Fenichel, N. (1977). Asymptotic stability with rate conditions II, Indiana Univ. Math. J, 26:81–93.

[10]Fenichel, N. (1979). Geometric singular perturbation theory for ordinary differential equations, J. Differ. Equ., 31: 53–98. 

[11]Ginoux, J.M.and Llibre, J. (2011). The flow curvature method applied to canard explosion, J. Phys. A Math. Theor., 44: 465203. 

[12]Ginoux, J.M. (2009). Differential geometry applied to dynamical systems, In: World Scientific Series on Nonlinear Science, Series A, 66, World Scientific, Singapore. 

[13]Ginoux, J.M. and Rossetto, B. (2006). Differential geometry and mechanics applications to chaotic dynamical systems, Int. J. Bifurc. Chaos, 4(16): 887–910. 

[14]Ginoux, J.M., Rossetto, B. and Chua, L.O. (2008). Slow invariant manifolds as curvature of the flow of dynamical systems, Int. J. Bifurc. Chaos, 11(18): 3409–3430. 

[15]Ginoux, J.M., Llibre, J. and Chua, L.O. (2013). Canards from Chua’s circuit, Int. J. Bifurc. Chaos, 23(4): 1330010. 

[16]Ginoux, J. M. (2014). The slow invariant manifold of the Lorenz–Krishnamurthy model, Qualitative theory of dynamical systems, 13(1): 19–37.

[17]Ginoux, J. M., & Rossetto, B. (2014). Slow invariant manifold of heartbeat model, arXiv preprint arXiv:1408.4988.

[18]Jin-hu, L., Zi-fan, Z. and Suo-chun, Z. (2003). Bifurcation analysis of a mitotic model of frog eggs. Applied Mathematics and Mechanics, 24(3): 284-297.

[19]Novak  B,  Tyson  J  J. (1993). Numerical  analysis  of  a  comprehensive  model  of M-phase  control  in  Xenopus oocyte  extracts  and  intact  embryos,  Journal  of  Cell  Science, 106(4) : 1153  -  1168.

[20]Novak  B,  Tyson  J  J. (1993). Modeling  the  cell  division  cycle:  M-phase  trigger,  osculations,  and  size  control,  Journal  of Theoretical  Biology,  165(1)  :  101  -  134. 

[21]FENG  Bei-ye,  ZENG  Xuan-wu. (2002). Qualitative  analysis  of  a  mitotic  model  of  frog  eggs, Acta Mathematicae  Applicatae  Sinica, 25(3) :460  -  468.  (in  Chinese)

[22]Tyson, J. J., & Novak, B. (2015). Bistability, oscillations, and traveling waves in frog egg extracts, Bulletin of mathematical biology, 77(5): 796-816.

[23]Borisuk,  M  T and  Tyson,  J  J.  (1998). Bifurcation analysis  of a  model  of  mitotic  control  in  frog  eggs, Journal  of  Theoretical  Biology,  195(1) :69 - 85.

[24]Murray, A. and Hunt T. (1993). The Cell Cycle. An Introduction. New York: W. H. Freeman Co.