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Journal of Applied Nonlinear Dynamics 11(1) (2022) 139--152 | DOI:10.5890/JAND.2022.03.008

Souma Mazumdar , Gautam Gangopadhyay

Department of Theoretical Sciences, Department of Chemical, Biological and Macro-molecular Sciences,

S.N.Bose National Centre For Basic Sciences, Block - JD, Sector - III, Salt Lake City, Kolkata - 700106

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Abstract

Centre Manifold analysis of a $3-D$ nonlinear system with general second order nonlinearities have been worked out. The system is shown to possess two fixed points on the reduced $2-D$ centre manifold. By introducing a $2-D$ centre manifold one can show how an oscillatory dynamics may be generated in the system. We also state and prove a theorem to find the stability of the resultant centre manifold equation apriori from the parity of the nonlinear terms in the original equations. For a $2-D$ nonlinear model with the example picked up from biochemistry, the protein molecules in assembly, kinetic stability analysis is provided for the chosen example and establish herewith the validity of the theorem for our chosen example.

References

1. [1]	Liu, L., Wong, Y.S., and Lee, B.H.K. (2000), Application of the centre manifold theory in non-linear aeroelasticity, Journal of Sound and Vibration, 234(4), 641-659.

2. [2]	Dessi, D., Mastroddi, F., and Morino, L. (2002), Limit-cycle stability reversal near a Hopf bifurcation with aeroelastic applications, Journal of Sound and Vibration, 256(2), 347-365.

3. [3]	Dessi, D. and Mastroddi, F. (2004), Limit-cycle stability reversal via singular perturbation and wing-flap flutter, Journal of Fluids and Structures, 19(6), 765-783.

4. [4]	Grzedzinski, J. (2005), Limitation of application of the center manifold reduction in aeroelasticity, Journal of Fluids and Structures, 21(2), 187-209.

5. [5]	Sanchez-Ruiz, J.M. (2010), Protein kinetic stability, Biophysical chemistry, 148(1-3), 1-15.

6. [6]	Manning, M. and Colon, W. (2004), Structural basis of protein kinetic stability: resistance to sodium dodecyl sulfate suggests a central role for rigidity and a bias toward $\beta$-sheet structure, Biochemistry, 43(35), 11248-11254.

7. [7]	Rodriguez-Larrea, D., Minning, S., Borchert, T.V., and Sanchez-Ruiz, J.M. (2006), Role of solvation barriers in protein kinetic stability, Journal of Molecular Biology, 360(3), 715-724.

8. [8]	Colon, W., Church, J., Sen, J., Thibeault, J., Trasatti, H., and Xia, K. (2017), Biological roles of protein kinetic stability, Biochemistry, 56(47), 6179-6186.

9. [9]	Broering, J.M. and Bommarius, A.S. (2005), Evaluation of Hofmeister effects on the kinetic stability of proteins, The Journal of Physical Chemistry B, 109(43), 20612-20619.

10. [10]	Polizzi, K.M., Bommarius, A.S., Broering, J.M., and Chaparro-Riggers, J.F. (2007), Stability of biocatalysts, Current opinion in chemical biology, 11(2), 220-225.

11. [11]	Strogatz, S.H. (2018), Nonlinear dynamics and chaos with student solutions manual: With applications to physics, Biology, Chemistry, and Engineering, CRC press.

12. [12]	Thompson, J.M.T. and Stewart, H.B. (2002), Nonlinear dynamics and chaos, John Wiley \& Sons.

13. [13]	Stephen, W. (2003), Introduction to applied nonlinear dynamical systems and chaos, Springer Science \& Business Media.

14. [14]	Tsuruyama, T. (2017), Kinetic stability analysis of protein assembly on the center manifold around the critical point, BMC Systems Biology, BioMed Central.