Remote Access Transactions of the Moscow Mathematical Society

Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)

Request Permissions   Purchase Content 
 
 

 

Local dynamics of two-component singularly perturbed parabolic systems


Authors: I. S. Kashchenko and S. A. Kashchenko
Translated by: E. Khukhro
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 77 (2016), vypusk 1.
Journal: Trans. Moscow Math. Soc. 2016, 55-68
MSC (2010): Primary 35K67; Secondary 35B10, 35B25, 35C20, 35K40
DOI: https://doi.org/10.1090/mosc/252
Published electronically: November 28, 2016
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the local dynamics in a neighbourhood of a stationary state of a two-component system of parabolic equations with periodic boundary conditions. In the critical cases we construct families of special equations--quasinormal forms whose solutions in principle give asymptotic solutions, up to the residual, of the original singularly perturbed system.


References [Enhancements On Off] (What's this?)

  • 1. J. E. Marsden and M. McCracken, The Hopf bifurcation and its applications, Appl. Math. Sci., vol. 19, Springer-Verlag, New York, 1976. MR 0494309
  • 2. V. I. Arnol'd, Geometrical methods in the theory of ordinary differntial equations, Moscow Centre for Contin. Math. Educ., Moscow, 2012; English transl. of 1st ed., Grundl. Math. Wissens., vol. 250, Springer-Verlag, New York, 1988 MR 947141
  • 3. A. D. Bryuno, Local method of nonlinear analysis of differential equations, Nauka, Moscow, 1979; English transl., Springer Ser. Soviet Math., Springer-Verlag, Berlin, 1989. MR 0542758
  • 4. S. A. Kashchenko, The local dynamics of two-component contrast structures in the neighborhood of a bifurcation point, Dokl. Akad. Nauk SSSR 312 (1990), no. 2, 345-350; English transl., Soviet Phys. Dokl. 35 (1990), no. 5, 420-422 MR 1072879
  • 5. S. A. Kashchenko, Construction of normalized systems for the investigation of the dynamics of hybrid and hyperbolic equations, Zh. Vychisl. Mat. Mat. Fiz. 34 (1994), no. 4, 564-575; English transl., Comput. Math. Math. Phys. 34 (1994), no. 4, 479-489. MR 1272902
  • 6. S. A. Kaschenko, Normalization in the systems with small diffusion. Nonlinear dynamics, bifurcations and chaotic behavior, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 6 (1996), no. 6, 1093-1109. MR 1409411
  • 7. I. S. Kashchenko, Multistability in nonlinear parabolic systems with low diffusion, Dokl. Akad. Nauk 435 (2010), no. 2, 164-167; English transl., Dokl. Math. 82 (2010), no. 3, 878-881. MR 2790504
  • 8. I. S. Kashchenko and S. A. Kashchenko, Quasi-normal forms for parabolic systems with strong nonlinearity and weak diffusion, Zh. Vychisl. Mat. Mat. Fiz. 52 (2012), no. 8, 1482-1491; English transl., Comput. Math. Math. Phys. 52 (2012), no. 8, 1163-1172. MR 3245239
  • 9. I. S. Kashchenko and S. A. Kashchenko, Quasinormal forms of two-component singularly perturbed systems, Dokl. Akad. Nauk 447 (2012), no. 4, 376-381; English transl., Dokl. Math. 86 (2012), no. 3, 865-870. MR 3077454
  • 10. V. I. Arnol'd, Lectures on bifurcations and versal families. A series of articles on the theory of singularities of smooth mappings, Uspekhi Mat. Nauk 27 (1972), no. 5, 119-184. (Russian) MR 0413191
  • 11. S. A. Kashchenko, Quasinormal forms for parabolic equations with small diffusion, Dokl. Akad. Nauk SSSR 299 (1988), no. 5, 1049-1052; English transl., Soviet Math. Dokl. 37 (1988), no. 2, 510-513. MR 947229
  • 12. R. I. Bogdanov, Bifurcations of a limit cycle of a certain family of vector fields on the plane, Trudy Sem. Petrovsk., 1976, no. 2, 23-35; English transl., Selecta Math. Soviet. 1 (1981), no. 4, 373-388. MR 0442988
  • 13. S. A. Kashchenko, Spatial singularities of high-mode bifurcations of two-component systems with small diffusion, Differentsial'nye Uravneniya 25 (1989), no. 2, 262-270; English transl., Differential Equations 25 (1989), no. 2, 193-199. MR 994709

Similar Articles

Retrieve articles in Transactions of the Moscow Mathematical Society with MSC (2010): 35K67, 35B10, 35B25, 35C20, 35K40

Retrieve articles in all journals with MSC (2010): 35K67, 35B10, 35B25, 35C20, 35K40


Additional Information

I. S. Kashchenko
Affiliation: Yaroslavl’ State University
Email: iliyask@uniyar.ac.ru

S. A. Kashchenko
Affiliation: Yaroslavl’ State University, National Research Nuclear University (Moscow Engineering Physics Institute)
Email: kasch@uniyar.ac.ru

DOI: https://doi.org/10.1090/mosc/252
Keywords: Parabolic equation, quasinormal form, small parameter.
Published electronically: November 28, 2016
Additional Notes: This research was supported by project no. 984 within the framework of the basic part of the state programme for scientific research of Yaroslavl’ State University and a grant from the President of the Russian Federation (contract no. 14.124.13.5948-MK)
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society