Ukrainian Mathematical Journal

, Volume 65, Issue 1, pp 21–46 | Cite as

Dynamics of periodic modes for the phenomenological equation of spin combustion

  • E. P. Belan
  • A. M. Samoilenko

We consider a scalar parabolic equation on a circle of radius r. The analyzed problem is a phenomenological model of gasless combustion on the surface of a cylinder of radius r. We study the problems of existence of traveling waves, their asymptotic form and stability and the nature of gaining and losing their stability.


Periodic Solution Stationary Solution Spin Mode Phenomenological Equation Gasless Combustion 
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  1. 1.
    A. P. Aldushin, Ya. B. Zel’dovich, and B. A. Malomed, “On the phenomenological theory of spin combustion,” Dokl. Akad. Nauk SSSR, 251, No. 5, 1102–1106 (1980).Google Scholar
  2. 2.
    A. P. Aldushin and B. A. Malomed, “Phenomenological description of nonstationary inhomogeneous combustion waves,” Fiz. Gor. Vzryv., 17, No. 1, 3–12 (1981).Google Scholar
  3. 3.
    Ya. B. Zel’dovich and B. A. Malomed, “Complex wave modes in distributed dynamical systems,” Izv. Vyssh. Uchebn. Zaved., Ser. Radiofiz., 15, No. 6, 591–618 (1982).MathSciNetGoogle Scholar
  4. 4.
    A.Yu. Kolesov and N. Kh. Rozov, “Buffering phenomenon in the combustion theory,” Dokl. Akad. Nauk, 396, No. 2, 170–173 (2004).MathSciNetzbMATHGoogle Scholar
  5. 5.
    E. F. Mishchenko, V. A. Sadovnichii, A.Yu. Kolesov, and N. Kh. Rozov, Autowave Processes in Nonlinear Media with Diffusion [in Russian], Fizmatlit, Moscow (2005).Google Scholar
  6. 6.
    A. M. Samoilenko and E. P. Belan, “Dynamics of traveling waves for the phenomenological equation of spin combustion,” Dokl. Akad. Nauk, 406, No. 6, 738–741 (2006).MathSciNetGoogle Scholar
  7. 7.
    A. Bayliss, B. J. Matkowsky, and A. P. Aldushin, “Dynamics of hot spots in solid fuel combustion,” Physica D, 166, 114–130 (2002).MathSciNetCrossRefGoogle Scholar
  8. 8.
    T. P. Ivleva and A. G. Merzhanov, “Description of the modes of propagation of solid flame,” Dokl. Akad. Nauk, 378, No. 1, 62–64 (2003).Google Scholar
  9. 9.
    T. S. Akhromeeva, S. P. Kurdyumov, G. G. Malinetskii, and A. A. Samarskii, Structure and Chaos in Nonlinear Media [in Russian], Nauka. Moscow (2007).Google Scholar
  10. 10.
    D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin (1981).zbMATHGoogle Scholar
  11. 11.
    N. N. Bogolyubov and Yu. A. Mitropol’skii, Asymptotic Methods in the Theory of Nonlinear Mechanics [in Russian], Nauka, Moscow (1974).Google Scholar
  12. 12.
    A. B. Vasil’eva, S. A. Kashchenko, Yu. S. Rozov, and N. Kh. Rozov, “Bifurcation of the autooscillations of nonlinear parabolic equations with weak diffusion,” Mat. Sb., 130(172), No. 4, 488–499 (1986).MathSciNetGoogle Scholar
  13. 13.
    A.V. Gaponov-Grekhov, A. S. Lomov, G. V. Osipov, and M. I. Rabinovich, “Pattern formation and the dynamics of two-dimensional structures in nonequilibrium dissipative systems,” in: A.V. Gaponov-Grekhov and M. I. Rabinovich (editors), Nonlinear Waves. Dynamics and Evolution [in Russian], Nauka, Moscow (1989), pp. 61–73.Google Scholar
  14. 14.
    A.V. Babin and M. I. Vishik, Attractors of Evolutionary Equations [in Russian], Nauka, Moscow (1989).Google Scholar
  15. 15.
    V. A. Pliss, “Principle of reduction in the theory of stability of motion,” Izv. Akad. Nauk SSSR, Ser. Mat., 28, No. 4, 1297–1324 (1964).MathSciNetzbMATHGoogle Scholar
  16. 16.
    J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer, New York (1976).zbMATHCrossRefGoogle Scholar
  17. 17.
    E. P. Belan, “On the dynamics of traveling waves in a parabolic equation with the transformation of shift of the space variable,” Zh. Mat. Fiz. Analiz. Geometr., 1, No. 1, 3–34 (1964).MathSciNetGoogle Scholar
  18. 18.
    O. V. Shiyan, “On the dynamics of traveling waves in a system of Van Der Pol equations with weak diffusion,” Dop. Nats. Akad. Nauk. Ukr., No. 7, 27–32 (2007).Google Scholar
  19. 19.
    O. V. Shiyan, “On the dynamics of traveling waves in a system of Van-Der Pol-type equations with weak diffusion,” in: Proc. of the Institute of Applied Mathematics and Mechanics, Ukrainian National Academy of Sciences [in Russian], 16 (2008), pp. 208–222.Google Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  • E. P. Belan
    • 1
  • A. M. Samoilenko
    • 2
  1. 1.Tavriya National UniversitySimferopolUkraine
  2. 2.Institute of MathematicsUkrainian National Academy of SciencesKyivUkraine

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