Geodesic
The problem of birth of autowaves in parabolic
Sbornik. Mathematics, Tome 198 (2007) no. 11, pp. 1599-1636 Cet article a éte moissonné depuis la source Math-Net.Ru

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A parabolic reaction-diffusion system with zero Neumann boundary conditions at the end-points of a finite interval is considered under the following basic assumptions. First, the matrix diffusion coefficient in the system is proportional to a small parameter $\varepsilon>0$, and the system itself possesses a spatially homogeneous cycle (independent of the space variable) of amplitude of order $\sqrt\varepsilon$ born by a zero equilibrium at an Andronov–Hopf bifurcation. Second, it is assumed that the matrix diffusion depends on an additional small parameter $\mu\geqslant0$, and for $\mu=0$ there occurs in the stability problem for the homogeneous cycle the critical case of characteristic multiplier 1 of multiplicity 2 without Jordan block. Under these constraints and for independently varied parameters $\varepsilon$ and $\mu$ the problem of the existence and the stability of spatially inhomogeneous auto-oscillations branching from the homogeneous cycle is analysed. Bibliography: 16 titles. 
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A. Yu. Kolesov; N. Kh. Rozov; V. A. Sadovnichii. The problem of birth of autowaves in parabolic. Sbornik. Mathematics, Tome 198 (2007) no. 11, pp. 1599-1636. http://geodesic.mathdoc.fr/item/SM_2007_198_11_a3/

[1] J. E. Marsden, M. McCracken, The Hopf bifurcation and its applications, Springer-Verlag, New York–Heidelberg–Berlin, 1976 | MR | MR | Zbl | Zbl

[2] B. D. Hassard, N. D. Kazarinoff, Y.-H. Wan, Theory and applications of Hopf bifurcation, London Math. Soc. Lecture Note Ser., 41, Cambridge Univ. Press, Cambridge–New York, 1981 | MR | MR | Zbl | Zbl

[3] E. F. Mishchenko, Yu. S. Kolesov, A. Yu. Kolesov, N. Kh. Rozov, Asymptotic methods in singularly perturbed systems, Monogr. Contemp. Math., Consultants Bureau, New York, 1994 | MR | MR | Zbl | Zbl

[4] A. Yu. Kolesov, E. F. Mishchenko, N. Kh. Rozov, “Asymptotic methods of investigation of periodic solutions of nonlinear hyperbolic equations”, Proc. Steklov Inst. Math., 222:3 (1998), 1–188 | MR | Zbl | Zbl

[5] A. Yu. Kolesov, N. Kh. Rozov, Invariantnye tory nelineinykh volnovykh uravnenii, Fizmatlit, M., 2004 | Zbl

[6] E. F. Mischenko, V. A. Sadovnichii, A. Yu. Kolesov, N. Kh. Rozov, Avtovolnovye protsessy v nelineinykh sredakh s diffuziei, Fizmatlit, M., 2005

[7] N. N. Bogolyubov, Yu. A. Mitropol'skii, Asymptotic methods in the theory of non-linear oscillations, Hindustan Publ., Delhi; Gordon and Breach, New York, 1961 | MR | MR | Zbl | Zbl

[8] P. Glansdorff, I. Prigogine, Thermodynamic theory of structure, stability and fluctuations, Wiley, London, 1971 | MR | Zbl | Zbl

[9] A. Yu. Kolesov, “The structure of a neighborhood of a homogeneous cycle in a medium with diffusion”, Math. USSR-Izv., 34:2 (1990), 355–372 | DOI | MR | Zbl

[10] Yu. S. Kolesov, D. I. Shvitra, Avtokolebaniya v sistemakh s zapazdyvaniem, Mokslas, Vilnyus, 1979 | MR | Zbl

[11] A. Yu. Kolesov, “Migratsionnye effekty v odnovidovom biotsenoze”, Nelineinye kolebaniya i ekologiya, Izd-vo YarGU, Yaroslavl, 1984, 34–61 | MR

[12] Ju. S. Kolesov, V. V. Maiorov, “A new method of investigating the stability of solutions of linear differential equations with almost-periodic coefficients that are almost constant”, Differential Equations, 10 (1976), 1363–1370 | MR | Zbl | Zbl

[13] Ju. S. Kolesov, “On the stability of solutions of linear differential equations of parabolic type with almost periodic coefficients”, Trans. Moscow Math. Soc., 36 (1979), 1–25 | MR | Zbl

[14] L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev, L. O. Chua, Methods of qualitative theory in nonlinear dynamics. Part I, World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises, 4, World Sci. Publ., River Edge, NJ, 1998 | MR | Zbl | Zbl

[15] S. D. Glyzin, A. Yu. Kolesov, “Ustanovivshiesya rezhimy uravneniya Khatchinsona s maloi diffuziei v sluchae kvadrata”, Kachestvennye metody issledovaniya operatornykh uravnenii, Izd-vo YarGU, Yaroslavl, 1988, 44–54 | Zbl

[16] M. Kholodniok, A. Klich, M. Kubichek, M. Marek, Metody analiza nelineinykh dinamicheskikh modelei, Mir, M., 1991