0
2013
Impact Factor

# One-Dimensional Reaction-Diffusion Equations and Simple Source-Sink Arcs on a Circle

2018, Vol. 14, no. 3, pp.  325-330

Author(s): Pochinka O. V., Loginova A. S., Nozdrinova E. V.

This article presents a number of models that arise in physics, biology, chemistry, etc., described by a one-dimensional reaction-diffusion equation. The local dynamics of such models for various values of the parameters is described by a rough transformation of the circle. Accordingly, the control of such dynamics reduces to the consideration of a continuous family of maps of the circle. In this connection, the question of the possibility of joining two maps of the circle by an arc without bifurcation points naturally arises. In this paper it is shown that any orientation-preserving source-sink diffeomorphism on a circle is joined by such an arc. Note that such a result is not true for multidimensional spheres.
Keywords: reaction-diffusion equation, source-sink arc
Citation: Pochinka O. V., Loginova A. S., Nozdrinova E. V., One-Dimensional Reaction-Diffusion Equations and Simple Source-Sink Arcs on a Circle, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 3, pp.  325-330
DOI:10.20537/nd180303

 Download File PDF, 229.57 Kb

## References

[1] Banyaga, A., “On the Structure of the Group of Equivariant Diffeomorphisms”, Topology, 16:3 (1977), 279–283
[2] Bonatti, C., Grines, V.Ż., Medvedev, V. S., and Pochinka, O. V., “Bifurcations of Morse – Smale Diffeomorphisms with Wildly Embedded Separatrices”, Proc. Steklov Inst. Math., 256:1 (2007), 47–61        ; Tr. Mat. Inst. Steklova, 256 (2007), 54–69 (Russian)
[3] Hirsch, M. W., Differential Topology, Grad. Texts in Math., 33, Springer, New York, 1976, X, 222 pp.
[4] Milnor, J., Lectures on the $h$-Cobordism Theorem, Princeton Univ. Press, Princeton, N.J., 1965
[5] Pesin, Ya. B. and Yurchenko, A. A., “Some Physical Models Described by the Reaction-Diffusion Equation, and Coupled Map Lattices”, Russian Math. Surveys, 59:3 (2004), 481–513          ; Uspekhi Mat. Nauk, 59:3(357) (2004), 81–114 (Russian)

This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License