On secondary bifurcations for some nonlinear convolution equations

Authors:
F. Comets, Th. Eisele and M. Schatzman

Journal:
Trans. Amer. Math. Soc. **296** (1986), 661-702

MSC:
Primary 58E07; Secondary 45G10, 82A25, 92A09

DOI:
https://doi.org/10.1090/S0002-9947-1986-0846602-7

MathSciNet review:
846602

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Abstract | References | Similar Articles | Additional Information

Abstract: On the -dimensional torus , we study the nonlinear convolution equation

For a general function , we show by the standard theory of local bifurcation that, if the eigenspace of the linearized problem is of dimension , a branch of solutions bifurcates at from the zero solution, and we show that it can be extended to infinity.

For special simple forms of , we show that the first bifurcating branch has no secondary bifurcation, but the other branches can.

These results are related to the theory of spin models on in statistical mechanics, where they allow one to show the existence of a secondary phase transition of first order, and to some models of periodic structures in the brain in neurophysiology.

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DOI:
https://doi.org/10.1090/S0002-9947-1986-0846602-7

Article copyright:
© Copyright 1986
American Mathematical Society