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Transactions of the American Mathematical Society

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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
MathSciNet review: 846602
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Abstract: On the $ d$-dimensional torus $ {{\mathbf{T}}^d} = {({\mathbf{R}}/{\mathbf{Z}})^d}$, we study the nonlinear convolution equation

$\displaystyle u(t) = g\{ \lambda \cdot w \ast u(t)\} , \quad t \in {{\mathbf{T}}^d}, \lambda > 0.$

where $ \ast$ is the convolution on $ {{\mathbf{T}}^d}$, $ w$ is an integrable function which is not assumed to be even, and $ g$ is bounded, odd, increasing, and concave on $ {{\mathbf{R}}^ + }$. A typical example is $ g = {\text{th}}$.

For a general function $ w$, we show by the standard theory of local bifurcation that, if the eigenspace of the linearized problem is of dimension $ 2$, a branch of solutions bifurcates at $ \lambda = {(g\prime(0)\hat w(p))^{ - 1}}$ from the zero solution, and we show that it can be extended to infinity.

For special simple forms of $ w$, 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 $ {{\mathbf{T}}^d}$ 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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Article copyright: © Copyright 1986 American Mathematical Society

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