On secondary bifurcations for some nonlinear convolution equations

Abstract:

On the $d$-dimensional torus ${{\mathbf {T}}^d} = {({\mathbf {R}}/{\mathbf {Z}})^d}$, we study the nonlinear convolution equation \[ u(t) = g\{ \lambda \cdot w \ast u(t)\} , \quad t \in {{\mathbf {T}}^d}, \lambda > 0.\] 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.