On the
d
d
-dimensional torus
T
d
=
(
R
/
Z
)
d
{{\mathbf {T}}^d} = {({\mathbf {R}}/{\mathbf {Z}})^d}
, we study the nonlinear convolution equation
\[
u
(
t
)
=
g
{
λ
⋅
w
∗
u
(
t
)
}
,
t
∈
T
d
,
λ
>
0.
u(t) = g\{ \lambda \cdot w \ast u(t)\} , \quad t \in {{\mathbf {T}}^d}, \lambda > 0.
\]
where
∗
\ast
is the convolution on
T
d
{{\mathbf {T}}^d}
,
w
w
is an integrable function which is not assumed to be even, and
g
g
is bounded, odd, increasing, and concave on
R
+
{{\mathbf {R}}^ + }
. A typical example is
g
=
th
g = {\text {th}}
. For a general function
w
w
, we show by the standard theory of local bifurcation that, if the eigenspace of the linearized problem is of dimension
2
2
, a branch of solutions bifurcates at
λ
=
(
g
′
(
0
)
w
^
(
p
)
)
−
1
\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
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
T
d
{{\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.