The mean field equation for the Ising model (without external field) is the following:
$$m=\mbox{tanh}\left(\frac{m}{T}\right)$$
where we consider all equivalent couplings and equal to $1$, $m$ is the global magnetization and $T$ is the temperature. The solution $m=0$ is valid for every value of $T$. On the other hand this solution is unstable for $T<1$. This can be seen plotting $m$ and $\mbox{tanh}\left(\frac{m}{T}\right)$ and looking for intersections. In this case we can graphically see that, for $T<1$ the solution $m=0$ is unstable, as depicted in the following figure
Graphical solution of the mean field equation
My problem is more complicated:
I want to solve the following system of integral equations numerically
$$q(J_0,T)=\int_{-\infty}^{\infty} \frac{1}{2\pi}e^{-\frac{z^2}{2}}\,\mbox{tanh}^2\left(\frac{z\sqrt{q(J_0,T)}+J_0 m(J_0,T)}{T}\right) dz$$
$$m(J_0,T)=\int_{-\infty}^{\infty} \frac{1}{2\pi}e^{-\frac{z^2}{2}}\,\mbox{tanh}\left(\frac{z\sqrt{q(J_0,T)}+J_0 m(J_0,T)}{T}\right) dz$$
and plot the values of $q(J_0,T)$ and $m(J_0,T)$ in the plane $J_0$ vs $T$. To do so I used the function FindRoot of Mathematica. In particular I got the $m$
and the $q$
Nevertheless the results for $q$ and $m$ are dependent on the initial value provided to FindRoot. Mathematically it is easy to see that the solution $q=0$ and $m=0$ is always available, but I expect, in general, non-trivial solutions. As for the simple Ising model this solution becomes unstable for some values of $J_0$ and $T$.
Is there a graphical way, similar to the previous one, to see that the solution $(m=0,q=0)$ is unstable under some conditions?
You don't need to plot a graphic to conclude that the fixed point $m=0$ is not attracting. This comes from the fact that $\left(\tanh (m/T)\right)'_{|m=0} = 1$. In order to $m=0$ being attractive one should have "< 1". In your example, you must properly define the functions spaces and corresponding norms and try to estimate the Lipschitz constant in that case (equivalent to the derivative in the simpler problem).