I have a system of equations:
\begin{cases} f\left(x_{1}\right)+f\left(x_{2}\right)+P=0\\ \\ g\left(x_{1}\right)+g\left(x_{2}\right)=0 \end{cases}
where $f,g$ are some functions, $\boldsymbol{x}=\left[\begin{array}{c}x_{1}\\x_{2}\end{array}\right]$ is the unknown, and $P$ is a parameter that I want to vary. Let's say that for $P<P^{*}$ the solution of my equation is a stable $\underline{\boldsymbol{x}}=\left[\begin{array}{c}\alpha\left(P\right) \\\alpha\left(P\right) \end{array}\right]$, while for $P>P^{*}$ the solution $\underline{\boldsymbol{x}}$ becomes unstable, and I also got two new stable solutions $\overline{\boldsymbol{x}}=\left[\begin{array}{c}\beta\left(P\right) \\\gamma\left(P\right) \end{array}\right]$, $\overline{\overline{\boldsymbol{x}}}=\left[\begin{array}{c}\gamma\left(P\right)\\\beta\left(P\right)\end{array}\right]$ (here $\alpha\left(\cdot\right)$, $\beta\left(\cdot\right)$ and $\gamma\left(\cdot\right)$ are functions that depend on $f\left(\cdot\right)$ and $g\left(\cdot\right))$.
Clearly the system of equations is always (i.e. $\forall P$) symmetric under exchange of $x_{1}$ and $x_{2}$, but the solutions (i.e. $\overline{\boldsymbol{x}}$ and $\overline{\overline{\boldsymbol{x}}}$) are not for $P>P^{*}$. Moreover, for $P>P^{*}$, in numerical simulations the solution converge to one between $\overline{\boldsymbol{x}}$ and $\overline{\overline{\boldsymbol{x}}}$. In other terms, I think this is a pitchfork bifurcation. Is this an example of spontaneous symmetry-breaking? Or just a symmetry breaking?
Thanks in advance!