I'm looking for some sets of solutions for this nonlinear system. I only have to respect the conditions, getting numerical values to use for another thing:
$$\begin{align} \delta_i &> 10 |\lambda_i|, \quad i=1,2 \tag{1,2} \\[4pt] \Delta_i &> 10 |\Omega_i|, \quad i = 1,2,3 \tag{3,4,5} \end{align}$$
$$\frac{|\lambda_1|^2}{\delta_1} = \frac{|\lambda_2|^2}{\delta_2} \tag{6}$$
$$\Delta_1 + \frac{|\Omega_1|^2}{\Delta_1} +\frac{|\Omega_3|^2}{\Delta_3} = \delta_1 + \frac{|\Omega_2|^2}{\Delta_2} \tag{7}$$
$$\delta_2 + \frac{|\Omega_1|^2}{\Delta_1} +\frac{|\Omega_3|^2}{\Delta_3} = \Delta_2 + \frac{|\Omega_2|^2}{\Delta_2} \tag{8}$$
$$\begin{align} \alpha &= \frac{\lambda_1 \Omega_1}{\delta_1} - \frac{\lambda_2 \Omega_2}{\delta_2} \tag{9} \\[4pt] \beta &= \frac{\lambda_1 \Omega_1}{\Delta_1} - \frac{\lambda_2 \Omega_2}{\Delta_2} \tag{10} \end{align}$$
$$\begin{align} \alpha &< 0.1 \tag{11} \\[4pt] \beta &< 0.1 \tag{12} \\[4pt] |\lambda_1| &=1.0 \tag{13} \end{align}$$
Naturally, $\delta_1, \delta_2, \Delta_1, \Delta_2$ and $\Delta_3$ are in $\mathbb{R}$.
I'm using Mathematica to solve this nonlinear system of parameters (image below) by using NSolve, where I give some parameters and apply the conditions, but it's not working.
I know it is far away from a good strategy to attack this problem. Could someone help me with a code in Mathematica that works setting a few parameters and the computer calculates the others (respecting all the conditions)?
I am not (from far away) an advanced user of Mathematica but I am not sure that NSolve can handle inequality constraints.
Let us consider that have $n$ equations $(f_1,f_2,\cdots,f_n)$. What I should do is to define $$\Phi=\sum_{i=1}^n f_i^2$$ and use NMinimize for $\Phi$ where we can add as many constraints of any kind as we want.
Try it and let me know.